The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations

We can think of the study of asymmetry as a resource theory. Any resource theory is specified by a convex set of free states and a semi-group of free transformations (which must map the set of free states to itself). Any non-free state is called a resource. The resource theory is the study of manipulations of resources under the free transformations. As we will explain, there are several types of questions and arguments that are relevant for all resource theories and so this point of view can help to achieve a better understanding of a specific resource theory by emphasizing its analogies with other resource theories.

A well-known example of a resource theory is the theory of entanglement. The free transformations in this case are those which can be implemented by local operations and classical communications (LOCC). The set of free states is the set of unentangled states. This set is closed under LOCC, i.e. an unentangled state cannot be transformed to an entangled one via LOCC [5]. More generally, given two quantum states one cannot necessarily transform the first one to the second with LOCC. Here the relevant properties of the states which determine whether such a transformation is possible or not are their entanglement properties. In the case of pure bipartite states it is a well-known fact that the entanglement properties of a state are uniquely specified by its Schmidt coefficients [5]. For example, Nielsen's theorem provides the necessary and sufficient condition for the existence of LOCC operations which transform one given state to another in terms of their Schmidt coefficients [6]. Entangled states are also a resource in the sense that they can be used to implement tasks that are impossible by LOCC and unentangled states alone. For example, one can use entangled states for teleportation, which can be interpreted as consuming a resource (entanglement) to simulate a non-free transformation (a quantum channel) via free transformations (LOCC).

Similarly, we can think of the study of asymmetry relative to a given representation of a group G as a resource theory. In this resource theory the time evolutions which respect the symmetry (G-covariant time evolutions) are free transformations and the states which do not break the symmetry (G-invariant states) are the free states. This is a consistent choice because G-covariant time evolutions form a semi-group under which the set of G-invariant states is mapped to itself. Similarly to entanglement theory, a resource (an asymmetric state) can be used to simulate a non-free transformation (non-G-covariant time evolution) via a free transformation (G-covariant time evolution).

In the resource theory of asymmetry, we seek to classify different types of resources and to find the rules governing their manipulations. For every question in entanglement theory, it is useful to ask whether there is an analogous question in the resource theory of asymmetry. In this paper, we will show that all the asymmetry properties of a pure state ψ relative to the group G and the unitary representation {U(g),g∈G} are specified by its characteristic function χ_ψ(g) ≡ 〈ψ|U(g)|ψ〉. This is analogous to how all the entanglement properties of a pure bipartite state are specified by its Schmidt coefficients.

We then proceed to find the complete set of selection rules for pure states under deterministic and stochastic single-copy operations, that is, the necessary and sufficient conditions under which one pure state can be converted to another by a G-covariant operation either deterministically or non-deterministically. These results are the analogues within the resource theory of asymmetry of, respectively, Nielsen's theorem [6] and Vidal's theorem in entanglement theory. Finally, we consider the case of catalysis of asymmetry transformations, wherein a state with asymmetry can be used to assist in the conversion but must be returned intact at the end of the protocol. We show that a finite catalyst is useless in the case of compact connected Lie groups, while in the case of a finite group, there exists for any state interconversion problem a finite catalyst that makes it possible.

We now summarize the structure of the paper. In section 2 we review some elementary concepts. We also formally define G-equivalence classes of states. Appendix A includes a short review of projective unitary representations and appendix B includes a discussion about the situations where the input and output Hilbert space of a time evolution are different. In section 3, we introduce the idea of two dual points of view to asymmetry, constrained-dynamical and information-theoretic. We also show how these two dual points of view arise naturally in the study of quantum reference frames. In section 4, we define the notion of unitary G-equivalence, another equivalence relation over states that is slightly stronger than G-equivalence. Using the constrained-dynamical and information-theoretic perspectives, we find two different ways to characterize the unitary G-equivalence classes of states: the characteristic function and the reduction to the irreducible projective unitary representations (irreps). Section 4.3 extends these considerations to the case of approximate unitary G-equivalence, in which one state should be transformed to a state that is close to (but not necessarily exactly equal to) a second. The proofs for this section are presented in appendix E.

In section 5, we show that the two different characterizations of the unitary G-equivalence classes are in fact two different representations of the same object, the reduction of the state to the associative algebra and that these representations can be transformed one to the other via Fourier and inverse Fourier transforms. We further outline several nice mathematical properties of the characteristic function of a state, properties which make it particularly useful for the study of the asymmetry of pure states. We also show, in appendix C, that both the amplitude and the phase of the characteristic function are important for specifying the asymmetry of a state, while in appendix D we explain more about characteristic functions and their connection with the classical characteristic function of probability distributions.

In section 6, we present our main result, the characterization of the G-equivalence classes. Specifically, we show that for compact Lie groups, the G-equivalence class of a state is uniquely specified by its characteristic function up to a one-dimensional (1D) representation of the group. In the important case of semi-simple Lie groups, we show that it is uniquely specified by the characteristic function alone.

Finally, the results on single-copy transformations are presented in the three short sections: deterministic transformations in section 7, state-to-ensemble transformations and stochastic transformations in section 9, and catalysis in section 8. We end with a general discussion.

For any given symmetry group, there are some states which are invariant under some or all symmetry transformations in the group. For example, for any symmetry and for any representation of the symmetry, the completely mixed state is invariant under all symmetry transformations.

Definition 1. The symmetry subgroup of a state ρ relative to the group G, denoted Sym_G(ρ), is the subgroup of G under which ρ is invariant,

$$\begin{equation} {\rm Sym}_G(\rho)\equiv \{ g\in G:\ \mathcal{U}_g[\rho] = \rho \}. \end{equation} \tag{ 4 }$$

If the symmetry subgroup contains only the identity element, it is said to be trivial. In this case, it is often said that the state has no symmetries (meaning no non-trivial symmetries). If the symmetry subgroup of a state ρ is the entire group G, so that it is invariant under all symmetry transformations g∈G, i.e.

$$\begin{equation} \forall g\in G:\ \mathcal{U}_{g} (\rho) = \rho, \end{equation} \tag{ 5 }$$

then we say that the state is G-invariant⁶.

We say that a time evolution is G-covariant if it commutes with all symmetry transformations in the group G, that is, for any initial state and any symmetry transformation, the final state is independent of the order in which the symmetry transformation and the time evolution are applied (figure 1)⁷. We will sometimes refer to an operation that is G-covariant as a symmetric operation. (It is important not to confuse symmetry transformations, which correspond to a particular group action, with symmetric transformations, which commute with all group actions.) We provide the rigorous form of the notion of G-covariance first for closed system evolutions and then for open system evolutions.

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Closed system dynamics are described by unitary operators over the Hilbert space. However, noting that the global phase of a vector in Hilbert space has no physical significance, it is useful to describe the dynamics in terms of its effect on density operators (every parameter of which has physical significance). Closed system dynamics are then described by linear maps $\mathcal {V}$ on the operator space that are of the form $\mathcal {V}[\rho ]=V\rho V^{\dagger }$, where V is a unitary operator. A closed system dynamics associated with the unitary V is G-covariant if

$$\begin{equation} \forall g\in G,\quad \forall \rho: V U(g)\rho U^{\dagger}(g) V^{\dagger} = U(g) V \rho V^{\dagger} U^{\dagger}(g), \end{equation} \tag{ 6 }$$

or equivalently,

$$\begin{equation} \forall g\in G:\ [\mathcal{V},\mathcal{U}_g]=0, \end{equation} \tag{ 7 }$$

where $[\mathcal {V},\mathcal {U}_g]:=\mathcal {V}\circ \mathcal {U}_g-\mathcal {U}_g\circ \mathcal {V}$. In other words, the map $\mathcal {V}$ commutes with every element of the (superoperator) representation of the group $\{ \mathcal {U}_g:g \in G\}$. This implies that

$$\begin{equation} \forall g\in G:\ V U(g)= U(g) V \omega(g), \end{equation} \tag{ 8 }$$

where ω(g) is a phase factor that can easily be shown to be a 1D representation of the group. In the case of finite-dimensional Hilbert spaces (which is the case under consideration in this paper), we can argue that ω(g) = 1 if the closed system dynamics is required to be continuous and symmetric at all times (in contrast to requiring only that the effective operation from initial to final time be symmetric) [9].

This argument justifies the common definition in the literature of when a closed system dynamics respects the symmetry, namely, when

$$\begin{equation} \forall g\in G:\ [V, U(g)]=0. \end{equation} \tag{ 9 }$$

We call any unitary V which satisfies this property a G-invariant unitary because ∀g∈G: U(g)V U^†(g) = V . More generally, any operator which commutes with the representation of group G on the Hilbert space of the system will be called G-invariant.

Clearly, if a Hamiltonian is G-invariant then all the unitaries it generates are G-invariant. Finally, note that if V is an isometry rather than a unitary, then it is said to be G-invariant if ∀g∈G: V U_in(g) = U_out(g)V , where U_in(g) and U_out(g) are the representations of the group on the input and output spaces of the isometry.

In general, a system might be open, i.e. it may interact with an environment. In this case, the time evolution cannot be described by the Hamiltonian of the system alone. Rather, to describe the time evolution we need the Hamiltonian of system and environment together. In the study of open systems we usually restrict our attention to the situations where the initial state of the system and environment are uncorrelated, in which case we can describe the evolution by a deterministic quantum channel $\mathcal {E}$, that is, a completely positive⁸, trace-preserving, linear map from $\mathcal {B}(\mathcal {H}_{\rm in})$ to $\mathcal {B}(\mathcal {H}_{\rm out})$ where $\mathcal {H}_{\rm in}$ and $\mathcal {H}_{\rm out}$ are the input and output Hilbert spaces and $\mathcal {B}(\mathcal {H})$ are the bounded operators on $\mathcal {H}$. After a time evolution described by quantum channel $\mathcal {E}$, the initial state ρ evolves to the final state $\mathcal {E}(\rho )$. Note that a general quantum channel may have input and output spaces that are distinct. This possibility is useful for describing transformations wherein the system of interest may grow (by incorporating into its definition parts of the environment) or shrink (by having some of its parts incorporated into the environment).

We now state the conditions for a general quantum operation (which may represent open or closed system dynamics) to be G-covariant.

Definition 2. The quantum operation $\mathcal {E}$ is said to be G-covariant if

$$\begin{equation} \forall g\in G:\ \mathcal{E}(U_{\rm in}(g)(\cdot)U_{\rm in}^\dagger(g))=U_{\rm out}(g)\mathcal{E}\left(\cdot\right)U_{\rm out}^\dagger(g), \end{equation} \tag{ 10 }$$

where {U_in(g):g∈G} and {U_out(g):g∈G} are the representations of G on the input and output Hilbert spaces of $\mathcal {E}$.

If the input and output spaces are equivalent then the condition of G-covariance can be expressed as

$$\begin{equation} \forall g\in G:\ \mathcal{E}\left(U(g)(\cdot)U^\dagger(g)\right)=U(g)\mathcal{E}\left(\cdot\right)U^\dagger(g), \end{equation} \tag{ 11 }$$

or equivalently,

$$\begin{equation} \forall g\in G:\ [\mathcal{E},\mathcal{U}_g]=0, \end{equation} \tag{ 12 }$$

where $\mathcal {U}_g[\cdot ]= U(g)(\cdot )U^{\dagger }(g)$.

As we demonstrate in appendix B, any G-covariant operation for which the input and output Hilbert spaces are different can always be modeled by one wherein the input and output Hilbert spaces are the same. The reason is that the input and output Hilbert spaces can always be taken to be two different sectors of a single larger Hilbert space, $\mathcal {H}_{\rm in}\bigoplus \mathcal {H}_{\rm out}$, and any operation from $\mathcal {B}(\mathcal {H}_{\rm in})$ to $\mathcal {B}(\mathcal {H}_{\rm out})$ that is G-covariant relative to the representations {U_in(g)} and {U_out(g)} can always be extended to an operation on $\mathcal {B}(\mathcal {H}_{\rm in}\bigoplus \mathcal {H}_{\rm out})$ that is G-covariant relative to the representation $\{U_{\rm in}(g)\bigoplus U_{\rm out}(g)\}$.

Similarly, any G-invariant isometry (a reversible operation where the input and output Hilbert spaces may differ) can always be modeled by a G-invariant unitary (where the input and output Hilbert spaces are the same). Again, this is shown in appendix B. It follows that without loss of generality, we can restrict our attention in the rest of this paper to G-covariant operations where the input and output spaces are the same.

Clearly, G-covariant quantum operations include those induced by G-invariant unitaries, that is, operations of the form $\mathcal {V}(\cdot )=V(\cdot )V^{\dagger }$ where ∀g∈G: [V,U(g)] = 0. As another example, consider a channel of the form

$$\begin{equation} \mathcal{K}\equiv\int_{K}\mathrm{d}k\ \mathcal{U}_k, \end{equation} \tag{ 13 }$$

where K is a subgroup of G and dk is the uniform measure over K. We refer to this as the uniform twirling overK.⁹ The uniform twirling over any normal subgroup of G is a G-covariant operation. First, recall that if K is a normal subgroup of G then ∀g∈G: gKg⁻¹ = K, where $g Kg^{-1}\equiv \left \{ gkg^{-1}:k\in K\right \} .$ It follows that

$$\begin{equation} \forall g\in G:\ \mathcal{U}_g\circ\mathcal{K}\circ\ \mathcal{U}_{g^{-1}}=\int _{K}\mathrm{d}k\ \mathcal{U}_{gkg^{-1}}=\mathcal{K}, \end{equation} \tag{ 14 }$$

and consequently that $\mathcal {K}$ is G-covariant. In particular any group is the normal subgroup of itself, therefore uniform twirling over any group G is a G-covariant channel.

Furthermore, if we couple the object system to an environment using a Hamiltonian which has the symmetry G and if the environment is initially uncorrelated with the system and prepared in a state that is G-invariant, and finally some proper subsystem is discarded, then the total effect of this time evolution is described by a G-covariant quantum operation. (Intuitively this is clear, because there is nothing in such a dynamics that can break the symmetry.) Here by proper subsystem we mean a subsystem which is closed under the action of the symmetry transformations, i.e. under this action any vector in that subsystem is mapped to a vector in the same subsystem.

As it turns out, every G-covariant quantum operation can in fact be realized in this way, i.e. by first coupling the system to an uncorrelated environment in a G-invariant state via a G-invariant unitary and secondly discarding a proper subsystem of the total system. This is sometimes called the Stinespring dilation theorem for G-covariant channels and was first proved in [17].¹⁰ This result provides an operational prescription for realizing every such operation.

In the theory of asymmetry we study the consequences of the fact that a (possibly open) dynamics has a symmetry. In particular, we are interested to know, for a given initial state of a G-covariant dynamics, which kind of constraints one can put on the possible final states based on the symmetries of dynamics. Equivalently, we are interested to know, for a given pair of states ρ and σ, whether there exists a G-covariant dynamics which transforms ρ to σ or not. We use the notation $\rho \xrightarrow {G{\raise -1pt\hbox{-}}{\mathrm { cov}}} \sigma$ to denote that state ρ can be transformed to state σ under a G-covariant time evolution.

For instance, a simple consequence of the symmetry of dynamics is that every symmetry of the initial state is a symmetry of the final state, i.e.

Proposition 1. If ρ transforms to σ by a G-covariant quantum operation ($\rho \xrightarrow {G{\raise -1pt\hbox{-}}{\mathrm{cov}}} \sigma$), then ${\rm Sym}_G (\rho) \subseteq {\rm Sym}_G(\sigma)$.

Proof. If g_s∈G is a symmetry of ρ then $\mathcal {U}_{g_{s}}(\rho )=\rho$. Since the operation $\mathcal {E}$ taking ρ to σ is G-covariant, it follows that

$$\begin{eqnarray*} &&\mathcal{E}(\rho)=\mathcal{E}\circ \mathcal{U}_{g_{s}}(\rho)= \mathcal{U}_{g_{s}}\circ\mathcal{E}(\rho). \end{eqnarray*}$$

So $\mathcal {U}_{g_{s}} (\sigma )=\sigma$.   □

In particular, therefore, one cannot generate an asymmetric state starting from a symmetric one. This proposition highlights a simple example of restrictions one can put on the final states of a possibly open system dynamics based on the initial state of the system and symmetry of dynamics. For instance, it implies that under rotationally-covariant time evolutions, a spin pointing along $\hat {z}$ cannot evolve to one pointing along $\skew3\hat{x}$ because the first state is invariant under the group of rotations around $\hat {z}$ while the second one is not. This result can be understood as a cognate of Curie's principle, which states that symmetric causes cannot have asymmetric effects [18]. Also, note that this proposition suggests a simple characterization of the asymmetry of states relative to a group G by characterizing the largest subgroup of G which leaves each state invariant. Indeed, this simple characterization is very useful, for example, in condensed matter theory. However, finding a more fine-grained characterization of asymmetry of states can also be useful, for example, to study the consequences of symmetry of an open system dynamics.

On the other hand, for any arbitrary pair of G-invariant states ρ and σ there always exist G-covariant channels which transform one to the other. A trivial instance of these G-covariant channels, is the one which discards the input state and generates the G-invariant state σ as the output, i.e. the channel described by

$$\begin{equation} \mathcal{E}_{\sigma}(X)={\rm tr}(X) \sigma. \end{equation} \tag{ 15 }$$

Finding the necessary and sufficient condition to determine for any given pair of states ρ and σ whether $\rho \xrightarrow {G-{\rm cov}}\sigma$ or not, turns out to be a hard problem and is still open. However, in this paper, we will answer this question for the special case where both ρ and σ are pure states. In the rest of this section we present two physical examples of channels which are covariant with respect to the group U(1), the group formed by all phases {e^iθ:θ∈(0,2π]}.

For concreteness, it is worth examining a specific example of symmetric operations, namely, those that are covariant under a unitary representation of the U(1) group. Here, we present two different physical scenarios in which a restriction to U(1)-covariant channels is natural.

2.3.1. Axially symmetric channels

U(1)-covariant quantum operations are relevant for describing a dynamics which has rotational symmetry around some axis, or axially symmetric dynamics. The set of all rotations around a fixed axis forms the group called SO(2) which is isomorphic to the group U(1). So the unitary representation of the rotations around a fixed axis forms a representation of U(1), e.g. if L_z is the operator of angular momentum in the z direction then

$$\begin{eqnarray*} &&{\mathrm{e}}^{{\mathrm{i}}\theta}\rightarrow {\mathrm{e}}^{{\mathrm{i}}\theta L_{z}} \end{eqnarray*}$$

is a representation of the group U(1). In general the eigenvalues of L_z are degenerate. But to simplify the notation here we assume L_z has no degeneracy. So {|m〉:m∈{ − j,−j + 1,...,j}}, the eigenbasis of L_z, is a basis for the Hilbert space of the system, where j is the angular momentum of the system and so is either half integer or integer and where L_z|m〉 = m|m〉 (taking ℏ = 1). Note that in the case of half-integer spins the representation e^iθ → e^iθL_z is a projective representation, i.e. the cocycle of the representation is non-trivial.

First, we consider the symmetries of a few different states. The state $(|0\rangle + |1\rangle )/\sqrt {2}$ has no symmetries, while the state $(|0\rangle + |2\rangle )/\sqrt {2}$ has a non-trivial symmetry subgroup because it is invariant under a π phase shift. Meanwhile, all the elements of the basis {|m〉:m∈{ − j,−j + 1,...,j}} are U(1)-invariant states. The set of all states (pure and mixed) that are U(1)-invariant are those which commute with all elements of the set $\{\exp {(\mathrm {i} \theta {L_{z}})}:\theta \in (0,2\pi ]\}$ and so commute with L_z and are therefore diagonal in the {|m〉:m∈{ − j,−j + 1,...,j}} basis.

Next we consider symmetric operations. First note that the U(1)-invariant unitaries are those that are diagonal in the {|m〉:m∈{ − j,−j + 1,...j}} basis and are therefore of the form

$$\begin{equation} V_{U(1)\mbox{-}{\rm inv}}=\sum_{m=-j}^{j} {\mathrm{e}}^{{\mathrm{i}}\beta_{m}} |m\rangle\langle m|. \end{equation} \tag{ 16 }$$

These unitaries all commute with each other. (Note, however, that if there is multiplicity in the representations, then the U(1)-invariant unitaries have a more complicated structure and do not necessarily commute with each other.)

Now one can easily see that using U(1)-invariant unitaries we cannot transform one arbitrary state to another. For example, we cannot transform |0〉 to $(|0\rangle +|1\rangle )/\sqrt {2}$: the first state is a symmetric state while the second has some asymmetry. Similarly we can easily see that $(|0\rangle +|1\rangle )/\sqrt {2}$ cannot be transformed to $(|2\rangle +|3\rangle )/\sqrt {2}$ using U(1)-invariant unitaries. However, this transformation is possible using a U(1)-covariant channel. Consider the quantum operation $\mathcal {E}$ described by the following Kraus operators:

$$\begin{eqnarray*} &&K_{0}=\sum_{m=-j}^{j-1} |m+1\rangle\langle m|\quad {\rm and}\quad K_{1}= |-j\rangle\langle j|, \end{eqnarray*}$$

where K^†₀K₀ + K^†₁K₁ = I. One can easily check that this quantum operation is covariant under rotations around $\hat {z}$, i.e.

$$\begin{equation} \forall\theta\in (0,2\pi]:\ \mathcal{E}({\mathrm{e}}^{{\mathrm{i}} \theta L_{z}} \rho \,{\mathrm{e}}^{-{\mathrm{i}} \theta L_{z}} )={\mathrm{e}}^{{\mathrm{i}} \theta L_{z}} \mathcal{E}(\rho ) {\mathrm{e}}^{-{\mathrm{i}} \theta L_{z}}. \end{equation} \noindent \tag{ 17 }$$

Furthermore, it maps the state $(|m-1\rangle + |m\rangle )/\sqrt {2}$ to $(|m\rangle + |m+1\rangle )/\sqrt {2}$ for all m < j. So, although the transformation is not possible via U(1)-invariant unitaries, it can be done by U(1)-covariant quantum operations. Similarly we can show that there is a U(1)-covariant quantum operation which transforms $(|m\rangle + |m+1\rangle )/\sqrt {2}$ to $(|m-1\rangle + |m\rangle )/\sqrt {2}$.

2.3.2. Phase-covariant channels in quantum optics

Another physical example of U(1)-covariant quantum operations comes from quantum optics (for more discussion see [2]). Consider a harmonic oscillator whose Hilbert space is spanned by the orthonormal basis $\{|n,\alpha \rangle : n\in \mathbb {N}\}$ with the number operator N such that N|n,α〉 = n|n,α〉 where n is a non-negative integer and α labels possible degeneracies. Then the operator which shifts this oscillator in its cycle by phase θ is $\exp {(\mathrm {i} \theta {N})}$. For example, this operator transforms the coherent state |γ〉 to |e^iθγ〉.

Now a quantum operation $\mathcal {E}$ is phase-covariant if

$$\begin{equation} \forall\theta\in (0,2\pi]:\ \mathcal{E}({\mathrm{e}}^{{\mathrm{i}} \theta {N}} \rho \,{\mathrm{e}}^{-{\mathrm{i}} \theta {N}} )={\mathrm{e}}^{{\mathrm{i}} \theta {N}} \mathcal{E}(\rho ) {\mathrm{e}}^{-{\mathrm{i}} \theta {N}}. \end{equation} \noindent \tag{ 18 }$$

For a particular physical scenario, there may be additional constraints on the accessible states and unitaries beyond those that are implied by the symmetry. For instance, here in this example, unlike the previous example, there is no invariant state which under the action of the symmetry group transforms as e^iNθ|ψ〉 = e^−iθ|ψ〉; all eigenvalues of the number operator are non-negative. This is a restriction relative to what occurs for our first example where to realize a particular axially symmetric operation an experimenter can couple the system to an ancilla in state {|m〉} for arbitrary positive or negative m.

However, it turns out that a restriction of the accessible irreps of U(1) to the non-negative does not have any impact on the set of operations one can implement—all U(1)-covariant operations are still physically accessible [10]. In other words, any phase-invariant quantum operation can be realized by coupling the system to another ancillary system which is initially in |n〉 for some non-negative n and the coupling can be chosen to be a phase-invariant unitary¹¹. For the rest of this paper, we will assume that all G-covariant operations are physically accessible (including in the quantum optics examples).

In this section we introduce another perspective to the notion of asymmetry of states which we call the information-theoretic perspective¹². Recall that a quantum state breaks a symmetry, say rotational symmetry, if for some non-trivial rotations, the rotated version of the state is not the same as the state itself, i.e. they are distinguishable. In this case, the ensemble of states corresponding to the orbit of the state under rotations can act as an encoding when the message to be encoded is an element of the rotation group. This suggests that information-theoretic concepts are also useful for the study of asymmetry.

Consider a set of communication protocols in which one chooses a message g∈G according to a measure over the group and then sends the state $\mathcal {U}_g[\rho ]$ where ρ is some fixed state. The goal of the sender is to inform the receiver about the specific chosen group element. We claim that the asymmetry properties of a state ρ can be defined as those that determine the effectiveness of using the signal states $\{ \mathcal {U}_g[\rho ] : g\in G \}$ to communicate a message g∈G. To get an intuition for this, note that if ρ is invariant under the effect of some specific group element h then the state used for encoding h would be the same as the state used for encoding the identity element e, ($\mathcal {U}(h)[\rho ]=\mathcal {U}(e)[\rho ]=\rho$), such that the message h cannot be distinguished from e. In the extreme case where ρ is invariant under all group elements this encoding does not transfer any information.

So from this point of view, the asymmetry properties of ρ can be inferred from the information-theoretic properties of the encoding $\{\mathcal {U}_g[\rho ]: g\in G\}$. To compare the asymmetry properties of two arbitrary states ρ and σ, we have to compare the information content of two different encodings: $\{\mathcal {U}_g[\rho ]: g\in G\}$ (encoding I) and $\{\mathcal {U}_g[\sigma ]: g\in G\}$ (encoding II). If each state $\mathcal {U}_g[\rho ]$ can be converted to $\mathcal {U}_g[\sigma ]$ for all g∈G, then encoding I has as much or more information about g than encoding II. If the opposite conversion can also be made, then the two encodings have precisely the same information about g. Consequently, in an information-theoretic characterization of the asymmetry properties, it is the reversible interconvertibility of the sets (defined by the two states) that defines equivalence of their asymmetry properties.

As it turns out, our two different approaches lead to the same definition of asymmetry properties, as the following lemmas imply.

Lemma 1. The following statements are equivalent:

• (A)  
  There exists a G-covariant quantum operation $\mathcal {E}_{G{\raise -1pt\hbox{-}}{\mathrm { cov}}}$ (as defined in equation (11)) which maps ρ to σ, i.e. $\mathcal {E}_{G{\raise -1pt\hbox{-}}{\mathrm { cov}}}(\rho )=\sigma$.
• (B)  
  There exists a quantum operation $\mathcal {E}$ which maps $\mathcal {U}_g[\rho ]$ to $\mathcal {U}_g[\sigma ]$ for all g∈G, i.e.

  $$\begin{equation} \forall g\in G:\ \mathcal{E}(\mathcal{U}_g[\rho])=\mathcal{U}_g[\sigma]. \end{equation} \noindent \tag{ 21 }$$

For pure states, we have

Lemma 2. The following statements are equivalent:

• (A)  
  There exists a G-invariant unitary V_G-inv (i.e. ∀g∈G: [V_G-inv,U(g)] = 0) which maps |ψ〉 to |ϕ〉, i.e. V_G-inv|ψ〉 = |ϕ〉.
• (B)  
  There exists a unitary operation V which maps U(g)|ψ〉 to U(g)|ϕ〉 for all g∈G, i.e.

  $$\begin{equation} \forall g\in G:\ V U(g)|{\psi}\rangle=U(g)|{\phi}\rangle. \end{equation} \noindent \tag{ 22 }$$

Note that in both of these lemmas, the condition (A) concerns whether it is possible to transform a single state to another under a limited type of dynamics. On the other hand, in the (B) condition, there is no restriction on the dynamics, but now we are asking whether one can transform a set of states to another set such that each state in the former set is mapped to its corresponding state in the latter set under this dynamics.

Adopting the latter perspective enables us to use the machinery of quantum information theory to study asymmetry and, via the lemmas, the consequences of symmetric dynamics. This technique has many other applications in the study of asymmetry. For instance, the information-theoretic approach is used in [10] to quantify the amount of asymmetry of states. In this paper we will find the characterization of the G-equivalence classes of pure states using both the constrained-dynamical and the information-theoretic approaches and we will show how these two characterizations are in fact equivalent via the Fourier transform. Also in the next section we explain how these two different perspectives on asymmetry naturally arise in the study of uncorrelated reference frames. First however, we present the proofs of the lemmas.

Proof of lemma 1. Condition (A) can be seen to imply (B) by taking $\mathcal {E}=\mathcal {E}_{G{\raise -1pt\hbox{-}}{\mathrm { cov}}}$. To show the reverse, note that (B) implies the existence of a quantum operation $\mathcal {E}$ which satisfies equation (21). Now we can define

$$\begin{equation} \mathcal{E}'\equiv \int \mathrm{d}g\ \mathcal{U}^\dagger_{g} \circ \mathcal{E} \circ\mathcal{U}_g. \end{equation} \tag{ 23 }$$

One can then easily check that $\mathcal {E}'$ is a G-covariant operation and that $\mathcal {E}'(\rho ) = \int \mathrm {d}g\ \mathcal {U}^\dagger _{g}\circ \mathcal {E} \circ \mathcal {U}_g(\rho )= \int \mathrm {d}g\ \mathcal {U}^\dagger _{g}\circ \mathcal {U}_g(\sigma )=\sigma$, such that we can choose $\mathcal {E}_{G{\raise -1pt\hbox{-}}{\mathrm { cov}}}=\mathcal {E}'.$ So (B) also implies (A).   □

Proof of lemma 2. Condition (A) can be seen to imply (B) by taking V =V_G-inv. In the following we prove that (B) also implies (A). Assume there exists a unitary V such that ∀g∈G,

$$\begin{equation} V U(g) |\psi\rangle=U(g)| \phi \rangle. \end{equation} \noindent \tag{ 24 }$$

First note that this implies |ϕ〉 = V |ψ〉. Furthermore it implies that for all g,h∈G we have

$$\begin{eqnarray*} V U(g) U(h)|{\psi}\rangle&=&\omega(g,h)V U(gh)|{\psi}\rangle\\ &=&\omega(g,h) U(gh) |{\phi}\rangle\\ &= & U(g)U(h) |{\phi}\rangle\\ &=& U(g) V U(h)|{\psi}\rangle, \end{eqnarray*}$$

where we have used the fact that g → U(g) is a projective representation of G and so U(g)U(h) = ω(g,h)U(gh) for a phase ω(g,h). Now suppose Π is the projector to the subspace spanned by all the vectors {U(h)|ψ〉,∀h∈G}. Then the above equation implies that

$$\begin{equation} \forall g\in G:\ V U(g)\Pi= U(g)V \Pi. \end{equation} \tag{ 25 }$$

Now by definition of the projector Π it is clear that it commutes with all {U(g):g∈G}. So the above equation implies

$$\begin{equation} \forall g \in G:\ [V\Pi,U(g)]=0. \end{equation} \tag{ 26 }$$

The operator V Π unitarily maps a subspace of the Hilbert space to another subspace and it commutes with all {U(g)}. Using lemma B.1 we conclude that this G-invaraint isometry can always be extended to a G-invariant unitary V_G-inv such that V_G-invΠ = V Π and therefore

$$\begin{equation} V_{G{\raise -1pt\hbox{-}}{\rm inv}} U(g)|\psi\rangle= V\Pi U(g)|\psi\rangle=U(g)|\phi\rangle. \end{equation} \tag{ 27 }$$

   □

Interestingly these two points of view to asymmetry naturally arise in the study of a communication scenario when the two distant parties lack a shared reference frame for some degree of freedom.

Specifically, consider a degree of freedom that transforms according to the group G. Passive transformations of the reference frame for this degree of freedom will then also be described by the group G, as will the relative orientation of any two such frames. Consider two parties, Alice and Bob, that each have a local reference frame but where these are related by a group element g∈G that is unknown to either of them. For instance, they might each have a local Cartesian frame, but do not know their relative orientation. (See [2] for a discussion.)

Now consider the following state interconversion task. Alice prepares a system in the state ρ relative to her local reference frame and sends it, along with a classical description of ρ, to Bob. She also sends him a classical description of a state σ, and asks him to try and implement an operation that leaves the system in the state σ relative to her local frame. In effect, Alice is asking Bob to transform ρ to σ but without the benefit of having a sample of her local reference frame. For instance, she may ask him to transform a spin aligned with her $\hat {z}$-axis to one that is aligned with her $\hat {y}$-axis. We consider how the task is described relative to each of their local frames.

3.2.1. Description relative to Alice's frame

3.2.2. Description relative to Bob's frame

We here find a characterization of the unitary G-equivalence classes within the constrained-dynamical perspective. We begin by determining the most general form of a G-invariant unitary.

Suppose {U(g):g∈G} is a projective unitary representation of a finite or compact Lie group G on the Hilbert space $\mathcal {H}$. We can always decompose this representation to a discrete set of finite-dimensional irreps. This suggests the following decomposition of the Hilbert space [2]:

$$\begin{equation} \mathcal{H}=\bigoplus_{\mu} \mathcal{M}_{\mu}\otimes \mathcal{N}_{\mu}, \end{equation} \tag{ 29 }$$

where μ labels the irreps and $\mathcal {N}_{\mu }$ is the subsystem associated to the copies of representation μ (the dimension of $\mathcal {N}_{\mu }$ is equal to the multiplicity of the irrep μ in this representation). Then U(g) can be written as

$$\begin{equation} U(g) =\bigoplus_\mu U_\mu(g) \otimes \mathbb{I}_{\mathcal{N}_\mu}, \end{equation} \tag{ 30 }$$

where U_μ(g) acts on $\mathcal {M}_{\mu }$ irreducibly and where $\mathbb {I}_{\mathcal {N}_\mu }$ is the identity operator on the multiplicity subsystem $\mathcal {N}_{\mu }$. We denote by Π_μ the projection operator onto the subspace $\mathcal {M}_{\mu }\otimes \mathcal {N}_{\mu }$, the subspace associated to the irrep μ.

Now we are ready to characterize the unitary G-equivalence classes:

Theorem 1. Two pure states |ψ〉 and |ϕ〉 are unitarily G-equivalent if and only if

$$\begin{equation} \forall \mu:\ {\rm tr}_{\mathcal{N}_{\mu}}({\Pi}_{\mu} |\psi\rangle\langle\psi| {\Pi}_{\mu})={\rm tr}_{\mathcal{N}_{\mu}}({\Pi}_{\mu} |\phi\rangle\langle\phi|{\Pi }_{\mu}). \end{equation} \tag{ 31 }$$

Proof. First, we find a simple characterization of G-invariant unitaries. It is shown in appendix A that any operator that commutes with all unitaries U(g) has the form of equation (A.3), which implies that any G-invariant unitary is of the form [2],

$$\begin{equation} V_{G{\raise -1pt\hbox{-}}{\rm inv}}=\bigoplus_\mu \mathbb{I}_{\mathcal{M}_\mu} \otimes V_{\mathcal{N}_\mu}, \end{equation} \tag{ 32 }$$

where $V_{\mathcal {N}_\mu }$ acts unitarily on ${\mathcal {N}_\mu }$.

Now suppose state |ψ〉 can be transformed to another state |ϕ〉 by a G-invariant unitary V_G-inv. Then given equation (32), it follows that for all μ,

$$\begin{equation} \Pi_\mu |{\phi}\rangle= \Pi_\mu V_{G{\raise -1pt\hbox{-}}{\rm inv}} |{\psi}\rangle= \mathbb{I}_{\mathcal{M}_\mu} \otimes V_{\mathcal{N}_\mu} \Pi_\mu |{\psi}\rangle. \end{equation} \noindent \tag{ 33 }$$

Equation (31) then follows from the cyclic property of the trace and the unitarity of $V_{\mathcal {N}_\mu }$.

Now we prove the converse. If equation (31) holds, then there exists a G-invariant unitary which transforms |ψ〉 to |ϕ〉. First note that we can think of the two vectors Π_μ|ψ〉 and Π_μ|ϕ〉 as two different purifications of ${\rm tr}_{\mathcal {N}_\mu }(\Pi _\mu |{\psi }\rangle \langle \psi | \Pi _\mu )={\rm tr}_{\mathcal {N}_\mu }( \Pi _\mu |{\phi }\rangle \langle \phi | \Pi _\mu )$. So Π_μ|ψ〉 can be transformed to Π_μ|ϕ〉 by a unitary acting on $\mathcal {N}_\mu$, denoted by $V_{\mathcal {N}_\mu }$, such that

$$\begin{equation} \mathbb{I}_{\mathcal{M}_\mu} \otimes V_{\mathcal{N}_\mu} \Pi_\mu |{\psi}\rangle=\Pi_\mu |{\phi}\rangle \end{equation} \noindent \tag{ 34 }$$

(see e.g. [5]). By defining

$$\begin{equation} V\equiv \bigoplus_\mu \mathbb{I}_{\mathcal{M}_\mu} \otimes V_{\mathcal{N}_\mu}, \end{equation} \noindent \tag{ 35 }$$

we can easily see that V is a G-invariant unitary and moreover V |ψ〉 = |ϕ〉. This completes the proof.   □

For an arbitrary state ρ we call the set of operators $\{ {\rm tr}_{ \mathcal {N}_{\mu }} ({\Pi }_{\mu } \rho {\Pi }_{\mu } )\}$, the reduction onto irreps of ρ. So in the above theorem we have proven that the unitary G-equivalence class of a pure state is totally specified by its reduction onto irreps. Note, however, that as we will see in section 5.1, this is not true for general mixed states.

Example 1. Recall the quantum optics example studied in section 2.3.2 where the set of all phase shifts forms a representation of group U(1). There the representation of group U(1) is e^iθ → U(θ) where the phase shift operator U(θ) is

$$\begin{equation} U(\theta)\equiv {\mathrm{e}}^{{\mathrm{i}} N\theta}= \sum_{n} {\mathrm{e}}^{{\mathrm{i}} n \theta } \sum_{\alpha} |n,\alpha\rangle\langle n,\alpha|, \end{equation} \tag{ 36 }$$

where N is the number operator with integer eigenvalues such that N|n,α〉 = n|n,α〉 and where α is a multiplicity index. In this case all irreps are 1D. It follows that the reduction onto irreps of a pure state $|\psi \rangle =\sum _{n,\alpha } \psi _{n,\alpha } |n,\alpha \rangle$ is simply given by

$$\begin{equation} p_{\psi}(n) \equiv \langle \psi| \Pi_n |\psi \rangle=\sum_{\alpha} |\psi_{n,\alpha}|^{2}, \end{equation} \tag{ 37 }$$

where Π_n is the projector to the eigen-subspace corresponding to the eigenvalue n of N. That is, the reduction onto irreps is the probability distribution over the spectrum of the number operator induced by |ψ〉. Consequently, two pure states are unitarily U(1)-equivalent if and only if they define the same probability distribution over number.

We will show that by taking the information-theoretic point of view, one finds that the unitary G-equivalence class of a pure state is specified entirely by its characteristic function, which is defined as follows.

Definition 5 The characteristic function of a state ρ relative to a projective unitary representation {U(g):g∈G} of a group G is a function $\chi _\rho : G \to \mathbb {C}$ of the form

$$\begin{equation} \chi_\rho(g)\equiv {\tr}\left(\rho U(g)\right). \end{equation} \tag{ 38 }$$

Specifically, we have

Theorem 2. Two pure states |ψ〉 and |ϕ〉 are unitarily G-equivalent if and only if their characteristic functions are equal,

$$\begin{equation} \forall g\in G:\ \langle\psi|U(g)|\psi\rangle=\langle\phi|U(g)|\phi\rangle . \end{equation} \tag{ 39 }$$

The benefit of trying to characterize the G-equivalence classes using the information-theoretic perspective is that we can make use of known results concerning the unitary interconvertibility of sets of pure states. We express the condition for such interconvertibility as a lemma, after recalling the definition of the Gram matrix of a set of states.

Definition 6 Consider the set of states {|ψ_θ〉}. If θ is a discrete parameter, then we define the Gram matrix of the set {|ψ_θ〉} by X_θ,θ' ≡ 〈ψ_θ|ψ_θ'〉. If θ is a continuous parameter, then we can define the function X(θ,θ') ≡ 〈ψ_θ|ψ_θ'〉, which, with a slight abuse of terminology, we will also call the Gram matrix of the set {|ψ_θ〉}.

Lemma 3. There exists a unitary operator V which transforms each member of {|ψ_θ〉} to its corresponding member in {|ϕ_θ〉}, that is, ∀θ: V |ψ_θ〉 = |ϕ_θ〉, if and only if the Gram matrices of the two sets of states are equal, i.e.

$$\begin{eqnarray*}&&\forall \theta,\theta':\ \ \langle{\psi_\theta}|{\psi_{\theta'}}\rangle=\langle{\phi_\theta}|{\phi_{\theta'}}\rangle.\end{eqnarray*}$$

A simple proof of this lemma is provided in the footnote¹³.

It is now straightforward to prove theorem 2.

Proof of theorem 2. By definition 4, |ψ〉 and |ϕ〉 are unitarily G-equivalent if there exists a unitary transformation V_G-inv which takes |ψ〉 to |ϕ〉. By lemma 2 there exists such a unitary if and only if there exists a unitary V such that ∀g∈G: V U(g)|ψ〉 = U(g)|ϕ〉. By lemma 3, the necessary and sufficient condition for the existence of such a unitary is the equality of the Gram matrices of the set {U(g)|ψ〉:g∈G} and the set {U(g)|ϕ〉:g∈G}. Given that the elements of these matrices are, respectively,

$$\begin{eqnarray*} &&[X_\psi]_{g_1,g_2}=\langle\psi|U^\dagger(g_1)U(g_2)|{\psi}\rangle=\omega(g_{1}^{-1},g_{2}) \langle\psi| U(g^{-1}_1g_2)|{\psi}\rangle, \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&[X_\phi]_{g_1,g_2}=\langle\phi|U^\dagger(g_1)U(g_2)|{\phi}\rangle=\omega(g_{1}^{-1},g_{2}) \langle\phi|U(g^{-1}_1g_2)|{\phi}\rangle, \end{eqnarray*}$$

where we have used the fact g → U(g) is a projective unitary representation and so

$$\begin{eqnarray*} &&U^\dagger(g_1)U(g_2)=U(g^{-1}_{1})U(g_2)=\omega(g^{-1}_{1},g_{2}) \end{eqnarray*}$$

for the cocycle ω. Equality of the Gram matrices is equivalent to

$$\begin{equation} \forall g\in G:\ \ \langle\psi|U(g)|\psi\rangle=\langle\phi|U(g)|\phi\rangle, \end{equation} \tag{ 40 }$$

and this is simply the statement that the characteristic functions of ψ and ϕ are equal.   □

Example 2. In example 1 we found the characterization of unitary equivalence classes based on the reduction of states to irreps in the case of group U(1) with representation e^iθ → e^iθN where N is the number operator with non-negative integer eigenvalues. Here, we use the result of lemma 3 to find another characterization of these unitary equivalence classes in terms of characteristic functions of states. In this case, for arbitrary state $|\psi \rangle =\sum _{n,\alpha } \psi _{n,\alpha } |n,\alpha \rangle$, the characteristic function is given by the expectation value of the phase shift operator, i.e.

$$\begin{equation} \chi_{\psi}(\phi) \equiv \langle \psi | \exp(\mathrm{i}\phi {N}) | \psi \rangle= \sum_n p_{\psi}(n) {\mathrm{e}}^{{\mathrm{i}} n\phi}, \end{equation} \tag{ 41 }$$

where $p_{\psi }(n) = \sum _{\alpha } |\psi _{n,\alpha }|^2$ is the reduction onto irreps.

It follows that in the U(1) case, the reduction onto irreps and the characteristic function are related by a Fourier transform. The Fourier transform can also be defined for arbitrary compact Lie groups or for finite groups (which might be non-Abelian) and in these cases as well, it describes the relation between the reduction onto irreps and the characteristic function, as will be shown in section 5.

We have found the necessary and sufficient condition for the existence of a G-invariant unitary which transforms a pure state ψ to another pure state ϕ. This is the condition for exact transformation. But there might be situations in which we cannot transform ψ to ϕ but we can transform it to some state close to ϕ.

In the following we demonstrate that if the reductions onto irreps of two pure states ψ and ϕ are close in some sense (or equivalently their characteristic functions are close) then there exists a G-invariant unitary which transforms ψ to a state close to ϕ (see appendix E for a discussion about the relevant notion of distance in this context).

Recall that the fidelity of two positive operators A₁ and A₂ is defined as

$$\begin{equation} {\rm Fid}(A_{1},A_{2})\equiv \|\sqrt{A_1}\sqrt{A_2}\|={\rm tr}\left(\sqrt{ \sqrt{A_{1}} A_{2} \sqrt{A_{1}}}\right), \end{equation} \tag{ 42 }$$

where ∥·∥ denotes the trace norm.

Theorem 3. Suppose {F^(μ)₁} and {F^(μ)₂} are respectively the reductions onto irreps of ψ₁ and ψ₂, two arbitrary pure states in the same Hilbert space. Then for any G-invariant unitary V acting on this space

$$\begin{equation} |\langle\psi_2|V|\psi_1\rangle|\leqslant \sum_\mu {\rm Fid}(F_{1}^{(\mu)},F_{2}^{(\mu)}). \end{equation} \tag{ 43 }$$

Furthermore there exists a G-invariant unitary V for which the equality holds.

According to this theorem if the fidelities of the reductions onto irreps is high then there exists a G-invariant unitary which transforms one of the states to a state very close to the other. On the other hand, if these fidelities are low we can never transform one of the states to a state close to the other via G-invariant unitaries.

Remark 1. For {F^(μ)₁} and {F^(μ)₂} the reductions of an arbitrary pair of states it holds that $\sum _\mu {\rm Fid}(F_{1}^{(\mu )},F_{2}^{(\mu )})\leqslant 1$ and the equality holds iff ∀μ:F^(μ)₁ = F^(μ)₂. So theorem 3 is a special case of theorem 3.

We present the proof of theorem 3 as well as some other versions of it and the proof of remark 1 in appendix E.

Example 3. Recall our quantum optics example where the set of all phase shifts forms a representation of the group U(1) (see example 1). Let p_ψ and p_ϕ be the probability distributions over integers which describe the reductions onto irreps of the states ψ and ϕ respectively. Then theorem 3 implies that for any U(1)-invariant unitary V ,

$$\begin{equation} |\langle\psi|V|\phi\rangle|\leqslant \sum_{n} \sqrt{p_{\psi}(n)p_{\phi}(n)} \end{equation} \tag{ 44 }$$

and furthermore there exists a U(1)-invariant unitary for which the equality holds.

If we are interested in only some particular degree of freedom of a quantum system then we do not need the full description of the state in order to infer the statistical features (expectation values, variances, correlations between two different observables, etc) of that degree of freedom. In particular suppose we are interested in the statistical properties of the set of operators $\{O_i\in \mathcal {B}(\mathcal {H})\}$. Closing this set under the operator product and sum yields the associative algebra generated by {O_i}, which is the set of all polynomials in {O_i}. We denote this associative algebra by Alg{O_i}. To specify all the statistical properties of the state $\rho \in \mathcal {B}(\mathcal {H})$ for the set of observables {O_i} it is necessary and sufficient to specify the expectation values of all the operators in Alg{O_i} under the state ρ. The object that contains all and only this information is called the reduction of the state to the associative algebra, denoted ρ|_Alg{Oi}.

Alg{O_i}, considered as a finite-dimensional C*-algebra, has a unique decomposition (up to unitary equivalence) of the form

$$\begin{equation} \bigoplus_J \mathfrak{M}_{m_J} \otimes \mathbb{I}_{n_J}, \end{equation} \tag{ 45 }$$

where $\mathfrak {M}_{m_J}$ is the full matrix algebra $\mathcal {B}(\mathbb {C}^{m_J})$ and $\mathbb {I}_{n_J}$ is the identity on $\mathbb {C}^{n_J}$ [20]. This means that any element A of the algebra can be written as

$$\begin{equation} A= \bigoplus_J A^{(J)}\otimes \mathbb{I}_{n_J}, \end{equation} \tag{ 46 }$$

where $A^{(J)}\in \mathcal {B}(\mathbb {C}^{m_J})$. Furthermore, if we consider the set of all elements of the algebra, that is, all A∈Alg{O_i}, and look at the set of corresponding A^(J) for fixed J, this set of operators acts irreducibly on $\mathbb {C}^{m_J}$ and spans $\mathcal {B}(\mathbb {C}^{m_J})$. Clearly this decomposition induces the following structure on the Hilbert space:

$$\begin{equation} \mathcal{H}=\bigoplus_J \mathcal{M}_{J} \otimes \mathcal{N}_{J}, \end{equation} \tag{ 47 }$$

where $\mathcal {M}_{J}$ is isomorphic to $\mathbb {C}^{m_J}$ and $\mathcal {N}_{J}$ is isomorphic to $\mathbb {C}^{n_J}$.

Suppose Π_J is the projective operator to the subspace $\mathcal {M}_{J} \otimes \mathcal {N}_{J}$. Then to specify all the relevant information about the observables in the algebra for the given state ρ it is necessary and sufficient to know all of the operators

$$\begin{equation} \rho^{(J)}\equiv \mathrm{tr}_{\mathcal{N}_J}(\Pi_J \rho \Pi_J). \end{equation} \tag{ 48 }$$

Then for any arbitrary observable A in the algebra we have

$$\begin{equation} {\rm tr}(A\rho)= \sum_{J} {\rm tr}(A^{(J)}\rho^{(J)}) \end{equation} \tag{ 49 }$$

and so specifying the set {ρ^(J)} we know all the relevant information about the state. In other words, {ρ^(J)} uniquely specifies the reduction to the algebra $\rho|_{{\rm Alg}\{O_{ i}\}}$.

The above discussion applies to any arbitrary set of observables. Here, we will be interested in the case where this set describes the degree of freedom associated to some symmetry transformation. If the symmetry transformation is associated with the symmetry group G and projective unitary representation {U(g):g∈G} on the Hilbert space of the system, then the set of observables to consider are all those in the linear span of {U(g):g∈G}. In particular, in the case of Lie groups this set contains the representation of all generators of the Lie algebra (associated to the group) and all the polynomials formed by these generators. For example, in the case of SO(3) the set includes all the observables in the linear span of {U(Ω):Ω∈SO(3)} and so it clearly contains all the generators, which in this case are angular momentum operators, as well as all polynomials of these.

Decomposition of this algebra in the form of equation (45) in fact coincides with the decomposition of the unitary projective representation {U(g):g∈G} to irreps

$$\begin{equation} U(g)\cong \bigoplus_\mu U^{(\mu)}(g)\otimes \mathbb{I}_{N_\mu}, \end{equation} \tag{ 50 }$$

where μ labels the irreps and $\mathbb {I}_{N_\mu }$ is the identity acting on the multiplicity subsystem associated to irrep μ (remember that G is by assumption a finite or compact Lie group and so it is completely reducible). Here we can think of μ playing the same role as J in the decomposition of the arbitrary algebra in equation (45). Each irrep index μ appearing in the decomposition of {U(g):g∈G} corresponds to one J in equation (45) and the set {U^μ(g):g∈G} for a fixed μ spans the full matrix algebra $\mathfrak {M}_{m_J}$ of the corresponding J. Consequently, the spaces on which the projective unitary representation of G acts irreducibly are simply the $\mathcal {M}_{J}$. So it follows that in this case, where the associative algebra coincides with the span of the elements of the projective unitary representation of the group, {U(g):g∈G}, the set of operators {ρ^(J)} (defined by equation (48)) is simply the reduction onto the irreps of the state ρ, the generalization to mixed states of the notion defined in the section 4.1, and therefore we can conclude that the reduction onto the irreps is a representation of the reduction onto the associative algebra.

Another way to specify the reduction of the state onto the associative algebra is to specify the Hilbert–Schmidt inner product of ρ with each of the U(g), namely, tr(ρU(g)) for all g∈G. So if we define the characteristic function associated to the state ρ as the function $\chi _\rho : G \to \mathbb {C}$ defined by χ_ρ(g) ≡ tr(ρU(g)), then the characteristic function is a particular representation of the reduction to the associative algebra. It is clear that this definition constitutes a generalization to mixed states of the notion of characteristic functions introduced in the section 4.2.

To summarize, we have

Remark 2. For a state $\rho \in \mathcal {B}(\mathcal {H})$ and a projective unitary representation U of a group G, the reduction of ρ to the associative algebra Alg{U(g):g∈G} can be represented either in terms of the reduction onto irreps of ρ, defined as

$$\begin{equation} \{ \rho^{(\mu)} \equiv {\rm tr}_{\mathcal{N}_{\mu}}(\Pi_{\mu} \rho \Pi_{\mu})\} \end{equation} \tag{ 51 }$$

(where the Hilbert space decomposition induced by U is $\mathcal {H}= \bigoplus _{\mu } \mathcal {M}_{\mu } \otimes \mathcal {N}_{\mu }$ and Π_μ projects onto $\mathcal {M}_{\mu } \otimes \mathcal {N}_{\mu }$), or in terms of the characteristic function of ρ, defined as

$$\begin{equation} \chi_\rho(g) \equiv {\rm tr}(\rho U(g)). \end{equation} \tag{ 52 }$$

Finally, we note that the relationship between these two representations is the Fourier transform over the group.

Proposition 2. The characteristic function and reduction onto irreps can be computed one from the other via

$$\begin{equation} \chi_\rho(g)=\sum_\mu {\tr}(\rho^{(\mu)}U^{(\mu)}(g)) \end{equation} \tag{ 53 }$$

and

$$\begin{equation} \rho^{(\mu)}=d_\mu \int \mathrm{d}g \chi_\rho(g^{-1}) U^{(\mu)}(g). \end{equation} \tag{ 54 }$$

Proof. The expression for χ_ρ(g) in terms of {ρ^(μ)}, equation (53), follows directly from equations (50) and (52). Conversely, to find the {ρ^(μ)} in terms of χ_ρ(g) we use the Fourier transform over the group. The idea is based on the following orthogonality relations which are part of the Peter–Weyl theorem (see e.g. [21]):

$$\begin{equation} \int_G \mathrm{d}g\, U_{i,j}^{(\mu)}(g) {{\overline{U}^{(\nu)}_{k,l}}}(g)=\frac{\delta_{\mu,\nu}\delta_{i,k}\delta_{j,l}}{ d_\mu}, \end{equation} \tag{ 55 }$$

where {U^μ_i,j} are the matrix elements of U^μ(g), dg is the unique Haar measure on the group, bar denotes the complex conjugate and d_μ is the dimension of irrep μ. According to this theorem any continuous function on a compact Lie group can be uniformly approximated by linear combinations of matrix elements U^(μ)_i,j(g). Note that for the finite groups, we can get the same orthogonality relations by replacing the integral with a summation. Furthermore any function over a finite group can be expressed as a linear combination of the matrix elements of irreps. So basically all the properties we use hold for finite groups as well as compact Lie groups.

An arbitrary operator A^(μ) in $\mathcal {B}(\mathcal {M}_{\mu })$ can be written as a linear combination of elements of {U^(μ)(g): g∈G}. The above orthogonality relations imply that this expansion has a simple form as

$$\begin{equation} A^{(\mu)}=d_{\mu}\int \mathrm{d}g\ U^{\mu}(g)\, {\rm tr}(A^{(\mu)} U^{\mu}(g^{-1})). \end{equation} \tag{ 56 }$$

Clearly this can be considered as a completeness relation where we have decomposed the identity map on $\mathcal {B}(\mathcal {M}_{\mu })$ as the sum of projections to a basis (which is generally overcomplete). Also note that the orthogonality relations imply that for ν ≠ μ

$$\begin{equation} \int \mathrm{d}g\ U^{\nu}(g) {\rm tr}(A^{(\mu)} U^{\mu}(g^{-1}))=0\quad (\nu \neq \mu). \end{equation} \tag{ 57 }$$

Using these orthogonality relations, we obtain equation (54).   □

We should emphasize that the reduction onto the associative algebra, though sufficient for deciding G-equivalence of pure states, is not in general sufficient for deciding G-equivalence of arbitrary states, i.e. mixed and pure. Its sufficiency in the case of pure states follows from its sufficiency for deciding unitary G-equivalence (proven in section 4.2) and the fact that the unitary G-equivalence classes are a fine-graining of the G-equivalence classes. Its insufficiency in the case of mixed states can be established by the following simple example of two states (one pure and one mixed) that have the same characteristic function but fall in different G-equivalence classes. The example is for the case of U(1)-covariant operations, and the two states are $\frac {1}{2}(|0\rangle +|1\rangle )(\langle 0|+\langle 1|)$ and $\frac {1}{2}(|0\rangle \langle 0| +|1\rangle \langle 1|)$. The second is clearly U(1)-invariant while the first is not and so they must lie in different U(1)-equivalence classes. Nonetheless, the characteristic function for both equals $\chi (\theta )=1/2(1+\exp (\mathrm {i}\theta ))$.

We close this section by mentioning another consequence of the orthogonality relations equation (55) which is useful later. Suppose A,B are arbitrary operators in $\mathcal {B}(\mathcal {M}_{\mu })$ and

$$\begin{eqnarray*} &&\fl \chi_A(g)\equiv {\rm tr}(AU^{(\mu)}(g)),\ \ \chi_B(g)\equiv {\rm tr}(BU^{(\mu)}(g)),\quad {\rm and}\quad \chi_{AB}(g)\equiv {\rm tr}(ABU^{(\mu)}(g))\end{eqnarray*}\noindent$$

are respectively the characteristic functions of A,B and AB. Then

$$\begin{equation} \chi_{AB}=d_\mu\ \chi_{A} \ast \chi_{B}, \end{equation} \tag{ 58 }$$

where * is the convolution of two functions¹⁴

$$\begin{equation} f_1\ast f_2(g)\equiv\ \int \mathrm{d}h f_1(gh^{-1})f_2(h). \end{equation} \tag{ 59 }$$

In particular, since tr(AB) = χ_AB(e) (where e is the identity of the group) the above formula can be used to find tr(AB) in terms of the characteristic functions of A and B. Using equation (58) we get

$$\begin{eqnarray*} {\rm tr}(AB)&=&\chi_{AB}(e)=d_\mu\ [\chi_{A} \ast \chi_{B}](e)\\\ms\ms &=&d_\mu\ \int \mathrm{d}h\ \chi_A(h) \chi_B(h^{-1}). \end{eqnarray*}$$

The characteristic functions introduced here are quantum analogues of those used in classical probability theory. The connection is discussed in detail in appendix D. Here we simply summarize some useful properties of characteristic functions.

• 1.  
  A function ϕ(g) from the finite or compact Lie group G to complex numbers is the characteristic function of a physical state iff it is continuous (in the case of Lie groups) positive definite (as defined in appendix D) and normalized (i.e. ϕ(e) = 1 where e is the identity of the group). (This property assumes that all irreps are physically accessible.)
• 2.  
  The characteristic function of a state is invariant under G-invariant unitaries acting on that state,

  $$\begin{eqnarray*} &&\chi_{\mathcal{V}_{G{\raise -1pt\hbox{-}}{\rm inv}}[\rho]}(g)=\chi_{\rho}(g), \end{eqnarray*} \noindent$$

  where $\mathcal {V}_{G{\raise -1pt\hbox{-}}{\mathrm { inv}}}[\cdot ]=V_{G{\raise -1pt\hbox{-}}{\rm inv}}(\cdot )V^{\dagger }_{G{\raise -1pt\hbox{-}}{\rm inv}}$ and [V_G-inv,U(g)] = 0 for all g∈G.
• 3.  
  Characteristic functions multiply under tensor product,

  $$\begin{equation} \chi_{\rho\otimes\sigma}(g)= \chi_{\rho}(g)\chi_{\sigma}(g). \end{equation} \noindent \tag{ 60 }$$
• 4.  
  |χ_ρ(g)| ⩽ 1 for all g∈G and χ_ρ(e) = 1 where e is the identity of the group.
• 5.  
  If |χ_ρ(g_s)| = 1 for g_s∈G then g_s is a symmetry of ρ. If ρ is a pure state, then g_s is a symmetry of ρ if and only if |χ_ρ(g_s)| = 1.
• 6.  
  So |χ_ρ(g)| = 1 for all g∈G implies that the state is invariant; in this case χ_ρ(g) is a 1D representation of the group.
• 7.  
  Suppose L is the representation of a generator of a Lie group on the Hilbert space of a system such that {e^iθL:θ∈(0,2π]} is the representation of a U(1)-subgroup of the group. Then we can find all moments of L using the characteristic function

  $$\begin{equation} {\rm tr}(\rho L^{k})=\mathrm{i}^{-k}\frac{\partial^k}{\partial\theta^k} \chi_{\rho}({\mathrm{e}}^{{\mathrm{i}}\theta L} ) \mid_{\theta=0}. \end{equation} \tag{ 61 }$$

  (Note that by χ_ρ(e^iθL) we really mean χ_ρ(g) for the group element g∈G which is represented by e^iθL.)

Proof. Item 1 is proven in appendix D.2. All the rest of these properties can simply be proved by using the definition of the characteristic function, χ_ρ(g) = tr(ρU(g)), and group representation properties. For example to prove item 3 we use the fact that if the representation of the symmetry G on the systems A (with state ρ) and B (with state σ) are g → U_A(g) and g → U_B(g) then the representation of the symmetry on the joint system AB is g → U_A(g)⊗U_B(g). Then

$$\begin{eqnarray*} &&\chi_{\rho\otimes \sigma}(g)={\rm tr}\left(\rho\otimes\sigma U_{A}(g)\otimes U_{B}(g)\right)={\rm tr}\left(\rho U_{A}(g)\right){\rm tr}\left(\sigma U_{B}(g)\right)= \chi_{\rho}(g) \chi_{\sigma}(g). \end{eqnarray*} \noindent$$

To prove item 5 we note that if |χ_ρ(g_s)| = 1 for g_s∈G then all eigenvectors of ρ are eigenvectors of U(g_s) with the same eigenvalue. As a result we get [ρ,U(g_s)] = 0 and so the state has the symmetry g_s. On the other hand, [ρ,U(g_s)] = 0 does not imply that |χ_ρ(g_s)| = 1. For instance, the state $\frac {1}{2}|0\rangle \langle 0|+\frac {1}{2}|1\rangle \langle 1|$ where |n〉 is a number eigenstate is U(1)-invariant, but nonetheless, for ϕ ≠ 0, |χ_ρ(ϕ)| ≠ 1. Therefore the points for which the amplitude of the characteristic function is one are a subset of the symmetries of the state. Meanwhile, if a pure state |ψ〉 has symmetry g_s, such that U(g_s)|ψ〉 = e^iθ|ψ〉 for some θ, then obviously |χ_ψ(g_s)| = 1. So for pure states the points for which the amplitude of the characteristic function is one are exactly the state's symmetries.

To prove item 6, we first note that if |χ_ρ(g)| = 1 for all g∈G, then the symmetry subgroup of ρ is the entire group G, which is the definition of ρ being G-invariant. Furthermore, for each g, the eigenvectors of ρ all live in the same eigenspace of U(g). Since the eigenvalue of a unitary is a phase factor, each such eigenvector |ν〉 must satisfy U(g)|ν〉 = e^iθ(g)|ν〉 for some phase e^iθ(g). It is then clear that χ_ρ(g) = e^iθ(g) and is a 1D representations of the group.   □

Among the above properties, the fact that the tensor product of states is represented by the product of their characteristic functions (property 3) turns out to be particularly useful. This is because the alternative representation, in terms of reductions onto irreps, does not provide a simple expression for the reduction of ρ⊗σ in terms of the reduction of ρ and the reduction of σ. It involves Clebsch–Gordan coefficients and is generally quite complicated for non-Abelian groups.

For this and other reasons, the characteristic function is generally our preferred way of representing the reduction of the state onto the algebra, and consequently we will make heavy use of it to answer various questions about the manipulation of asymmetry of pure states.

Recall our quantum optics example where the set of all phase shifts forms a representation of group U(1) (see example 1). According to theorem 6 there exists a deterministic U(1)-covariant map transforming ψ to ϕ if and only if there exists a positive definite f(θ) such that

$$\begin{equation} \chi_{\psi}(\theta)=f({\theta})\chi_{\phi}({\theta}). \end{equation} \tag{ 73 }$$

Since f(θ) is positive definite all Fourier components of this function {q_n} are positive. Furthermore, since χ_ψ(0) = χ_ϕ(0) = 1 we conclude that f(0) = 1 which implies that $\sum _{n} q_{n}=1$ and so the set {q_n} is also a probability distribution. Suppose the probability distributions over integers p_ψ and p_ϕ are the Fourier transforms of χ_ψ and χ_ϕ respectively. Then the Fourier transform of equation (73) yields

$$\begin{equation} p_{\psi}(n)=\sum_k p_{{\phi}}({n-k}) q(k). \end{equation} \tag{ 74 }$$

So the U(1)-covariant transformation from ψ to ϕ exists iff there exists a probability distribution q over integers which satisfies the above equality. This is indeed the condition for deterministic interconversion in the U(1) case found in [3].

Suppose the group under consideration is the group Z_N, the cyclic group of order N. For any N, the group Z_N is isomorphic to the group of integers {0,...,N − 1} where the group action is addition modulo N. We use this isomorphism to denote the group elements. These groups are clearly Abelian and so all of their irreps are 1D. We can easily see that these irreps can be identified by an integer J in the set {0,...,N − 1} such that the irrep labeled by J is

$$\begin{equation} k\in Z_{N}\rightarrow U_{J}(k)={\mathrm{e}}^{{\mathrm{i}}2\pi Jk/N}. \end{equation} \tag{ 75 }$$

So an arbitrary (non-projective) unitary representation of Z_N, k∈Z_N → U(k), can be decomposed as

$$\begin{equation} U(k)=\bigoplus_{J,\alpha} {\mathrm{e}}^{{\mathrm{i}} Jk 2\pi/N } |J,\alpha\rangle\langle J,\alpha|, \end{equation} \tag{ 76 }$$

where α labels copies of irrep J and {|J,α〉} is a basis for the Hilbert space. An arbitrary state ψ in this basis can be expanded as

$$\begin{equation} |\psi\rangle=\sum _{J,\alpha} \psi(J,\alpha) |J,\alpha\rangle. \end{equation} \tag{ 77 }$$

As with any other Abelian group, the reduction of the state onto the irreps is simply the probability distribution that the state induces over the irreps. So the reduction of ψ is specified by the probability distribution

$$\begin{eqnarray*} &&\{p_\psi(J)\equiv \sum_\alpha |\psi(J,\alpha)|^2: J=0,\ldots,N\}. \end{eqnarray*}$$

On the other hand, the characteristic function of ψ is by definition the function k∈{0,...,N − 1} → 〈ψ|U(k)|ψ〉, that is,

$$\begin{equation} \chi_\psi(k)=\sum _{J,\alpha} |\psi(J,\alpha)|^2 \,{\mathrm{e}}^{{\mathrm{i}}2\pi Jk/N}. \end{equation} \tag{ 78 }$$

Clearly the characteristic function is the discrete Fourier transform of the reduction of the state onto the irreps.

Now we are interested to know whether there exists a Z_N-covariant quantum operation which transforms ψ to ϕ. Assuming the characteristic function of ϕ, χ_ϕ(k), is non-zero for all k's, it follows from theorem 6 that such a Z_N-covariant map exists iff χ_ψ(k)/χ_ϕ(k) is a positive definite function. But this function is positive definite iff its Fourier transform is always positive, i.e. iff

$$\begin{equation} q(J)\equiv \sum_{k} \frac{ \chi_\psi(k)}{\chi_\phi(k)} {\mathrm{e}}^{{\mathrm{i}}2\pi Jk/N} \end{equation} \tag{ 79 }$$

is positive for all J = 0,...,N. So to summarize, the necessary and sufficient condition for the existence of a Z_N-covariant channel which transforms ψ to ϕ is that

$$\begin{equation} \forall J\in\{0,\ldots ,N\}: \ \ \ \sum_{k} \frac{ \chi_\psi(k)}{\chi_\phi(k)} {\mathrm{e}}^{{\mathrm{i}}2\pi Jk/N}\geqslant 0. \end{equation} \tag{ 80 }$$

Consider the case of Z₂ which has only two group elements denoted by {e,π} where e is the identity of the group and π² = e. Using the above convention we denote e by k = 0 and π by k = 1. This group has only two inequivalent irreps: the trivial representation (J = 0) in which

$$\begin{eqnarray*} &&U_{J=0}(0)=U_{J=0}(1)=1 \end{eqnarray*}$$

and the non-trivial (J = 1) in which

$$\begin{eqnarray*} &&U_{J=1}(1)=-U_{J=1}(0)=-1. \end{eqnarray*}$$

Then the reduction of ψ onto irreps is specified by the probability assigned to each of these irreps and because there are only two irreps we only need to specify one of the probabilities, say p_ψ(J = 0). The characteristic function of ψ is

$$\begin{equation} \chi_{\psi}(k)= p_\psi(J=0)+(-1)^{k} p_\psi(J=1). \end{equation} \tag{ 81 }$$

So χ_ψ(0) = 1 and χ_ψ(1) = 2p_ψ(J = 0) − 1. Then equation (80) implies that the transformation $\psi \xrightarrow {G{\raise -1pt\hbox{-}}{\rm cov}} \phi$ is possible iff

$$\begin{equation} q(0)= \frac{ \chi_\psi(0)}{\chi_\phi(0)}+ \frac{ \chi_\psi(1)}{\chi_\phi(1)} \geqslant 0 \end{equation} \tag{ 82 }$$

and

$$\begin{equation} q(1)= \frac{ \chi_\psi(0)}{\chi_\phi(0)}- \frac{ \chi_\psi(1)}{\chi_\phi(1)} \geqslant 0. \end{equation} \tag{ 83 }$$

Since $\frac { \chi _\psi (0)}{\chi _\phi (0)}$ is always equal to one it turns out that the above two inequalities are equivalent to |χ_ψ(1)| ⩽ |χ_ϕ(1)|, i.e.

$$\begin{equation} \left| p_{\psi}(J=0)-p_{\psi}(J=1) \right| \leqslant \left| p_{\phi}(J=0)-p_{\phi}(J=1) \right|. \end{equation} \tag{ 84 }$$

Since

$$\begin{eqnarray*} &&p_{\psi}(J=0)+p_{\psi}(J=1)=p_{\phi}(J=0)+p_{\phi}(J=1)=1, \end{eqnarray*}$$

the above condition is equivalent to the condition

$$\begin{equation} \min\{p_{\phi}(J=0),p_{\phi}(J=1)\} \leqslant \min\{p_{\psi}(J=0),p_{\psi}(J=1)\} \end{equation} \tag{ 85 }$$

which is exactly the same condition previously obtained in [3] using a totally different approach. Equation (80) is the generalization of this specific result for arbitrary cyclic group Z_N.

In the case of compact connected Lie groups, using the above argument and by virtue of the analyticity of characteristic functions one can argue that catalysts cannot help, i.e. if a transformation is possible with a catalyst, it is also possible without any catalyst. To see this, first note that for any finite-dimensional representation of a compact Lie group there is a neighborhood around the identity element of the group within which the characteristic functions of all pure states are non-zero (otherwise there would be a unitary which is arbitrarily close to identity for which 〈ψ|U|ψ〉 = 0 for some state ψ, but in a finite-dimensional Hilbert space this is not possible). This implies that in this neighborhood, if equation (87) holds then the following equation holds:

$$\begin{equation} \chi_{\psi}(g)=\chi_{\phi}(g)f'(g). \end{equation} \noindent \tag{ 88 }$$

But since all these functions are analytic and since the group G is connected, if the above equality is true for a neighborhood around the identity element of G then it will be true for all G. Then by theorem 6 we can conclude that there exists a G-covariant channel which transforms ψ to ϕ (without the help of any catalyst). So if this transformation is possible with the use of a catalyst then it is also possible without using the catalyst. So to summarize we have proven that

Theorem 7. For symmetries associated with compact connected Lie groups, there are no examples of non-trivial catalysis using a finite catalyst.

The above argument clearly does not work in the case of finite groups. Indeed, as we will see in the following, in the case of finite groups there are states for which the characteristic function is zero for all g∈G except the identity. If we use such a state as a catalyst, equation (87) holds for all group elements and consequently for any pair of states ψ and ϕ, one can always transform one to the other using the catalyst. (Indeed as we show in the following one can always transform any mixed state to any other mixed state using such a catalyst.)

For a group with a finite number of elements, it is possible for the catalyst to consist of a system with a Hilbert space $\mathcal {H}$ having dimension greater than or equal to the order of the group. In this case, the representation of the group can be the left regular representation on the Hilbert space $\mathcal {H}$, g → T_L(g), such that

$$\begin{equation} \forall g\in G:\ \ \ T_{\mathrm{L}}(g)|h\rangle=|gh\rangle, \end{equation} \tag{ 89 }$$

where {|h〉:h∈G} is an orthonormal basis for $\mathcal {H}$. Now note that the characteristic function of any state |h〉 is χ_h(g) = 〈h|T_L(g)|h〉 = δ_e,g, the Kronecker-delta function centered on the identity group element. Equation (87) then implies that such a state can catalyze any pure-to-pure transformation.

Also it is straightforward to show that for any pair of states ρ and σ (pure or mixed) there exists a G-covariant channel which transforms ρ⊗|h〉〈h| to σ⊗|h〉〈h|. One realization of this G-covariant map is the following:

$$\begin{equation} \mathcal{E}_h(X) \equiv \sum_{g\in G}{\rm tr}\left( \left[\mathbb{I}\otimes|g\rangle\langle g|\right] X \right) U(gh^{-1})\sigma U^{\dagger}(gh^{-1})\otimes|g\rangle\langle g| , \end{equation} \tag{ 90 }$$

where g → U(g) is the representation of the symmetry on the space where σ lives and $\mathbb {I}$ is the identity operator acting on the Hilbert space of ρ.¹⁵

So unlike the case of connected compact Lie groups, in the case of a symmetry described by a finite group, catalysts can be helpful.

Recall our quantum optics example where the set of all phase shifts forms a representation of group U(1) (see example 1). Let Irreps_U(1)(ψ) be the set of eigenvalues of the number operator N to which the pure state ψ assigns non-zero weight. Assuming that ψ can be transformed to ϕ with non-zero probability under a U(1)-covariant operation, one can easily show that

• 1.  
  the cardinality of Irreps_U(1)(ψ) is larger than or equal to the cardinality of Irreps_U(1)(ϕ), i.e.

  $$\begin{equation} \left|{\rm Irreps}_{{\rm U}(1)}(\phi)\right|\leqslant \left|{\rm Irreps}_{{\rm U}(1)}(\psi)\right|, \end{equation} \tag{ 92 }$$
• 2.  
  $$\begin{eqnarray*}&&\fl {\rm max}\{{\rm Irreps}_{{\rm U}(1)}(\phi) \}-{\rm min}\{{\rm Irreps}_{{\rm U}(1)}(\phi) \}\leqslant {\rm max}\{{\rm Irreps}_{{\rm U}(1)}(\psi) \}-{\rm min}\{{\rm Irreps}_{{\rm U}(1)}(\psi) \}. \end{eqnarray*}$$

Here, we prove item 2 by contradiction. Assume this condition does not hold. Then for any positive definite function f(θ), χ_ϕ(θ)f(θ) has a non-zero component of e^imθ for some m such that m < n_min(ψ) or m > n_max(ψ). Since both χ_ϕ(θ) and f(θ) are positive-definite, the coefficient of e^imθ will be positive. This implies that for any non-zero probability p, the coefficient of e^imθ in χ_ψ(θ) − pχ_ϕ(θ)f(θ) is negative and so the function χ_ψ(θ) − pχ_ϕ(θ)f(θ) is not positive definite for any non-zero p. This proves the claim. Item 1 is proven similarly.

Item 2 was obtained by a different argument in [3].¹⁶

Let Irreps_SO(3)(ψ) be the set of all angular momentum quantum numbers j corresponding to irreps of SO(3) to which the pure state ψ assigns non-zero weight.

Using a similar argument to the one we used for the case of U(1), one can easily conclude that if ψ can be transformed to ϕ under an SO(3)-covariant channel, then

• 1.  
  the cardinality of Irreps_SO(3)(ψ) is larger than or equal to the cardinality of Irreps_SO(3)(ϕ), i.e.

  $$\begin{equation} \left|{\rm Irreps}_{{SO}(3)}(\phi)\right|\leqslant \left|{\rm Irreps}_{{SO}(3)}(\psi)\right|, \end{equation} \tag{ 93 }$$
• 2.  
  $$\begin{eqnarray*}&&\fl {\rm max}\{{\rm Irreps}_{{SO}(3)}(\phi) \}-{\rm min}\{{\rm Irreps}_{{SO}(3)}(\phi) \}\leqslant {\rm max}\{{\rm Irreps}_{{SO}(3)}(\psi) \}-{\rm min}\{{\rm Irreps}_{{SO}(3)}(\psi) \}, \end{eqnarray*}$$
• 3.  
  $$\begin{equation}{\rm max}\{{\rm Irreps}_{{SO}(3)}(\phi) \}\leqslant {\rm max}\{{\rm Irreps}_{{SO}(3)}(\psi) \}. \end{equation} \tag{ 94 }$$

The proofs of items 1 and 2 are similar to the case of U(1). To prove item 3 note that the maximum value of j to which χ_ϕ(θ)f(θ) assigns non-zero weight is greater than or equal to j_max(ϕ). So if j_max(ϕ) is strictly greater than j_max(ψ), then for any non-zero p, χ(ψ) − pχ_ϕ(θ)f(θ) cannot be positive definite.

Item 3 implies that if a pure state does not have any component of angular momentum higher than j then by rotationally covariant operations it cannot be transformed with non-zero probability to another pure state which assigns some amplitude to an angular momentum higher than j.

According to a version of the Stinespring dilation theorem, a general state-to-ensemble transformation can always be purified in the following way: first, the input system (with Hilbert space $\mathcal {H}_{\rm in}$) unitarily interacts with an ancillary system (with Hilbert space $\mathcal {H}_{\rm anc}$). Now we consider the total Hilbert space $\mathcal {H}_{\rm in}\otimes \mathcal {H}_{\rm anc}$ as

$$\begin{equation} \mathcal{H}_{\rm in}\otimes\mathcal{H}_{\rm anc}=\bigoplus_i \mathcal{H}_{i}\otimes \mathcal{H}_{i}' \otimes |i\rangle\langle i|. \end{equation} \tag{ 95 }$$

After the unitary time evolution we perform a projective measurement on the third subsystem in the basis {|i〉〈i|} and according to the outcome of measurement we discard the subsystem $\mathcal {H'}_{i}$. The output would be the system described by $\mathcal {H}_{i}$. This procedure realizes the most general state-to-ensemble transformation.

Suppose a transformation maps |ψ〉 to |ϕ_i〉 with probability p_i. Since the output is pure, clearly it cannot be entangled with the discarded system. In other words, after applying the unitary V which couples the system and ancilla the total state should be in the form

$$\begin{equation} V|\psi\rangle|\nu\rangle=\sum_i \sqrt{p_i} |\phi_i\rangle |\eta_i\rangle |i\rangle, \end{equation} \tag{ 96 }$$

where |ψ〉 is the initial state of the system and |ν〉 is the initial state of the ancilla.

Now according to an extension of Stinespring's dilation theorem for G-covariant quantum operations, if the state-to-ensemble transformation is G-covariant then one can choose the initial state |ν〉 of ancilla, the unitary V , and the basis {|i〉} to all be G-invariant [17] .

Assuming V is a G-invariant unitary then the characteristic function of the right hand side should be equal to the characteristic function of |ψ〉|ν〉. This implies

$$\begin{equation} \chi_{\psi}(g){\mathrm{e}}^{{\mathrm{i}}\theta(g)}=\sum_{i}p_{i} \chi_{\nu_{i}}(g)\chi_{\phi_{i}}(g) {\mathrm{e}}^{{\mathrm{i}}\alpha_i(g)}, \end{equation} \tag{ 97 }$$

where e^iθ(g) is the characteristic function of the G-invariant ancilla |ν〉 and {e^iα_i(g)} are the characteristic functions of the G-invariant states {|i〉}. Now because the product of two characteristic functions is also a characteristic function, χ_νi(g)e^iα_i(g) e^−iθ(g) is a valid characteristic function. So if there exists a G-covariant transformation which maps state ψ to ϕ_i with probability p_i, then the equation (91) should hold. This completes the proof of one direction of the theorem. To prove the other direction, we note that property (1) of characteristic functions listed in section 5.2 implies that there exists a set of states {|ν_i〉} which have characteristic functions equal to {f_i}. Now we choose |ν〉, the initial state of the ancilla, to be a G-invariant state and we assume that its characteristic function is equal to 1 for all group elements (i.e. any group element maps |ν〉 exactly to itself). Similarly we choose a basis {|i〉} to be a set of G-invariant orthonormal states and assume the characteristic functions of all of them are constant and equal to 1. Then, equation (91) implies that the characteristic function of |ψ〉|ν〉 is equal to the characteristic function of $\sum _i \sqrt {p_i} |\phi _i\rangle |\eta _i\rangle |i\rangle$ and so there exists a G-invariant unitary which maps the former state to the latter. Now by performing a measurement in the basis {|i〉} and discarding the subsystem with the state |η_i〉 we can realize the desired map. This completes the proof.

Suppose $\mathcal {E}$ is a channel (completely-positive trace-preserving linear map) from $\mathcal {B}(\mathcal {H}_{\rm in})$ to $\mathcal {B}(\mathcal {H}_{\rm out})$ which is G-covariant, i.e. for all g∈G we have $U_{\rm out}(g)\mathcal {E}[\cdot ]U^{\dagger }_{\rm out}(g) =\mathcal {E}(U_{\rm in}(g)[\cdot ]U^{\dagger }_{\rm in}(g) )$. Then we can always extend this channel to $\tilde {\mathcal {E}}$, a G-covariant channel from $\mathcal {B}(\mathcal {H}_{\rm in}\oplus \mathcal {H}_{\rm out})$ to itself, by defining

$$\begin{equation} \tilde{\mathcal{E}}\equiv \mathcal{E}(\Pi_{\rm in}[\cdot] \Pi_{\rm in})+ \frac{ I_{\mathcal{H}_{\rm in}\oplus \mathcal{H}_{\rm out} } }{d_{\rm in}+d_{\rm out} } {\rm tr}(\Pi_{\rm out}[\cdot] \Pi_{\rm out})m, \end{equation} \noindent \tag{ B.1 }$$

where $I_{\mathcal {H}_{\rm in}\oplus \mathcal {H}_{\rm out} } /({d_{\rm in}+d_{\rm out} })$ is the completely mixed state on $\mathcal {H}_{\rm in}\oplus \mathcal {H}_{\rm out}$. Clearly by this definition $\tilde {\mathcal {E}}$ is completely-positive and trace-preserving and so, a valid channel, and moreover it is G-covariant. Furthermore the restriction of $\tilde {\mathcal {E}}$ to $\mathcal {H}_{\rm in}$, i.e. $\tilde {\mathcal {E}}(\Pi _{\rm in}[ \cdot ]\Pi _{\rm in} )$, is equal to $\mathcal {E}(\cdot )$.

On the other hand, if there is a G-covariant channel from $\mathcal {B}(\mathcal {H}_{\rm in}\oplus \mathcal {H}_{\rm out})$ to itself which maps all operators in $\mathcal {B}(\mathcal {H}_{\rm in})$ to operators in $\mathcal {B}(\mathcal {H}_{\rm out})$ then clearly by restricting its input to $\mathcal {B}(\mathcal {H}_{\rm in})$ we get a valid G-covariant channel from $\mathcal {B}(\mathcal {H}_{\rm in})$ to operators in $\mathcal {B}(\mathcal {H}_{\rm out})$.

Finally consider the situation where there is a G-covariant channel $\mathcal {E}$ from $\mathcal {B}(\mathcal {H})$ to itself which maps ρ_i to σ_i for a set of i's. Assume the representation of the group G on the Hilbert space is {U(g):g∈G}. Define Π_in and Π_out to be respectively the span of the supports of all operators {U(g)ρ_iU^†(g)} and {U(g)σ_iU^†(g)}. It is clear from this definition that both Π_in and Π_out commute with all {U(g):g∈G}. Therefore the subspaces associated to these projectors, $\mathcal {H}_{\rm in}$ and $\mathcal {H}_{\rm out}$, have a natural representation of the group G given by {Π_inU(g)Π_in} and {Π_outU(g)Π_out}. Now $\tilde {\mathcal {E}}\equiv \mathcal {E}(\Pi _{\rm in}[\cdot ]\Pi _{\rm in})$ is a new G-covariant quantum channel which maps states from $\mathcal {B}(\mathcal {H}_{\rm in})$ to $\mathcal {B}(\mathcal {H}_{\rm out})$ and $\tilde {\mathcal {E}}(\rho _{i})=\sigma _{i}$.

Basically we can repeat all of these observations to prove the equivalence of a G-invariant unitary where the input and output spaces are the same and a G-invariant isometry where the input and output spaces are different.

For example if there exists a G-invariant unitary on $\mathcal {H}_{\rm in}\oplus \mathcal {H}_{\rm out}$ which unitarily maps the subspace $\mathcal {H}_{\rm in}$ to (a subspace of) $\mathcal {H}_{\rm out}$ then clearly there exists a G-invariant isometry V from H_in to $\mathcal {H}_{\rm out}$ such that ∀g∈G: V U_in(g) = U_out(g)V and V^†V =I_in where I_in is the identity on $\mathcal {H}_{\rm in}$.

The only property which is less trivial in the case of unitary-isometry equivalences is the following: suppose V is an isometry from H_in to H_out which is G-invariant i.e. ∀g∈G: V U_in(g) = U_out(g)V . Then there exits a unitary V_ext on $\mathcal {H}_{\rm in}\oplus \mathcal {H}_{\rm out}$ such that ∀g:∈G: V_ext(U_in(g)⊕U_out(g)) = (U_in(g)⊕U_out(g))V_ext and moreover V =Π_outV_extΠ_in where Π_in/out is the projector to $\mathcal {H}_{\mathrm { in/out}}$. This is shown by the following lemma.

Lemma B.1. Suppose W maps the subspace of the support of the projector Π unitarily to another subspace such that ΠW^†WΠ = Π (in other words, WΠ is an isometry). Then if ∀g∈G: [WΠ,U(g)] = 0 there exits a unitary W_G-inv such that ∀g∈G: [W_G-inv,U(g)] = 0 and W_G-invΠ = WΠ.

Proof. WΠ commutes with all U(g) and so does ΠW^†. Therefore Π = ΠW^†WΠ also commutes with all U(g). Now we consider the decomposition of U(g) to irreps,

$$\begin{equation} U(g)=\bigoplus_{\mu} U_{\mu}(g) \otimes I_{\mathcal{N}_{\mu}}. \end{equation} \tag{ B.2 }$$

Since Π commutes with all {U(g):g∈G} it has a simple form in this basis:

$$\begin{equation} \Pi=\bigoplus_\mu I_\mu\otimes {\Pi}^{(\mu)}, \end{equation} \tag{ B.3 }$$

where Π² = Π implies ${{\Pi}^{(\mu)}}^{2}={\Pi}^{(\mu)}$ and so all Π^(μ)'s are projectors (note that for some μ, Π_μ might be zero). WΠ also commutes with all {U(g)}. Since WΠ = (WΠ)Π we conclude that the decomposition of WΠ should be in the following form:

$$\begin{equation} W\Pi=\bigoplus_\mu I_\mu\otimes (W^{(\mu)}{\Pi}^{(\mu)}). \end{equation} \tag{ B.4 }$$

ΠW^†WΠ = Π implies that ${\Pi}^{(\mu)} {W^{(\mu)}}^\dagger {W^{(\mu)}}{\Pi}^{(\mu)}={\Pi}^{(\mu)}$. Therefore W^(μ)Π^(μ) acts unitarily on the subspace of the support of Π^(μ). Now we can always find a unitary $\tilde {W}^{(\mu )}$ on this subsystem such that $\tilde {W}^{(\mu )}{\Pi }^{(\mu )}=W^{(\mu )}{\Pi }^{(\mu )}$. Finally define the unitary $\tilde {W}$ as

$$\begin{equation} {W}_{G{\raise -1pt\hbox{-}}{\rm inv}}=\bigoplus_\mu I_\mu\otimes \tilde{W}^{(\mu)}. \end{equation} \tag{ B.5 }$$

Clearly it commutes with all {U(g)} and $\tilde {W}\Pi =W\Pi$.   □

For a real random variable x with the distribution function F(x) the characteristic function is defined as the expectation value of the random variable e^itx i.e.

$$\begin{equation} f_{x}(t)=\int \mathrm{d}F(x) {\mathrm{e}}^{{\mathrm{i}}tx}. \end{equation} \tag{ D.1 }$$

The distribution function is uniquely determined by its characteristic function. Moreover if the probability density exists then it will be equal to the inverse Fourier transform of the characteristic function. One particularly useful property of the characteristic function is the multiplicative property according to which the characteristic function of the sum of two independent random variables is equal to the product of their characteristic functions:

$$\begin{equation} f_{x+y}(t)=f_{x}(t)f_{y}(t). \end{equation} \tag{ D.2 }$$

There exists a remarkably simple proof of the central limit theorem using this multiplicative property of characteristic functions.

The derivative of characteristic functions at the origin determines the moments of the random variable

$$\begin{equation} \langle x^n \rangle = \mathrm{i}^{-n} \frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}} f_{x}(t)\mid_{t=0}. \end{equation} \tag{ D.3 }$$

Sometimes it is more favorable to use cumulants of the random variable instead where the nth order cumulant is defined as the nth order derivative of the logarithm of the characteristic function at the point 0, multiplied by i⁻ⁿ:

$$\begin{equation} \kappa^{(n)}\equiv \mathrm{i}^{-n} \frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}} \log(f_{x}(t))\mid_{t=0}. \end{equation} \tag{ D.4 }$$

The first and second cumulants are the mean and the variance of the random variable. By this definition, it turns out that cumulants of a sum of independent random variables is equal to the sum of the cumulants of the individual terms for all orders of cumulants.

The set of all classical characteristic functions is determined by Bochner's theorem, according to which a complex function f(t) is the characteristic function of a random variable if and only if (i) f(0) = 1, (ii) f(t) is continuous at the origin, and (iii) it is positive definite. Recall that a function f(t) is positive definite if for any integer n and for any string of real numbers t₁,...,t_n the matrix a_i,j ≡ f(t_i − t_j) is a positive definite matrix. Positive definiteness of a function guarantees that the inverse Fourier transform of this function is positive for all values of the random variable, which is clearly a necessary condition for a function to be a probability density.

For more discussion about the properties of characteristic functions of probability distributions, see e.g. [24].

As the characteristic function of a probability distribution determines all of its statistical properties, the characteristic function of a quantum state over the group G uniquely specifies all the statistical properties of observables in the algebra of observables which generates the projective unitary representation of G. For example, suppose L is the representation of a generator of the Lie group G, then we have

$$\begin{equation} {\rm tr}(\rho L^{k})= \mathrm{i}^{-k}\frac{\partial^{k}}{\partial \theta^{k}} \chi_{\rho}({\mathrm{e}}^{{\mathrm{i}}\theta L}) \mid_{{{\theta=0}}}. \end{equation} \tag{ D.5 }$$

In particular the first derivative (k = 1) determines the expectation value of the generator. This is just property 7 of characteristic functions from section 5.2.

Similarly we can define cumulants of the observable L, where the nth order cumulant is defined as the nth order derivative of the logarithm of the characteristic function at the identity element multiplied by i⁻ⁿ:

$$\begin{equation} \kappa_{L}^{(n)}\equiv \mathrm{i}^{-k}\frac{\partial^{k}}{\partial \theta^{k}} \log[\chi_{\rho}({\mathrm{e}}^{{\mathrm{i}}\theta L})] \mid_{{{\theta=0}}}. \end{equation} \tag{ D.6 }$$

The first and second cumulants are the mean and the variance of the observable. By this definition, it turns out that the cumulants of the tensor product of two states is equal to the sum of the cumulants of the individual states for all orders of cumulants.

In the rest of this appendix, we are interested to find the generalization of Bochner's theorem, i.e. the set of necessary and sufficient conditions for ϕ(g), a complex function over group, to be the characteristic function of some quantum state. We see that such a generalization can be found via both the non-commutative Fourier transform and the Gelfand–Naimark–Segal (GNS) construction theorem. As in the rest of the paper, we focus on the finite groups and compact Lie groups.

As the first necessary condition we note that tr(ρ) = 1 implies that χ(e) = 1 (where e is the identity of the group). We call the functions which satisfy this condition normalized functions. In the case of compact Lie groups, ϕ(g) should also be a continuous function. We also need a condition on ϕ(g) equivalent to the positivity of density operators. As we just saw in the case of probability distributions the condition of positivity of probabilities is equivalent to the positive definiteness of the characteristic function of the probability distribution. Similarly it turns out that the relevant condition on ϕ(g) to be the characteristic function of a positive operator is the natural generalization of positive definiteness for the functions defined on the group:

Definition D.1. A complex function ϕ(g) on a group G is positive definite if for all choices $m\in \mathbb {N}$, g₁,...,g_m∈G and $\alpha _{1},\ldots , \alpha _{m}\in \mathbb {C}$

$$\begin{equation} \sum_{{i,j=1}}^{m} \bar{\alpha_{i}}\alpha_{j} \phi(g^{-1}_{i}g_{j}) \geqslant 0. \end{equation} \tag{ D.7 }$$

For the case of compact Lie groups where the function should also be continuous we can express the condition as

Definition D.2. A continuous function ϕ(g) on a group G with the Haar measure dg is called positive definite if it satisfies

$$\begin{equation} \int\int \mathrm{d}g\,\mathrm{d}h\ \bar{f}(g)\phi(g^{-1}h)f(h)\geq\ 0 \end{equation} \tag{ D.8 }$$

for any f∈L¹(G).

Now using the Fourier transform, one can easily prove a theorem similar to Bochner's theorem [26, 27]:

Theorem D.1. A complex function ϕ(g) on the finite or compact Lie group G is the characteristic function of a quantum state in a finite-dimensional Hilbert space iff ϕ(e) = 1, ϕ(g) is positive definite and continuous (in the case of Lie groups).

Proof. We present the proof assuming that the group G is a compact Lie group (the same argument works for a finite group by replacing integrals with summation). We use the inverse Fourier transform. Suppose $B^{(\mu )}\equiv d_{\mu }\int \mathrm {d}g U^{(\mu )}(g^{-1})\phi (g)$. Then the set of operators {B^(μ)} is the reduction onto irreps of a valid quantum state iff (i) $\sum _{\mu } {\rm tr}(B^{(\mu )})=1$ and (ii) all operators {B^(μ)} are positive definite. The first condition expresses the fact that the trace of the state is 1 and is guaranteed by ϕ(e) = 1. On the other hand, B^(μ) is positive iff tr(FF^†B^(μ)) ⩾ 0 for all operators F acting on $\mathcal {M}_\mu$ (the subsystem on which U^μ acts irreducibly). Note that tr(FF^†B^(μ)) is equal to the Fourier transform of the operator FF^†B^(μ) at point e. So using the convolution property of characteristic functions, equation (58), we get

$$\begin{equation} {\rm tr}(FF^\dagger B^{(\mu)})=d^2_\mu \int \int \mathrm{d}h_1\, \mathrm{d}h_2 f(h_1) \overline{f(h_2)} \phi(h^{-1}_1h_2). \end{equation} \tag{ D.9 }$$

So if ϕ(g) is positive definite and therefore satisfies equation (D.8) then all B^(μ)'s are positive. We can prove the other direction of the theorem similarly.

Therefore the set of normalized positive definite functions (also continuous in the case of Lie groups) are exactly the set of characteristic functions of states.

We can also get this result using a more fundamental theorem in the representation theory of C* algebras, namely, the GNS construction. A specific form of this theorem states

Theorem D.2 With every (continuous) positive definite function ϕ(g) we can associate a Hilbert space $\mathcal {H}$, a unitary representation {U(g):g∈G} of G in $\mathcal {H}$ and a vector ψ, cyclic for {U(g):g∈G}, such that

$$\begin{equation} \phi(g)=\left\langle \psi\right\vert U(g)\left\vert \psi\right\rangle . \end{equation} \tag{ D.10 }$$

Moreover the representation {U(g)} is unique up to a unitary equivalence.

Note that a vector |ξ〉 is cyclic for the representation {U(g):g∈G} on the space $\mathcal {H}$ if the span of vectors {U(g)|ξ〉: g∈G} is a dense subset of the space $\mathcal {H}$.

Therefore the GNS construction theorem guarantees that for any given (continuous) normalized positive definite function there exists a corresponding pure cyclic state with that characteristic function. Note that for any arbitrary mixed or pure state there exists a pure state which is cyclic (for the representation on its Hilbert space) with exactly the same characteristic function. So the set of all (continuous) normalized, positive definite functions is exactly the same as the set of all characteristic functions of states.