Renormalized Von Neumann entropy with application to entanglement in genuine infinite dimensional systems

One of the basic, genuine quantum resources—that existing quantum information processing technology intensively exploits—is so-called quantum correlations [1]. For an exhaustive review of the present-day state of quantum hardware technology see, i.e. [2]. The interesting point here is that so-called continuous quantum systems (ions, atoms, lasers, ...) are becoming a very promising candidates for being a basic quantum ingredients of the future, full-scale quantum computers. This implies, that the better control on the quantum correlations in such systems may be crucial for developing these technologies. In particular, an appropriate qualitative and quantitative measures of quantum correlations have to be prepared. As it is well known, the phenomenon of quantum entanglement plays a crucial role for performing successfully several quantum protocols like teleportation, QKD protocols and many more [3, 4].

Several, qualitative and quantitative entanglement measures (obeying set of reasonable and natural—from the mathematical and information theory point of view—demands) contained in quantum states are being proposed [5, 6]. Many of them are based on the use and properties of the quantum von Neumann entropy. However, there does not exist a general straightforward passage with these mathematical formalism from the case of finite dimensional systems of qudits to the genuine, infinite-dimensional systems like ions, atoms, etc. It is the main purpose of the present paper to propose, how it is possible to fill up this gap. Majority of quantum states describing bipartite (and many partite systems as well) and infinite dimensional systems is characterized by the fact that the von Neumann entropy (and therefore, the corresponding entropy based on tangles measures for many partite systems) of the corresponding conditional quantum states (reduced density matrices) is taking infinite value [7, 8]. With the use of Fredholm determinants technique, it is possible to remove the arising infinities and thus, it is possible to extend several results known for the finite dimensional systems to the genuine, infinite dimensional continuous systems. It is the great Author’s hope that the presented here mathematical technique will find, besides those included here, many other applications in the field of Quantum Information Theory.

Let us consider the model of two spinless quantum particles interacting with each other and placed in three dimensional Euclidean space \(\mathbb {R}^3\). Generally, the states of such quantum systems are described by the density matrices which are non-negative, of trace class operators acting on the space \( {\mathcal {H}}=L_2 ( \mathbb {R}^3 ) \otimes L_2 ( \mathbb {R}^3 )\), see [9, 10]. The latter is, in fact, unitary equivalent to \(L_2 ( \mathbb {R}^6 )\). In particular, any pure state can be represented (up to the global phase calibration) by the corresponding wave function \(\psi (x,y)\in {\mathcal {H}}\); then the density matrix takes the form of the projector onto the ket vector \(| \psi \rangle \).

Using Schmidt decomposition theorem, cf[11, Thm. 26.8], we conclude that for any pure normalized state \(\psi \in {\mathcal {H}}\) there exist: a sequence of non-negative numbers \(\{\lambda _n\}_{n=1}^\infty \) (called the Schmidt coefficients of \(\psi \)) satisfying the condition \(\sum _{n=1}^\infty \lambda _n^2 =1\) and two complete orthonormal systems of vectors \(\{\varphi _n \}_{n=1}^\infty \), \(\{\omega _n \}_{n=1}^\infty \) in \(L_2 ( \mathbb {R}^3 )\) such that the following equality (in the \(L_2\) space sense):has to be satisfied.

In particular, we call the vector \(\psi \) a separable pure state iff there appears only one nonzero Schmidt coefficient in the decomposition (1.1). If the number of nonzero Schmidt coefficients is finite than we say that \(\psi \) is of finite Schmidt rank pure state. In this case, one can apply the standard and the most frequently used measure of amount of entanglement included in the state \(\psi \) which is given by the von Neumann formula:Although, the set of finite Schmidt rank pure states of the system under consideration is dense (in the \(L_2\)-topology) on the corresponding Bloch sphere (this time infinite-dimensional and given here modulo global phase calibration for simplification of the following discussion only) denoted as \(B= \{\psi \in L_2(\mathbb {R}^6)\,:\, \Vert \psi \Vert =1\}\), it appears that also the set of infinite Schmidt rank pure states is dense there. The situation is even more complicated as it can be shown that the set of pure states for which the value of von Neumann entropy is finite is dense in B but also the set of states with infinite entropy of entanglement is dense in this Bloch sphere [7].

Similar results on densities of the infinite/finite Schmidt rank states are also valid in the proper physical \(L_1\)-topologies on the corresponding Bloch sphere. Very roughly, the reason is that in infinite dimensions there are many (too many in fact) sequences \(\{\lambda _n\)}) such that: for all n, \(\lambda _n\ge 0\) and \(\sum _{n=1}^\infty \lambda _n^2 =1\) but \( \sum _{n=1}^\infty \lambda _n ^2 \log (\lambda _n ^2 )=-\infty \). In other words, the set of pure states for which the entropy is finite has no internal points and this fact causes serious problems in the fundamental question on continuity of the von Neumann entropy in genuine infinite dimensional setting [7, 8]. In finite dimensions the von Neumann entropy is a non-negative, concave, lower semi-continuous and also norm continuous function defined on the set of all quantum states. A lot of fundamental results on several quantum versions of entropy, in particular, on von Neumann entropy have been obtained in the last decades, cf[12‐20]. However, in the infinite dimensional setting, the conventionally defined von Neumann entropy is taking the value \(+\infty \) on a dense subset of the space of quantum states of the system under consideration cf[7, 8, 11, 13, 21‐26].

Nevertheless, defined in the standard way von Neumann entropy has continuous and bounded restrictions to some special (selected by some physically motivated arguments) subsets of quantum states. For example, the set of states of the system of quantum oscillators with bounded mean energy forms a set of states with finite entropy [7, 8, 27, 28]. Since, the continuity of the entropy is a very desirable property in the analysis of quantum systems, various, sufficient for continuity, conditions have been obtained up to now. The earliest one, among them, seems to be Simon’s dominated convergence theorems presented in [15‐17] and widely used in applications, see [12‐14]. Another useful continuity condition originally appeared in [7, 8] and can be formulated as the continuity of the entropy on each subset of states characterized by bounded mean value of a given positive unbounded operator with discrete spectrum, provided that its sequence of eigenvalues has a sufficient large rate of decrease. Some special conditions yielding the continuity of the von Neumann entropy are formulated in the series of papers by Shirokov [21‐26]. A stronger version of the stability property of the set of quantum states naturally called there as strong stability was introduced by Shirokov together with some applications concerning the problem of approximations of concave (convex) functions on the set of quantum states and a new approach to the analysis of continuity of such functions has been presented there. Several other attempts and ideas to deal with the noncommutative, infinite dimensional setting were published in the current literature also. Some of them are based, on a very sophisticated, tools and methods, such as, for example theory of noncommutative (versions of) the (noncommutative) log-Sobolev spaces of operators [29].

The main idea of the present paper is to introduce an appropriate renormalized version of the widely known von Neumann formula for the entropy in the non-commutative setting [12‐14]. The notion of von Neumann entropy is one of the basic concept introduced and applied in quantum physics. However formula proposed by von Neumann works perfectly well only in the context of finite dimensional quantum systems [7, 8]. The extension to the genuine infinite-dimensional setting is not straightforward and meets several serious obstacles as mentioned in the previous sentences. Our prescription for extracting finite part of the infinite valued (which is true typically in the sense of Baire category theory) standard von Neumann formula is very simple. For this goal, let Q be a quantum state, i.e. Q is non-negative, of trace class operator defined on some separable Hilbert space \({\mathcal {H}}\) and such that \(\text {Tr}(Q) = 1\). The standard definition of von Neumann entropy \(\textrm{EN}\) is given as:Our renormalization proposal, denoted as FEN, is given by:where \(1_{\mathcal {H}}\) stands for the unit operator in \({\mathcal {H}}\).

This means that the introduced mapis finite on the space \(E({\mathcal {H}})\) of the quantum states on \({\mathcal {H}}\). The detailed mathematical study of the basic properties of the introduced here renormalization of the von Neumann entropy is the main topic of this paper. Additionally, presentation of several applications of the introduced entropy FEN and addressed to the Quantum Information Theory [3, 4, 27, 28] are also included. To achieve all these goals, the theory of Fredholm determinants as given by Grothendick [32] is intensively used in the following presentation. Also certain results from the infinite dimensional majorisation theory [33‐38] have been used. A very preliminary and illustrative idea of von Neumann entropy renormalization was recently published by the Author in [39].

In the next Sect. 2, the technique of the Fredholm determinants is successfully applied to show that the proposed here renormalized version of von Neumann entropy formula in the genuine infinite-dimensional setting is finite and continuous (in the \(L_1\)-topology meaning) on the space of quantum states. Elements of the so-called multiplicative version of the standard majorization theory [3, 4, 33, 34, 40] are being introduced in Sect. 3. The main results reported there are: the rigorous proof of monotonicity of the introduced renormalization of von Neumann entropy under the semi-order relations (caused by the defined there multiplicative majorization) lifted to the space of quantum states. Additionally, an extension of the basic (in the present context) Alberti-Uhlmann theorem [33] is proved in Sect. 3. Also monotonicity of the introduced notion of renormalized von Neumann entropy under the action of a general quantum operations on quantum states is proved there. Section 4 is devoted to the study of two-partite quantum systems of infinite dimensions both (the case of one factor being finite dimensional is analysed in details see [41, 42]). In particular, the corresponding reduced density matrices are studied there and some useful formula and estimates of the corresponding renormalized entropies are included there. The particular case of pure bipartite states is analyzed from the point of view of majorization theory with the use of novel, local unitary and monotonous invariants perspective of Gram operators as introduced in another papers [43‐47]. The finite dimensional results of this type, presented in [43, 44, 46], are being extended to the infinite dimensional setting there with the use of Fredholm determinants theory [46]. At the end of this paper three appendices are attached to make this paper autonomous and also because some additional results which might be helpful in further developments of the ideas presented here are being formulated there. In appendix A, the Author have presented (after Grothendik [32], see also Simon [48]) crucial facts and estimates from the infinite dimensional Grassman algebra theory with the applications to control Fredholm determinants. Appendix B includes several results and formulas on the different types of combined Schmidt and spectral decompositions of a general bipartite quantum states. Finally in Appendix C, some useful remarks on the operator valued function \(\log (1 + Q)\) are collected.

Extensions of the approach to the renormalization of the von Neumann entropy presented in this paper to a very rich palette of intriguing questions, like for example renormalization of quantum relative entropy and quantum relative information notions [27, 28, 49‐55], possible applications to the renormalization of the quantum entropy in the context of general Quantum Field Theory (see i.e. the recent paper on this [56]) and also possible applications to the so called Continuous Variable Quantum Information Theory [57‐62] are also visible for the Author and some work on them is in progress.

Assume that \({\mathcal {H}}\) is a separable infinite dimensional Hilbert space.¹ In this paper, we use the following standard notation:The convention which is used in the present paper is that always spectrum of a Hermitian Q is ordered in an non-increasing order (this is possible to achieve by performing certain unitary operation on a given operator Q).

In further discussion, we will relay on the following inequalities, cf[30, 31],the latter inequality also holds for \(\Vert B A \Vert _1\) with obvious changes.

The following spaces of sequences will be used in further analysis

The most useful local invariants and local monotone quantities characterizing in the qualitative as well as quantitative way quantum correlations, as entanglement of states in the finite dimensional systems, are defined by means of the special versions of the entropy measures, cf. [3‐6, 63, 64]. The von Neumann quantum entropy measure is, without a doubt, the most common tool for these purposes.

Suppose that \(\underline{a} \in C^\infty \) and \(a_i \ne 0 \) for all i. Moreover, we assume that that the limit \( \lim _{n\rightarrow \infty }\prod _{i=1}^n a_i\) exists and it is nonzero. Then, we say that the product \(\prod _{i=1}^\infty a_i\) exists.

The continuity of \(x\mapsto \log x \) implies the following statement.

Let A be a compact operator in a separable Hilbert space \(\mathcal H\) and \(\underline{\sigma (A)}\) stands for the discrete eigenvalues of A counted with multiplicities and ordered into non-increasing sequence. On the other hand, let \(\underline{\lambda (A)}\) denote singular values of A counted with multiplicities and forming non-increasing sequence. If \(A\in L_1 ({\mathcal {H}} )\) then \(\lambda (A)= \sigma (|A^\dag A|^{1/2 })\) and \(\sum _{n=1}^\infty \lambda _n <\infty \). The Fredholm determinant takes the formWe remind the basic properties of the Fredholm determinants, cf. [32, 48] below.

In the further discussion, we will use the following quantity.

In order to relate the above definition with the results formulated in Theorem 1, we introduce the following entropy-generating operators \(S_{\pm }\).

Our definition (2.32) is the renormalized (due to the infinite dimension of the corresponding spaces) version of it is: “\((1+Q)^{-(1+Q)}-1\)”.

One of the main results reporting on this note is the following theorem.

The proof is based on the following sequence of Lemmas.

Let us define the scalar function

In order to prove that the renormalized entropy functions \(\textrm{FEN}_{\pm } \) are \(L_1\) continuous it is enough to prove that the operator valued maps \(S_{\pm } \) are \(L_1\) continuous. The latter is proved below.

The use of standard, not renormalized, definition of entropy of entanglement leads to the statement that it is taking values in interval \([0, \log (d)]\), which shows that there is no possible straightforward passage from the finite dimensional situation to the infinite dimensional systems. The widely used, another entropic measures of entanglement [3‐6] also must be suitable renormalized in order to be applied in infinite dimensions in a way that overcome the several discontinuity and divergences problems as well problems arising in the genuine infinite dimensional cases. The results on this will be presented in a separate note.

Let Q(n) be the sequence of \(L_1 ( {\mathcal {H}} )\) such that the n-th first eigenvalues of Q(n) is equal to 1/n and the rest of spectrum is equal to zero. The renormalized entropy of Q(n) is given byIt is easy to see that \(\lim _{n \rightarrow \infty } \textrm{FEN}_{+} (Q(n)) = 1\).

It would be interesting to describe in the explicit way the most mixed states i.e. the states for which the value of \(\textrm{FEN}_+ (Q) =1\).

Let us recall the finite dimensional formula for computing determinant of tensor product of matrices.

If one of the factors in (4.65) is infinite dimensional and the determinant (absolute value of) of the corresponding matrix Q is strictly bigger than one (or strictly smaller than one) then the value \(\textrm{det}\) of the product (4.65) is infinite, respectively equal to zero.

In order to understand better this problem, we define renormalized Kronecker productwhich formally can be written as:

The proof is an immediate consequence of the Theorem 1 (i) and Lemma (4.2) below.

Another renormalization of the tensor product can be achieved by the use of infinite dimensional Grassmann algebras as we have outlined in the Appendix A to this note. For this goal let us definewhere \(\wedge \) stands for skew (antisymmetric) tensor product and the right hand side here is defined as a one particle operator in the skew Grassmann algebras built on \({\mathcal {H}}_A\) and \({\mathcal {H}}_B\), see Appendix A. Using the unitary isomorphism map J in between the antisymmetric product of fermionic Fock spaces build on the spaces \({\mathcal {H}}_A\) and \({\mathcal {H}}_B\) (see Appendix A and the Theorem 27) and the antisymmetric Fock build on the space \({\mathcal {H}}_\oplus = {\mathcal {H}}_A \oplus {\mathcal {H}}_B\), we can define

Another interesting implication of Theorem 27 seems to be the following observation.

Also the following result seems to be interesting.

Let \({\mathcal {H}} = {\mathcal {H}} _A\otimes {\mathcal {H}}_B\) be the tensor product of two separable Hilbert spaces \({\mathcal {H}} _A\) and \({\mathcal {H}}_B \) of arbitrary dimensions. In this section, we assume that both spaces \({\mathcal {H}}_A\), \({\mathcal {H}}_B\) are infinite dimensional (everything works also in finite dimensional situations [41], and also in situation for which only one of the spaces \({\mathcal {H}}_i\) is finite dimensional as well [41]).

Let \(Q^A\) (respectively \(Q^B\)) be the corresponding reduced density matrices obtained from Q by tracing out the corresponding degrees of freedom. Then \(Q^A \ge 0\), \(\textrm{Tr}_{{\mathcal {H}}_A } [Q^A]=1\), and identically in the case of \(Q^B\). As is well known the spectrum \(\sigma (Q^A) =( \lambda _n )\) is purely discrete (we are presenting it always with the corresponding multiplicities and in nonincreasing order) and in general different from the spectrum of \(Q^B\) in the case of mixed states. For more on this see below and the Appendix B. In the case when, as in the introduction, \(Q=|\Psi \rangle \langle \Psi |\) for some \(\Psi \in {\mathcal {H}}\) the spectrum of \(Q^A\) and \(Q^B\) are equal to each other and equal to the list of squared Schmidt coefficients of the corresponding Schmidt decomposition of the vector \(\Psi \) [3, 4, 69]. The same is valid for the Hilbert–Schmidt level reduced density matrices when we consider these type of Schmidt decompositions of a given \(Q \in {\mathcal {H}}\), see Appendix B and [44, 46].

Let us recall now some well-known facts on the reduced density matrices. Let \(Q \in E({\mathcal {H}} )\). Let \(\{ |i\rangle \}\) be an arbitrary complete orthonormal system of vectors in \({\mathcal {H}}_B\). Then, we have canonical unitary equivalencewhere \(\cong \) means that \(\varphi \in {\mathcal {H}}\) is decomposed as \(|\varphi \rangle \cong \oplus _i ( \textrm{I}_{{\mathcal {H}}_A} \otimes | i\rangle \langle i |) |\varphi \rangle \).

Then for any \(A \in B({\mathcal {H}})\), we can write:whereis the bounded linear map from \({\mathcal {H}}_A \otimes | i \rangle \) to \({\mathcal {H}}_A \otimes | j \rangle \).

Using the Krauss decomposition Theorem 3.59, we have the following observation: the linear and bounded mapnamed partial trace map is a quantum operation in the sense of the previously introduced definition in Sect. 3.

Let \({\mathcal {H}}= {\mathcal {H}}_A\otimes {\mathcal {H}}_B\) be a bipartite separable Hilbert space and let \(Q \in E({\mathcal {H}} )\). It is well known that the spectrum of Q counted with multiplicities, denoted \(\sigma (Q) =(\lambda _1,...)\) is purely discrete and the following spectral decomposition holds:where the orthogonal (and normalized) system of eigenfunctions \(|\Psi _n \rangle \) of Q forms a complete system. Each eigenfunction \(|\Psi _n \rangle \) can be expanded further by the use of the Schmidt decomposition:where \(\tau _i^n \ge 0\) \(\sum _{n=1}^\infty (\tau _i^n)^2 =1 \) and the systems \( \{\psi ^n_i \}\) and \( \{\phi ^n_i \}\) form the complete orthonormal systems in \({\mathcal {H}} _A\) and, respectively, \({\mathcal {H}} _B\). Using (4.91) and (4.90), we can compute the corresponding reduced density matriceswhere the operatorsare the states on \({\mathcal {H}}_B\). Similarly, for the reduced density matrix connected to the observer localized with \({\mathcal {H}}_A\):where \(Q_n^A=\sum _{i=1}^\infty |\tau _i^n|^2 | \psi ^n_i \rangle \langle \psi ^n_i|\) are states on \({\mathcal {H}}_A\). The obtained systems of operators \(\{ Q_n^A\}\) and \(\{ Q_n^B\}\) consist of bounded non-negative self-adjoint, local operators of class \(L_1({\mathcal {H}}_A)\), respectively of class \(L_1({\mathcal {H}}_B )\) and therefore they are locally measurable. In particular the squares of the Schmidt coefficients \(\tau _i^n\) of the Schmidt decompositions of the eigenfunctions of the parent state Q are observable (measurable locally) quantities.

Let \({\mathcal {H}}= {\mathcal {H}}_A \otimes {\mathcal {H}}_B\) be a bipartite, separable Hilbert space and let \(Q \in E({\mathcal {H}})\) be such that tr\((Q^2)=1\). Then, there exists an unique, normalized vector \(|\Psi \rangle \in {\mathcal {H}}\) such that \(Q=|\Psi \rangle \langle \Psi |\).

Let \(\{e_i^A, i=1, \dots \}\), resp. \(\{e_j^B, j=1, \dots \}\) be some complete orthonormal systems in \({\mathcal {H}}_A\), resp. in \({\mathcal {H}}_B\).

Then, we can write:where \(\Psi _{ij}=\langle e_j^B \otimes e_i^A|\Psi \rangle \).

We start with the Schmidt decomposition (essentially SVD decomposition, see i.e. Thm. 26.8 in [11]) in the infinite dimensional setting.

The decomposition 4.100 is called the Schmidt decomposition of \(| \Psi \rangle \). The expansion formula 4.100 can be rewritten as:whereand alsowhereLet us define pair of linear maps \(J^A: {\mathcal {H}}_A \rightarrow {\mathcal {H}}_B\), resp. \(J^B: {\mathcal {H}}_B \rightarrow {\mathcal {H}}_A\) by the followingand then extended by linearity and continuity to the whole \({\mathcal {H}}_A\). In an identical way the map \(J^B\) is defined. Both of the introduced operators J are bounded as can be seen by simple arguments. Now, we define a pair of operators which plays an important role in the followingand similarlySome elementary properties of the introduced operators \(\Delta ^A\) and \(\Delta ^B\) are collected in the following proposition.

The interesting observation is that the explicite Gram matrix nature (it is well known fact [66] that any (semi)-positive matrix has a Gram matrix structure) of the operators \(\Delta \) can be flashed on.

In the finite dimensional case the following, nice geometrical picture is known [43]. Let \(\{v_i, i=1,..d\}\) be a system of linearly independent vectors in the space \(C^d\), where \(d=d\). Let us build on these vectors a d dimensional parallelepiped. Then, the Euclidean volume of this parallelepiped is equal to the determinant of the Gram matrix built on these vectors. The matrix elements of this Gram matrix are given by the scalar products \(\langle v_i|v_j \rangle \) for \(i,j=1:d\). Under the condition that the sum of the lengths of the spanning vectors \(v_i\) is equal to 1 the parallelepiped which has the maximal volume is that which is spanned by the system of orthogonal vectors of equal length. In this particular case, the corresponding Gram matrix elements are equal to \((1/d)\delta _{ij}\). In a general case, the volume of the parallelepiped spanned by the vectors forming some square matrix columns (or rows) can be estimated from above be several inequalities. The Hadamard inequality saying that this volume is no bigger than the product of the lengths of the spanning vectors \(v_i\) is the best known among them. For more on this see [43].

On the basis of results and facts presented in previous sections, we can define the following quantity (in fact entire function of z) that will be called gramian function of the state \(|\Psi \rangle \).In particular case when \(z=1\) the value of the gramian function G of state \(|\Psi \rangle \) will be called the gramian volume of \(|\Psi \rangle \) and denoted as \(G(\Psi )\). The logarithm of the gramian volume will be called the logarithmic (gramian) volume of \(|\Psi \rangle \) and denoted as \(g( \Psi )\). Using (4.111), it follows that

We can see that the Gramian volume of pure states is locally invariant (under the local unitary operations action) quantity. And what is also important, we have proved that the gramian volume defined in (4.111) do not increase under the action of any separable quantum operation. This is why we think that the Gramian volume might be a very good candidate for the entanglement measure included in pure quantum states.

Sometimes it is more useful to use logarithmic Gramian volume g instead of the Gramian volume G. Some basic properties of the logarithmic volume g are contained in the following Theorem.

Similar results are true for the renormalized von Neumann entropies. For this goal, let us recall the definitions of entropies generating operators:From which we obtain an estimate

Finally, we mention the monotonicity of the renormalised Entropy with respect to the by majorization relation introduced semi-order. Details will be presented elsewhere [46].