Maximally informative ensembles for SIC-POVMs in dimension 3

The most general description of quantum measurement is given by a positive operator valued measure (POVM). Mathematically, in the finite case, it is a set of Hermitian positive semidefinite operators Π_j summing up to identity $\sum _{j=1}^{k}{{\Pi }_{j}}=\mathbb{I}$. The outcomes of the measurement are given by $j\in \{1,\ldots ,k\}$ and the probability of obtaining the jth outcome, if the system before measurement was in the state ρ, is given by ${{p}_{j}}(\rho ,\Pi )={\rm Tr}(\rho {{\Pi }_{j}})$. A POVM is called informationally complete (IC POVM) if the probabilities ${{p}_{j}}(\rho ,\Pi )$ allow us to completely determine the initial state ρ. IC POVMs were first introduced in [19]. The minimal number of elements of an IC POVM is d², where d is the dimension of the Hilbert space describing our quantum system. A symmetric informationally complete POVM (SIC-POVM) consists of d² subnormalized rank-one projections ${{\Pi }_{j}}=\mid {{\phi }_{j}}\rangle \langle {{\phi }_{j}}\mid /d$ with equal pairwise Hilbert–Schmidt inner products: ${\rm Tr}(\Pi _{i}^{*}{{\Pi }_{j}})=\mid \langle {{\phi }_{i}}\mid {{\phi }_{j}}\rangle {{\mid }^{2}}/{{d}^{2}}=1/({{d}^{2}}(d+1))$ for $i\ne j$. The notion of SIC-POVMs has been introduced by Renes et al [22], but they were studied previously by Zauner in his PhD thesis [30] under the name of regular quantum designs with degree 1. Since then they have been gaining increased attention in the quantum community due to their applications in quantum state tomography [24, 32], quantum cryptography [21] and quantum communication [18]. The question of whether a SIC-POVM exists in any finite dimension is still open.

We consider the group-covariant SIC-POVMs with respect to the Weyl–Heisenberg (WH) group; see section 2. The group covariance means that a group G exists with projective unitary representation $G\;\ni \;g\to {{U}_{g}}\in {\rm U}(d)$ and a surjection $s:G\to \{1,\ldots ,k\}$ such that ${{U}_{g}}{{\Pi }_{s(h)}}U_{g}^{*}={{\Pi }_{s(gh)}}$. We call $\mid {{\phi }_{s(e)}}\rangle$ a fiducial vector (where e denotes the neutral element of G). Without loss of generality one can assume that $s(e)=1$. The assumption of WH covariance is not very restrictive since all known SIC-POVMs are group-covariant and most of them are WH-covariant. In particular, if d is prime then group covariance implies WH covariance [31, lemma 1].

Here, we determine the states of the system for which the uncertainty of the measurement outcomes is minimal before the measurement. One of the ways to determine this is to study the Shannon entropy of measurement Π, defined by:

$$\begin{eqnarray*}&&H(\rho ,\Pi )\;:=\;\mathop{\sum }\limits_{j=1}^{k}\eta ({{p}_{j}}(\rho ,\Pi )),\end{eqnarray*}$$

for an initial state ρ, where $\eta (x)\;:=\;-x{\rm ln} x$ ($x\gt 0$), $\eta (0)=0$. This quantity has already been considered, for example, in the context of entropic uncertainty principles [9, 16, 17, 29], and any lower bound for the entropy of measurement can be regarded as an entropic uncertainty relation for single measurement [16].

The problem of minimizing entropy is connected with the problem of maximization of the mutual information between ensembles of initial states (classical quantum states) and the POVM Π. For an ensemble $V\;:=\;\{{{\pi }_{i}},{{\rho }_{i}}\}_{i=1}^{l}$ of initial states ρ_i with a priori probabilities ${{\pi }_{i}}$ the mutual information between V and Π is given by

$$\begin{eqnarray*}&&I(V,\Pi )\;:=\;\mathop{\sum }\limits_{i=1}^{l}\eta \left( \mathop{\sum }\limits_{j=1}^{k}{{P}_{ij}} \right)+\mathop{\sum }\limits_{j=1}^{k}\eta \left( \mathop{\sum }\limits_{i=1}^{l}{{P}_{ij}} \right)-\mathop{\sum }\limits_{i=1}^{l}\mathop{\sum }\limits_{j=1}^{k}\eta ({{P}_{ij}}),\end{eqnarray*}$$

where ${{P}_{ij}}={{\pi }_{i}}{\rm Tr}({{\rho }_{i}}{{\Pi }_{j}})$ for $i=1,\ldots ,l$ and j = 1,..., k. The problem of maximization of $I(V,\Pi )$ consists of two dual aspects [3, 13, 15]: maximization over all possible measurements, providing the ensemble V is given (see e.g. [8, 12, 23, 28]), and the less explored maximization over ensembles, when the POVM Π is fixed [2, 18]. We are interested in the second one, which allows us to answer the question of how informative the measurement is, by looking for the quantity called informational power [4]:

$$\begin{eqnarray*}&&W(\Pi )\;:=\;\mathop{{\rm sup} }\limits_{V}\;I(V,\Pi ).\end{eqnarray*}$$

An ensemble that maximizes the mutual information is called maximally informative for Π. In fact, it is enough to take into consideration ensembles consisting of pure states only [4, 18]. What is more, if Π is group-covariant, then the maximizer can be found in the set of group-covariant ensembles; i.e., ensembles of the form $V(\rho )\;:=\;{{\{\mid G{{\mid }^{-1}},{{U}_{g}}\rho U_{g}^{*}\}}_{g\in G}}$, where ρ is a pure state [18]. Additionally, the problems of finding the informational power of group-covariant measurement and of minimizing the entropy of such a measurement are equivalent, since in such a situation we have

$$\begin{eqnarray*}&&I(V(\rho ),\Pi )={\rm ln} \mid G\mid -H(\rho ,\Pi )\;=:\;\tilde{H}(\rho ,\Pi ),\end{eqnarray*}$$

where $\tilde{H}(\rho ,\Pi )$ is the relative entropy of Π with respect to the uniform distribution; i.e., the relative entropy (or Kullback–Leibler divergence) of the probability distribution of measurement outcomes with respect to the uniform distribution. Note that $\tilde{H}$ measures non-uniformity of the distribution of the measurement outcomes and 'can be interpreted as a measure of knowledge, as against uncertainty' [27]. Indeed, the greater $\tilde{H}$ is, the more we know about the measurement outcomes.

Both the minimum entropy of Π and its informational power can be interpreted in terms of a quantum classical channel $\Phi :\mathcal{S}\left( {{\mathbb{C}}^{d}} \right)\to \mathcal{S}\left( {{\mathbb{C}}^{k}} \right)$ generated by Π and given by $\Phi \left( \rho \right)=\sum _{j=1}^{k}{\rm Tr}\left( \rho {{\Pi }_{j}} \right)\left| {{e}_{j}} \right\rangle \left\langle {{e}_{j}} \right|$, where $\left( \left| {{e}_{j}} \right\rangle \right)_{j=1}^{k}$ is any orthonormal basis in ${{\mathbb{C}}^{k}}$. The minimum output entropy of Φ is equal to the minimum entropy of Π; i.e., ${{{\rm min} }_{\rho }}S(\Phi (\rho ))={{{\rm min} }_{\rho }}H(\rho ,\Pi )$ [25], where S denotes the von Neumann entropy. Moreover, the informational power of Π can be identified [14, 18] as the classical capacity $\chi (\Phi )$ of the channel Φ; i.e.,

$$\begin{eqnarray*}&&W(\Pi )=\chi (\Phi )\;:=\;\mathop{{\rm max} }\limits_{V=\{{{\pi }_{i}},{{\rho }_{i}}\}}\left\{ S\left( \mathop{\sum }\limits_{i}{{\pi }_{i}}\Phi ({{\rho }_{i}}) \right)-\mathop{\sum }\limits_{i}{{\pi }_{i}}S(\Phi ({{\rho }_{i}})) \right\}.\end{eqnarray*}$$

While working on this paper, we discovered that the informational power for an exemplary SIC-POVM in dimension 3 had been independently calculated by DallʼArno et al [5]. Nevertheless, our results work for an arbitrary group-covariant SIC-POVM in dimension 3. Moreover, they seem to give a more complete characterization of the maximizers as they provide a deeper insight into their geometric and algebraic structure. In particular, the group-covariant maximally informative ensembles are described in detail.

Let us denote an orthonormal basis in ${{\mathbb{C}}^{d}}$ by $\mid {{e}_{0}}\rangle ,\mid {{e}_{1}}\rangle ,...\mid {{e}_{d-1}}\rangle$. We define the unitary operators T and S as follows:

$$\begin{eqnarray*}&&T\mid {{e}_{r}}\rangle \;:=\;{{\omega }^{r}}\mid {{e}_{r}}\rangle ,\quad S\mid {{e}_{r}}\rangle \;:=\;\mid {{e}_{r\oplus 1}}\rangle ,\end{eqnarray*}$$

where $r=0,\ldots ,d-1$, ⊕ denotes the addition modulo d and $\omega :={\rm exp} (2\pi {\rm i}/d)$. We define also ${{D}_{{\bf p}}}={{D}_{({{p}_{1}},{{p}_{2}})}}:={{\tau }^{{{p}_{1}}{{p}_{2}}}}{{S}^{{{p}_{1}}}}{{T}^{{{p}_{2}}}}$, where $\tau :=-{\rm exp} (\pi {\rm i}/d)$ and ${\bf p}\in {{\mathbb{Z}}^{2}}$. We have

$$\begin{eqnarray*}&&D_{{\bf p}}^{*}={{D}_{-{\bf p}}}\end{eqnarray*}$$

and

$$\begin{eqnarray*}&&{{D}_{{\bf p}}}{{D}_{{\bf q}}}={{\tau }^{\langle {\bf p},{\bf q}\rangle }}{{D}_{{\bf p}+{\bf q}}}\end{eqnarray*}$$

for all ${\bf p},{\bf q}\in {{\mathbb{Z}}^{2}}$, where $\langle {\bf p},{\bf q}\rangle ={{p}_{2}}{{q}_{1}}-{{p}_{1}}{{q}_{2}}$ is a symplectic form.

The finite Heisenberg group (also called the finite WH group or the generalized Pauli group) is irreducibly and faithfully represented by the elements of the form ${{\omega }^{{{p}_{3}}}}{{D}_{({{p}_{1}},{{p}_{2}})}}$, where ${{p}_{1}},{{p}_{2}},{{p}_{3}}\in {{\mathbb{Z}}_{d}}$. The map ${{\mathbb{Z}}_{d}}\times {{\mathbb{Z}}_{d}}\;\ni \;({{p}_{1}},{{p}_{2}})\mapsto {{D}_{({{p}_{1}},{{p}_{2}})}}$ defines the projective unitary representation of ${{\mathbb{Z}}_{d}}\times {{\mathbb{Z}}_{d}}$ on ${{\mathbb{C}}^{d}}$ and matrices ${{D}_{{\bf p}}}$ are called Weyl matrices or generalized Pauli matrices.

Let us consider the normalizer of the WH group in the group ${\rm UA}(d)$ of all unitary and antiunitary operators on ${{\mathbb{C}}^{d}}$, the so-called extended Clifford group ${\rm EC}(d)$. We denote by ${\rm ESL}(2,{{\mathbb{Z}}_{d}})$ the group of all 2 × 2 matrices over ${{\mathbb{Z}}_{d}}$ with determinant $\pm 1$ (mod d) and by ${\rm i}(d)$ the group of unitary multiples of the identity operator on ${{\mathbb{C}}^{d}}$. Appleby [1, theorem 2] has shown that for an odd dimension d there exists a unique isomorphism ${{f}_{E}}:{\rm ESL}(2,{{\mathbb{Z}}_{d}})\ltimes {{({{\mathbb{Z}}_{d}})}^{2}}\to {\rm EC}(d)/{\rm i}(d)$ fulfiling the condition

$$\begin{eqnarray*}&&U{{D}_{{\bf p}}}{{U}^{*}}={{\omega }^{\langle {\bf r},\mathcal{F}{\bf p}\rangle }}{{D}_{\mathcal{F}{\bf p}}}\end{eqnarray*}$$

for any $(\mathcal{F},{\bf r})\in {\rm ESL}(2,{{\mathbb{Z}}_{d}})\ltimes {{({{\mathbb{Z}}_{d}})}^{2}}$, $U\in {{f}_{E}}(\mathcal{F},{\bf r})$ and ${\bf p}\in {{({{\mathbb{Z}}_{d}})}^{2}}$.

Let us consider a unitary ${{U}_{(\mathcal{F},{\bf r})}}\in {{f}_{E}}(\mathcal{F},{\bf r})$, such that ${\rm det} \mathcal{F}=1$, ${\rm Tr}\mathcal{F}\equiv -1$ (mod d), $\mathcal{F}\ne \mathcal{I}$. One can choose the phase factor of ${{U}_{(\mathcal{F},{\bf r})}}$ in a way that ${{({{U}_{(\mathcal{F},{\bf r})}})}^{3}}=\mathbb{I}$ [1]. Such a unitary is called (following Appleby) a canonical order 3 unitary.

It has been conjectured by Zauner [30] that in every dimension there exists a fiducial vector for some WH-covariant SIC-POVM which is an eigenvector of the canonical order 3 unitary ${{U}_{\mathcal{Z}}}:={{U}_{(\mathcal{Z},0)}}$, where $\mathcal{Z}=\left( \begin{array}{ccccccccccccccc} 0 & -1 \\ 1 & -1 \\ \end{array} \right)$ has been later referred to as the Zauner matrix. This has been strengthened by Appleby [1] to the conjecture that in every dimension there exists a fiducial vector and every such vector is an eigenvector of a canonical order 3 unitary conjugated to ${{U}_{\mathcal{Z}}}$ (the conjugacy relation is considered up to a phase in the extended Clifford group). Grassl [11] gave a counter-example to the latter one in dimension 12, but there are no known counter-examples in any other dimensions. Still, another conjecture by Appleby, that in every dimension there exists a fiducial vector and every such vector is an eigenvector of a canonical order 3 unitary, remains open. Let us also recall that these two conjectures are equivalent in the prime dimensions greater than three, since in these dimensions all canonical order 3 unitaries are in the same conjugacy class [10].

We consider a group-covariant SIC-POVM in dimension 3. As mentioned in the introduction, every such SIC-POVM must be group-covariant with respect to the WH group.

The first result we present here is strictly connected with the geometry of SIC-POVM in dimension 3 and does not involve any algebraic structure, although the assumption of this theorem concerning linear dependency among the vectors defining a SIC-POVM is not by any means obvious. However, it follows from [7, theorem 1] that this assumption is fulfilled if a SIC-POVM is covariant with respect to the WH group and its fiducial vector is an eigenvector of a certain canonical order 3 unitary conjugated to ${{U}_{\mathcal{Z}}}$, which is not a huge restriction since all known SIC-POVMs in dimension 3 are of this form. The second theorem gives us a deeper insight into the algebraic structure of entropy minimizers.

Theorem 1. Let $\Pi =\{(1/3)\mid {{\phi }_{j}}\rangle \langle {{\phi }_{j}}\mid \}_{j=1}^{9}$ be a SIC-POVM in dimension 3 and let us assume that some 3 out of 9 vectors $\mid {{\phi }_{j}}\rangle$ are linearly dependent. Then the state $\mid \psi \rangle \langle \psi \mid$, where $\mid \psi \rangle$ is orthogonal to the two-dimensional subspace spanned by these vectors, minimizes (resp. maximizes) the entropy of Π (resp. the relative entropy of Π). Moreover, all global minimizers (resp. maximizers) are of this form.

Proof. Let us assume that $\mid {{\phi }_{1}}\rangle ,\mid {{\phi }_{2}}\rangle$ and $\mid {{\phi }_{3}}\rangle$ are linearly dependent. We will consider the Bloch representation of quantum states. We can represent our SIC-POVM on the generalized Bloch set ${\bf B}\subset {{S}^{7}}$ (unit sphere) by the set of vertices of a regular 8-simplex, which we denote by $B:=\{{{v}_{1}},{{v}_{2}},\ldots ,{{v}_{9}}\}$. Inner products of vectors ${{\psi }_{1}},{{\psi }_{2}}\in {{\mathbb{C}}^{3}}$ and the corresponding Bloch vectors ${{u}_{1}},{{u}_{2}}\in {{\mathbb{R}}^{8}}$ are related in the following way: $\mid \langle {{\psi }_{1}}\mid {{\psi }_{2}}\rangle {{\mid }^{2}}=(2({{u}_{1}}\cdot {{u}_{2}})+1)/3$. In particular, if $\mid {{\psi }_{1}}\rangle$ and $\mid {{\psi }_{2}}\rangle$ are orthogonal, then ${{u}_{1}}\cdot {{u}_{2}}=-1/2$. Let us consider the five-dimensional affine subspace ${{\pi }_{1}}$:

$$\begin{eqnarray*}&&{{\pi }_{1}}:=\{u\in {{\mathbb{R}}^{8}}\mid u\cdot {{v}_{1}}=u\cdot {{v}_{2}}=u\cdot {{v}_{3}}=-1/2\}\end{eqnarray*}$$

and the affine hyperplane ${{\pi }_{2}}$ tangent to the sphere S⁷ at the point $w:=-({{v}_{1}}+{{v}_{2}}+{{v}_{3}})/\parallel {{v}_{1}}+{{v}_{2}}+{{v}_{3}}\parallel$:

$$\begin{eqnarray*}&&{{\pi }_{2}}:=\{w+{{u}_{0}}\mid {{u}_{0}}\in {{\mathbb{R}}^{8}},{{u}_{0}}\cdot w=0\}=\{u\in {{\mathbb{R}}^{8}}\mid u\cdot w=1\}.\end{eqnarray*}$$

It is easy to check that $w\in {{\pi }_{1}}\subset {{\pi }_{2}}$. Thus w needs to be the Bloch vector corresponding to $\mid \psi \rangle .$ For $j\in \{4,\ldots ,9\}$ we get $w\cdot {{v}_{j}}=1/4$.

We now apply the method based on the Hermite interpolation described in detail in [26]. Firstly, we redefine the entropy of Π to be a function of Bloch vectors:

$$\begin{eqnarray*}&&{{H}_{{\bf B}}}(u)\;:=\;H(\rho ,\Pi )=\mathop{\sum }\limits_{j=1}^{9}\eta \left( \frac{1+2u\cdot {{v}_{j}}}{9} \right)=\mathop{\sum }\limits_{j=1}^{9}h(u\cdot {{v}_{j}}),\end{eqnarray*}$$

where u is the Bloch vector corresponding to ρ and $h:[-1/2,1]\to {{\mathbb{R}}^{+}}$ is given by $h(t)=\eta \left( \frac{1+2t}{9} \right)$. We are looking for the interpolating Hermite polynomial p such that $p(-1/2)=h(-1/2)$, $p(1/4)=h(1/4)$ and $p^{\prime} (1/4)=h^{\prime} (1/4)$. What is crucial here is that p interpolates h from below. The degree of p is at most 2, so it is the degree of the polynomial function $P:{\bf B}\to \mathbb{R}$ given by $P(u)=\sum _{j=1}^{9}p\left( u\cdot {{v}_{j}} \right)$ for $u\in {\bf B}$. As $\sum _{j=1}^{9}{{v}_{j}}=0$, the linear part vanishes. Additionally, since $\sum _{j=1}^{9}{{(u\cdot {{v}_{j}})}^{2}}=\frac{9}{8}\parallel u{{\parallel }^{2}}$ for any $u\;\in \;R$⁸ (vertices of regular N-simplex in ${{\mathbb{R}}^{N}}$ form a tight frame in ${{\mathbb{R}}^{N}}$ with bound $N/(N-1)$), P must be constant on any sphere. Knowing that $P(u)\leqslant {{H}_{{\bf B}}}(u)$ and $P(w)={{H}_{{\bf B}}}(w)\ (={\rm ln} 6)$ we conclude that the entropy attains its minimum value (and so the relative entropy $\tilde{H}$ attains its maximum value) at $\mid \psi \rangle \langle \psi \mid$.

In order to show that all global minimizers of the entropy (and so maximizers of the relative entropy) are of the same form; i.e., they are orthogonal to some 3 out of 9 vectors defining SIC-POVM, let us observe that if $\tilde{w}\in {\bf B}$ is a global minimizer for ${{H}_{{\bf B}}}$, then ${{H}_{{\bf B}}}(\tilde{w})=P(\tilde{w})={\rm ln} 6$. In consequence also $h(\tilde{w}\cdot {{v}_{j}})=p(\tilde{w}\cdot {{v}_{j}})$ for $j=1,\ldots ,9$, and so $\{\tilde{w}\cdot {{v}_{j}}\mid j=1,\ldots ,9\}\subset \{w\cdot {{v}_{j}}\mid j=1,\ldots ,9\}=\{-1/2,1/4\}$, since p agrees with h exactly in the points of interpolation. Under the constraint $\sum _{j=1}^{9}\tilde{w}\cdot {{v}_{j}}=0$ we get that $\{\tilde{w}\cdot {{v}_{j}}\mid j=1,\ldots ,9\}=\{-1/2,1/4\}$ and there are exactly three jʼs such that $\tilde{w}\cdot {{v}_{j}}=-1/2$; i.e., there are exactly three vectors $\mid {{\phi }_{j}}\rangle$ orthogonal to $\mid \tilde{\psi }\rangle$.□

Note that it is possible to give an alternative proof that the state $\mid \psi \rangle \langle \psi \mid$ minimizes the entropy. It is enough to notice that the lower bound for the Shannon entropy, namely ${\rm ln} 6$, provided by Rastegin [20, proposition 6] is satisfied here, as it was done in [5, corollary 2]. Moreover, we can consider as well some generalized entropies as it turns out that both the lower bounds for the Tsallis α-entropies for $\alpha \in (0,2]$, i.e. ${{(1-\alpha )}^{-1}}({{6}^{1-\alpha }}-1)$, given in [20, proposition 6] and the lower bound ${\rm ln} 6$ for the Rényi α-entropies for $\alpha \in (0,2]$, given as the corollary from [20, proposition 7], are achieved. However, dimension 3 can be exceptional, in the sense that the lower bounds may not be satisfied in higher dimensions, as indicated by some preliminary numerical calculations in dimensions 4 to 6. On the other hand, the method based on the Hermite interpolation seems to be applicable also in the higher dimensions.

Theorem 2. Let ${{U}_{(\mathcal{G},{\bf q})}}$ be a canonical order 3 unitary conjugated (up to a phase in the extended Clifford group) to ${{U}_{\mathcal{Z}}}$. Then the relative entropy of the three-dimensional WH SIC-POVM, whose fiducial vector $\mid {{\phi }_{1}}\rangle$ is an eigenvector of ${{U}_{(\mathcal{G},{\bf q})}}$ is maximized in the eigenstates of the Weyl matrix ${{D}_{{\bf s}}}$, where ${\bf s}\ne (0,0)$ satisfies $\mathcal{G}{\bf s}={\bf s}$.

Proof. Operators ${{U}_{(\mathcal{G},{\bf q})}}$ and ${{U}_{\mathcal{Z}}}$ are conjugated if and only if there exists $(\mathcal{F},{\bf r})\in ESL(2,{{\mathbb{Z}}_{3}})\ltimes {{({{Z}_{3}})}^{2}}$ such that $\mathcal{G}=\mathcal{F}\mathcal{Z}{{\mathcal{F}}^{-1}}$ and ${\bf q}=(\mathcal{I}-\mathcal{G}){\bf r}$. Now, if ${\bf p}$ is a non-zero fixed point of $\mathcal{Z}$ (thus ${\bf p}=(1,2)$ or ${\bf p}=(2,1)$), then ${\bf s}=\mathcal{F}{\bf p}$ is a non-zero fixed point of $\mathcal{G}$. Let us observe that ${{D}_{{\bf s}}}$ (and so $D_{{\bf s}}^{2}={{D}_{2{\bf s}}}$) commutes with ${{U}_{(\mathcal{G},{\bf q})}}$:

$$\begin{eqnarray*}&&{{U}_{(\mathcal{G},{\bf q})}}{{D}_{{\bf s}}}={{\omega }^{\langle {\bf q},\mathcal{G}{\bf s}\rangle }}{{D}_{{\bf s}}}{{U}_{(\mathcal{G},{\bf q})}}={{D}_{{\bf s}}}{{U}_{(\mathcal{G},{\bf q})}},\end{eqnarray*}$$

since $\langle {\bf q},\mathcal{G}{\bf s}\rangle =\langle {\bf r}-\mathcal{G}{\bf r},\mathcal{G}{\bf s}\rangle =\langle {\bf r},{\bf s}\rangle -\langle \mathcal{G}{\bf r},\mathcal{G}{\bf s}\rangle =\langle {\bf r},{\bf s}\rangle -\langle {\bf r},{\bf s}\rangle =0$. We consider the set S consisting of the vectors $\mid {{\phi }_{1}}\rangle$, ${{D}_{{\bf s}}}\mid {{\phi }_{1}}\rangle$ and ${{D}_{2{\bf s}}}\mid {{\phi }_{1}}\rangle$. By commutativity they all belong to the same eigenspace of ${{U}_{(\mathcal{G},{\bf q})}}$. Since ${{U}_{\mathcal{Z}}}$ has two eigenspaces (one-dimensional and two-dimensional), so has ${{U}_{(\mathcal{G},{\bf q})}}$, and we will refer to them as ${{\mathcal{H}}_{1}}$ and ${{\mathcal{H}}_{2}}$. Thus, vectors from S must be linearly dependent, and since they are not co-linear, they span ${{\mathcal{H}}_{2}}$. Let us take any $\mid \psi \rangle \in {{\mathcal{H}}_{1}}$. It is obviously orthogonal to the above vectors, and so from theorem 1 we know that it maximizes the relative entropy of Π. There exists a common eigenbasis for ${{D}_{{\bf s}}}$ and ${{U}_{(\mathcal{G},{\bf q})}}$, thus $\mid \psi \rangle$ is also an eigenvector of ${{D}_{{\bf s}}}$. Since the orbit under the action of the WH group of an eigenvector of any operator ${{D}_{{\bf p}}}$ is an eigenbasis of this operator [6, theorem 2.2], the theorem is proven. □

It is worth noting that the above theorems are not equivalent. Let us consider a family of SIC-POVMs parametrized by $t\in [0,\pi /3]$ and generated by the following vectors: $(0,1,-{{{\rm e}}^{{\rm i}t}}{{\eta }^{j}})$, $(-{{{\rm e}}^{{\rm i}t}}{{\eta }^{j}},0,1)$, $(1,-{{{\rm e}}^{{\rm i}t}}{{\eta }^{j}},0)$, $j=0,1,2$, where $\eta :={{{\rm e}}^{2\pi {\rm i}/3}}$. For every $t\in [0,\pi /3]$ the fiducial vector $\mid {{\phi }_{1}}\rangle \;:=\;(0,1,-{{{\rm e}}^{{\rm i}t}})$ is an eigenvector of unitary ${{U}_{(\mathcal{G},0)}}$ for $\mathcal{G}=\left( \begin{array}{ccccccccccccccc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$. Thus $\mid {{\phi }_{1}}\rangle ,{{D}_{{\bf s}}}\mid {{\phi }_{1}}\rangle$ and $D_{{\bf s}}^{2}\mid {{\phi }_{1}}\rangle$,where ${\bf s}:=(0,1)$ is a fixed point of $\mathcal{G}$, are linearly dependent and the maximum relative entropy is attained in the eigenstates of operator ${{D}_{{\bf s}}}$. In consequence, a WH-covariant maximally informative ensemble consists of the eigenbasis of ${{D}_{{\bf s}}}$. Nevertheless, there are two special cases: t = 0 and $t=2\pi /9$, described in detail in [7, section 3]. In the former, $\mid {{\phi }_{1}}\rangle$ is an eigenvector of every symplectic canonical order 3 unitary (i.e. one of the form ${{U}_{(\mathcal{F},0)}}$), so the maximum relative entropy is attained at the eigenstates of any Weyl matrix. Thus, the maximizers form a set of four mutually unbiased bases (MUBs) [6, theorem 2.3] and there are four WH-covariant maximally informative ensembles, each consisting of different eigenbases of Weyl matrices. In the latter case, we get additional linear dependencies that do not arise from the eigenspace of any canonical order 3 unitary; e.g. between vectors $\mid {{\phi }_{1}}\rangle ,{{D}_{(1,2)}}\mid {{\phi }_{1}}\rangle$ and ${{D}_{(2,0)}}\mid {{\phi }_{1}}\rangle$; see also [7]. It turns out that the orbit under the action of the WH group of the vector orthogonal to this additional subspace consists of three MUBs, and together with the maximizers described in theorem 2 form a set of four MUBs. Therefore there are two WH-covariant maximally informative ensembles: one consisting of an eigenbasis of ${{D}_{{\bf s}}}$ and the second one consisting of the three MUBs indicated above.

The natural question is whether the analogs of the results presented above hold in higher dimensions. Dang et al [7, theorems 1–3] have shown that some linear dependencies may arise in any dimension d, although if d is not divisible by 3 we may not find a fiducial vector in a proper eigenspace of a canonical order 3 unitary. Some numerical calculations in dimensions 4 to 6 show that one can find a vector of higher relative entropy than the ones predicted by the potential analogs of theorems presented in this paper. However, it is still possible that the global maximizers of the relative entropy of WH SIC-POVMs possess some special algebraic properties that may be revealed during further study.

Finally, let us take a look at a SIC-POVM in dimension 2 (vertices of the tetrahedron in the Bloch representation). It is known [4, 5, 18, 26] that in this case the maxima of the relative entropy are located in the states constituting the dual SIC-POVM. From the algebraic point of view, all these eight states lie on the single orbit under the action of the extended Clifford group ${\rm EC}(2)$; in consequence, they have the same stabilizers (up to conjugacy). Moreover, these stabilizers are maximal [26]. Note that in dimension 3 the maximizers are not located on the orbit of SIC-POVM, but they have larger stabilizers in the extended Clifford group ${\rm EC}(3)$ than the elements of SIC-POVM, with the only exception appearing when the fiducial vector is an eigenvector of all symplectic canonical order 3 unitaries (case t = 0 in the example presented at the end of the previous section).

The author is grateful to Wojciech Słomczyński for helpful discussions and valuable suggestions for improving this paper. This research was supported by Grant No. N N202 090239 of the Polish Ministry of Science and Higher Education.