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

A renormalized version of the von Neumann quantum entropy (which is finite and continuous in general, infinite dimensional case) which obeys several of the natural physical demands (as expected for a “good” measure of entanglement in the case of general quantum states describing bipartite and infinite-dimensional systems) is proposed. The renormalized quantum entropy is defined by the explicit use of the Fredholm determinants theory. To prove the main results on continuity and finiteness of the introduced renormalization, the fundamental Grothendick approach, which is based on the infinite dimensional Grassmann algebra theory, is applied. Several features of majorization theory are preserved under the introduced renormalization as it is proved in this paper. This fact allows to extend most of the known (mainly, in the context of two-partite, finite-dimensional quantum systems) results of the LOCC comparison theory to the case of genuine infinite-dimensional, two-partite quantum systems.


Introduction
Let us consider the model of two spinless quantum particles interacting with each other and living in three dimensional Euclidean space R 3 .Generally, the states of such quantum systems are described by the density matrices which are nonnegative, of trace class operators acting on the space H = L 2 (R 3 ) ⊗ L 2 (R 3 ), see [3,4].The latter is, in fact, unitary equivalent to L 2 (R 6 ).In particular, any pure state can be represented (up to the global phase calibration) by the corresponding wave function ψ(x, y) ∈ H; then the density matrix takes the form of the projector onto the ket vector |ψ .
Using Schmidt decomposition theorem, cf [14,Thm. 26.8], we conclude that for any pure normalized state ψ ∈ H there exist: a sequence of nonnegative numbers {λ n } ∞ n=1 (called the Schmidt coefficients of ψ) satisfying the condition ∞ n=1 λ 2 n = 1 and two complete orthonormal systems of vectors ) such that the following equality (in the L 2space sense) holds.
In particular, we call the vector ψ a separable pure state iff there appears only one non-zero Schmidt coefficient in the decomposition (1.1).If the number of non-zero Schmidt coefficients is finite than we say that ψ 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 ψ 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 = {ψ ∈ L 2 (R 6 ) : ψ = 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 [2].
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 {λ n }) such that: for all n, λ n ≥ 0 and 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 [2,54].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 [6,7,8,9,10,11,12,25,44].However, in the infinite dimensional setting, the conventionally defined von Neumann entropy is taking the value +∞ on a dense subset of the space of quantum states of the system under consideration cf [2,7,14,15,16,17,18,19,20,54].
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 [2,54,65,66].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 [9,10,11] and widely used in applications, see [6,7,8].Another useful continuity condition originally appeared in [2,54] 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 [15,16,17,18,19,20].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 1.1 The main idea of the paper 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 [69].

The main idea of the paper
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 [6,7,8].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 [2,54].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 (otherwise typically in the sense of Baire category theory) infinite valued 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 H and such that Tr(Q) = 1.The standard definition of von Neumann entropy EN is given as : Our renormalisation proposal, denoted as FEN, is given by: where 1 H stands for the unit operator in H.
Claim 1.1 For any such Q the value FEN(Q) is finite .
Using the elementary inequality together with functional calculus [5,13,14] we have the following estimate (1.6)

Organisation of the paper
This means that the introduced map is finite on the space E(H) of the quantum states on H.The detailed mathematical study of the basic properties of the introduced here renormalizations of von Neumann entropy is the main topic of the present paper.Additionally presentations of several applications of the introduced entropy FEN and adressed to the Quantum Information Theory [22,23,65,66] are included also.To achieve all these goals the theory of Fredholm determinants as given by Grothendick [1] is intensively used in the following below presentation.Also certain results from the infinite dimensional majorisation theory theory [26,27,28,29,67] have been used.Some, a very preeliminary and illustrative idea of von Neumann entropy renormalization was recently published by the Author in [45].

Organisation of the paper
In the next Section 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 continuos (in the L 1 -topology meaning) on the space of quantum states.Elements of the so-called multiplicative version of the standard majorization theory [22,23,26,27,39] is being introduced in Section 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 [26] is proved in Section 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 [46,70]).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 analysed 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 [32,47,50,52,68].The finite dimensional results of this type, presented in [32,47,52] are being extended to the infinite dimensional setting there with the use of Fredholm determinants theory [52].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 [1], see also B.Simon [21]) 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 renormalisation of the von Neumann entropy presented in this paper to a very rich palette of intriguing questions, like for example renormalisation of quantum relative entropy and quantum relative information notions [35,36,37,38,40,41,49,65,66] are also visible for the Author and some work on them is in progress.

Renormalized version of the von Neumann entropy 2.1 Some mathematical notation
Assume that H is a separable infinite dimensional Hilbert space 1 .In this paper we use the following standard notation: • L 1 (H) stands for the Banach space of trace class operators acting on H and equipped with the norm and the symbol † means the hermitian conjugation, • L 2 (H) denotes the Hilbert-Schmidt class of operators acting in H and with the scalar product • B(H) denotes the space of the bounded operators with norm defined as the operator norm • , • Let E(H) be the complete metric space of quantum states Q on the space H, i.e. the L 1 -completed intersection of the cone of nonnegative, trace class operators on H and L 1 -closed hyperplane described by the normalisation condition Tr[Q] = 1.
In further discussion we will relay on the following inequalities, cf [5,13], ) (2.9) the latter inequality also holds for BA 1 with obvious changes.
The following spaces of sequences will be used in further analysis )

The renormalized von Neumann entropy
The most useful local invariants and local monotone quantities characterising 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.[22,23,24,33,48,55].The von Neumann quantum entropy measure is, without a doubt, the most common tool for these purposes.Suppose that a ∈ C ∞ and a i = 0 for all i.Moreover we assume that that the limit lim n→∞ n i=1 a i exists and it is nonzero.Then we say that the product ∞ i=1 a i exists.The continuity of x → log x implies the following statement.

The renormalized von Neumann entropy
Proof.The claim follows directly from (2.14) Let A be a compact operator in a separable Hilbert space H and σ(A) stands for the discrete eigenvalues of A counted with multiplicities and ordered into non-increasing sequence.On the other hand, let λ(A) denote singular values of A counted with multiplicities and forming non-increasing sequence. (2.15) Below we remind the basic properties of the Fredholm determinants, cf.[1,21].21]] Let H be a separable Hilbert space.Then extends to the entire function which obeys the bound in particular det is the Lipschitz continuous.
iii) The following three equivalences hold: and where ∧ n (∆) stands for the antisymmetric tensor power of ∆, see Appendix A for more details, and Remarks 2.4 The last equivalence Eq. 2.21 determines so called Pelmelj expansion with |z| < 1.For larger values of |z| the analytic continuations are necessary to be performed.
In the Appendix A we outline the methods of infinite dimensional Grassmann algebras (the Fermionic Fock spaces in the physical notations) as introduced in the fundamental Grothendick memoir [1].
In the further discussion we will use the following quantity.
We define where This means that In order to relate the above definition with the results formulated in Thm.2.3 we introduce the following entropy-generating operators S ± .

The renormalized von Neumann entropy
where I means the unit operator here in the space H and spectral functional calculus has been used.
Remark 2.7 In the standard, finite dimensional situation [6,7,8], the corresponding entropic operator S − (Q), formally looks like (informally) as Our definition (2.25) is the renormalized (due to the infinite dimension of the corresponding spaces) version of it is : One of the main results reporting on in this note is the following theorem.
Theorem 2.8 For any Q ∈ E(H), FEN ± (Q) are finite and, moreover, The proof is based on the following sequence of Lemmas.
Let us define the scalar function Lemma 2.9 i) The function f + (x) is monotonously increasing and convex on [0, 1] and (2.28) ii) The function f − (x) is monotonously decreasing and concave on [0, 1],

The renormalized von Neumann entropy
Proof.For 0 ≤ x ≤ 1 the following estimate is valid (2.30) From where we used Th.2.3 and Lemma 2.1.
Proof.For 0 ≤ x ≤ 1 the following estimate is valid (2.33) where we have used Th.2.3 and Lemma 2.1.In order to prove that the renormalized entropy functions FEN ± are L 1 continuous it is enough to prove that the operator valued maps S ± are L 1 continuous.The latter is proved below.Lemma 2.12 Let H be a separable Hilbert space and let E(H) be a space of quantum states on H. Then the maps are L 1 continuous on E(H).
Proof.It is enough to present essential details of the proof for the case S + (Q).Let Q and Q ′ be the states on H.By the application of the Duhamel formula and equations (2.8) and (2.9) we get To complete the proof it suffices to prove the norm continuity of the operator valued function log Using again formulae (2.8) and (2.9) we have The analysis of further properties, together with the analysis of the case τ = 1 of the map Q → log(I + Q) we postpone to Appendix C.

Proposition 2.13
Let H be a separable Hilbert space and let E(H) be the space of quantum states on H. Then the L ∞ norms (spectral norm) of the entropy maps S ± (Q) = (I + Q) ±(1+Q) − I are given by: , where ) which tends monotonously, as d tends to infinity to the value 1 (resp.to the value -1).
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 [22,23,24,48] 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 (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 by Theorem 2. 15 For any sequence of states Proof.For any such sequence Q(n) we apply the Banach-Alaglou theorem first, concluding that the set {Q(n)} forms * -week precompact set and therefore, in the * -week topology lim Q(n) by subsequences do exists.However, these limits are all equal to zero.In order to obtain a non-trivial result we use the Banach-Saks theorem which tells us that there exists a subsequence n j such that the following Cesaro sum of It would be interesting to describe in the explicit way the most mixed states i.e. the states for which the value of FEN + (Q) = 1.

Some remarks on the majorization theory
The fundamental results obtained in Alberti and Uhlmann monograph [26] and applied so fruitfully to the quantum information theory by many researchers (see [22,23,26,27,30,42] and references therein, are known widely today under the name (S)LOCC majorization theory (in the context of quantum information theory).Presently this theory is pretty good understood in the context of bipartite, finite dimensional systems, (especially in the context of pure states), see [22,23,27].In papers [28,29,30,31,67] successful attempts are presented in order to extend this theory to the case of bipartite ,infinite dimensional systems.Below, we present some remarks which seems to be useful in this context.
For a given a ∈ C ∞ we apply the operation of ordering in non-increasing order and denote the result as a ≥ .Of particular interest will be the image of this operation, when applied pointwise to the infinite dimensional simplex . This will be denoted as C ≥ .Let us recall some standard definitions of majorisation theory.Let a, b ∈ C ≥ .Then we will say that b is majorising a iff for any n the following is satisfied If this holds then we denote this as a b.
We will say that b is majorizes multiplicatively a iff for any n the following is satisfied If this holds to be true then we denote this fact as a m-b.
Let F be any function (continuous, but not necessarily) on the interval [0, 1].The action of F on C ∞ + (and other spaces of sequences that do appear) will be defined (F (a i )).
Recall the well known result, see i.e. [5,6].The last result says that each linear chain of the semi-order relation min C ≥ is contained in some linear chain of the semi-order .It means that the semi-order m-is finer than those induced by .
Corollary 3.4 Any -maximal element in C ≥ is also m--maximal.
Proof.If a m-b then a b.Let a * be a -maximal in C ≥ and let us assume that there exists a * * such that a * m-a * * and the contradiction is present.
To complete this subsection we quote the infinite dimensional extension of the majorisation theory applications in the context of quantum information theory.
For this goal let us consider any Q ∈ E(H), where H is a separable Hilbert space.With any such Q we connect a sequence (P sp (N)) of finite dimensional projections P sp (Q) which we will call the spectral sequence of Q.This is defined in the following way: let Q = ∞ n=1 τ n E φn be the spectral decomposition of Q rewritten in such a way that eigenvalues τ n of Q are written in nonincreasing order.Then we define P sp (Q)(n) = ⊕ n i=1 E φn .Finally, we define a sequence of Gram numbers g n (Q) connected to Q: (3.45) for all n.This will be written as The standard definition of majorisation is the following: Proposition 3.6 Let H be separable Hilbert space and let Proof.The point ( 1) and ( 2) follows from the fact that majorisation in the sense of Definition 3.5 is equivalent to the m-majorisation of the considered entropy generating operators from which follows, using Corollary 3.3, that they are also in the standard majorisation relation.
To prove (3) and (4) let us introduce the following interpolation: if It is easy to see that assuming as function of t is smooth.Calculating the second derivative of its we find This completes the proof.Before we present (after [26,28,29] and with minor modifications) infinite dimensional generalisation of the fundamental in this context Alberti-Uhlmann theorem we briefly recall some definitions.
A completely positive map Φ on a von Neumann algebra L ∞ (H) is said to be normal if Φ is continuous with respect to the ultraweak ( * -weak) topology.Normal completely positive contractive maps on B(H) are characterized by the theorem of Kraus which says that Φ is a normal completely positive map if and only if there exists at least one sequence (A i ) i=1,... of bounded operators in L ∞ (H) such that for any Q ∈ L ∞ (H): where and where the limits are defined in the strong operator topology.A normal completely positive map Φ which is trace preserving is called a quantum channel.If a normal completely positive map Φ satisfies Φ(I H ) ≤ I H then called a quantum operation.A quantum operation Φ is called unital iff Φ(I H ) = I H which is equivalent to ∞ i=1 A i A † i = I H for some Krauss decomposition of Φ.
A quantum operation Φ is called bistochastic operation if it is both trace preserving and unital.Central notion for us is the notion of a mixed unitary operation.
A quantum operation Φ is called a (finite) mixed unitary operation iff there exists a (finite) ensemble {U i } i=1:n of unitary operators on H and a (finite) sequence p i ∈ [0, 1] such that n i=1 p 1 = 1 and Theorem 3.7 Let H be a separable Hilbert space and let Then there exists a sequence (Φ n ) of mixed unitary operations and a limiting bi-stochastic operation Φ * such that the sequence of states Proof.The only essential difference comparing to the original formulation of this result [26,28,29] is that instead of type majorisation m-is used.Also the following result holds Theorem 3.8 Let H be a separable Hilbert space and let Φ be any quantum operation acting on E(H).Then Proof.Let T be any non-expansive linear operator acting on H -this means that the operator norm of T , T ≤ 1.Using the Grothendick formula (A.7) and the following reasoning: where we have used the assumption that the norm T of is not bigger then 1 and positivity of Q.
Now, let us assume that we have a pair of bounded operators T 1 , T 2 and such that (3.57) Now, the general case follows by application of Krauss representation theorem for quantum operations (3.52) and some elementary inductive and continuity arguments.
Several additional results on renormalised version of von Neumann entropy, in particular on the invariance and monotonicity properties of von Neumann entropy in the infinite dimensional setting of conditional entropies are included in [49].4 The case of tensor product of states

Renormalized Kronecker products
Let us recall the finite dimensional formula for computing determinant of tensor product of matrices.Lemma 4.1 (Kronecker formula) Let H A and H B be a pair of finite dimensional Hilbert spaces with dimension N A , and resp.N B .Then, for any (4.58) Proof.(quick-argument based).Let stands I A , respectively I B stands for the unit operators in the corresponding spaces H. Then from which it follows easily the Kronecker formula (4.58).
If one of the factors in (4.58) is infinite dimensional and the determinant (absolute value of) of the corresponding matrix Q is strictly bigger then one (or strictly smaller then one) then the value det of the product(4.58)is infinite, respectively equal to zero.
In order to understand better this problem we define renormalized Kronecker product which formally can be written as: defines an entire function in the complex plane and such that the following estimate is valid: The proof is an immediate consequence of the Theorem 2.3 (i) and Lemma (4.3) below.
from which it follows: This completes the proof.Another renormalisation 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 define where ∧ 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 H A and H B , see Appendix A. Using the unitary isomorphism map J in between the antisymmetric product of fermionic Fock spaces build on the spaces H A and H B (see Appendix A and the Theorem A.9) and the antisymmetric Fock build on the space defines an entire function in the complex plane and such that the following estimate is valid: Proof.As we have proved in the Theorem (A.9) the right hand side of (4.68) is equal to the product det( Having this the claim of this theorem follows by a straightforward application of Theorem 2.3 (i).follows.
Remark 4.5 For an interesting paper on the influence of quantum statisics on the entanglement see i.e. [34].
Another interesting implication of Theorem A.9 seems to be the following observation.
Proof.Let us observe that the renormalized entropy operators S ± can be decomposed as: Therefore using Theorem A.10 we obtain (4.72) Also the following result seems to be interesting.

Theorem 4.7 Let H = H
A ⊗ H B be a separable Hilbert space and let Φ be a separable quantum operation on H, i.e.Φ = Φ A ⊗ Φ B , where Φ A , resp.Φ B are local quantum operations.Then for any Q ∈ E(H): Proof.Let K A i , i = 1, . .., resp.K B j , j = 1, . . .be the families of operator giving the Kraus representations, for and, resp.
Then, for any Q ∈ E(H): be the corresponding reduced density matrices.Then: Proof.Follows from the formula 4.81 which demonstrates that the operations of taking partial traces are quantum operations and application of Theorem 3.8.Let H = H A ⊗ H B be a bipartite separable Hilbert space and let Q ∈ E(H).It is well known that the spectrum of Q counted with multiplicities, denoted σ(Q) = (λ 1 , ...) is purely discrete and the following spectral decomposition holds: where the orthogonal (and normalised) system of eigenfunctions |Ψ n of Q forms a complete system.Each eigenfunction |Ψ n can be expanded further by the use of the Schmidt decomposition: where 1 and the systems {ψ n i } and {φ n i } form the complete orthonormal systems in H A and, respectively, H B .Using (4.84) and (4.83) we can compute the corresponding reduced density matrices where the operators are the states on H B .Similarly, for the reduced density matrix connected to the observer localized with H A : where be the corresponding reduced density matrices and let And corr.
be the corresponding operators as defined in (2.0 , corr.In 2.).Then, for any n: 2. The value G(Q A (n)) is invariant under the action of unitary group, for any unitary map U ∈ H A : is not increasing under the action of any local quantum operation Φ acting on E(H A ): Identical facts are valid for the reduced density matrices Q B (n) . Proof.Obvious.
Remark 4.10 The list Γ(Q) = (r n j ) associated with Q is locally SU(H A ) ⊗ SU(H B ) matrix valued invariant of Q (after taking care on the localisation in this 2d table of the corresponding Schmidts numbers).Therefore any scalar functions build on Γ will define a locally-unitary invariant of Q.Some of them are additionally also monotonous under the action of the local quantum operations and therefore are promising candidates for being a "good" [22,23,24,48] quantitative measures of quantum correlations included in Q.More on this is reported elsewhere [32,50].
Another approach to certain version of reduced density matrices structure is based on the use of the Schmidt decomposition method in the Hilbert-Schmidt space of operators build on the space H A ⊗ H B .Some details are presented in appendix B and in paper [47].
Systematic and much wider applications of the obtained forms of the reduced density matrices will be presented in an another publications (under preparations now).

The case of pure states
Let H = H A ⊗ H B be a bipartite, separable Hilbert space and let Q ∈ E(H) be such that tr(Q 2 ) = 1.Then there exists an unique, normalized vector |Ψ ∈ H such that Q = |Ψ Ψ|.
Then we can write : where Ψ ij = e B j ⊗ e A i |Ψ .We start with the Schmidt decomposition (essentially SVD decomposition, see i.e.Thm.26.8 in [14]) in the infinite dimensional setting.• two ,complete orthonormal systems of vectors {φ n } in H A and {ω n } in H B such that the following equality (in the L 2 -space sense) holds: The decomposition 4.93 is called the Schmidt decomposition of |Ψ .The expansion formula 4.93 can be rewritten as: where and also where and then extended by linearity and continuity to the whole 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 following  RDM2 The non-zero parts of the spectra of ∆ A and ∆ B coincides and are equal to squares τ 2 n of non-zero Schmidt numbers in (4.93).
RDM3 In particular the following formulas are valid: which means that the kets |φ n are eigenvectors of the reduced density matrix Q A , and similarly for The interesting observation is that the explicite Gram matrix nature (it is well known fact [42] that any (semi)-positive matrix has a Gram matrix structure) of the operators ∆ can be flashed on.
be given.Then the matrix elements of the corresponding operators ∆, given in the product base e A i ⊗ |e B j are given by the formulas below and similarly where the corresponding vectors F are given by (4.95) and (4.97).
In the finite dimensional case the following, niece geometrical picture is known [32].Let {v i , i = 1, ..d ′ } be a system of linearly independent vectors in the space C d , where d ′ = d.Let as build on the vectors 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 v i |v j for 4.3 The case of pure states 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 ′ )δ 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 then the product of the lengths of the spanning vectors v i is the best known among them.For more on this see [32].
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 |Ψ .

G(Ψ)(z) = det(I
We are seeing 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 (4.105) 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.
We have also some results about extensions of the results presented here to the many-partite systems as well [51].
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.

.108)
Then the logarithmic gramian volume g(Ψ) has the following properties; 1.For any |Ψ Similar results are true for the renormalized von Neumann entropies.For this goal let us recall the definitions of entropies generating operators : (4.112) From which we obtain an estimate Lemma 4. 16 For any pure state |Ψ ∈ H the renormalized entropy generator defined as S − (Ψ) in (4.112) obeys the bound:

.113)
Proof.We have used the following, rough estimate:  Finally we mention the monotonicity of the renormalised Entropy with respect to the by majorisation relation introduced semi-order.Details will be presented elsewhere [52].
The fermionic Fock space over H is defined as The corresponding antisymmetric Fock spaces in this situation are used for describing fermionic, discrete degrees of freedom .
Let T ∈ B(H).Then we lift the action of T onto the Fock space(s) as Γ(T ) : and similarly for f 1 ∧ . . .∧ f n case.Let us collect here some well known facts: He and two complete (in the corresponding He spaces), orthonormal systems of L 2 -class hermitean operators {Ω ⋆A n } ⊂ He(L 2 (H)), resp.{Ω ⋆B n } ⊂ He(L 2 (H)) such that: Proof.If dimensions of the spaces H A and H are both finite then the proof follows from the very definition of separability.For N < ∞ we define (modulo normalisation) using expansion (B.2) the following separable states: 3) The sequence Q N tends in the L 2 topology to the limiting state Q.Therefore we conclude that Q belongs to the L 2 closure of the set of separable states.But Q belongs to E(H) from the very assumptions made on it.
As it is well known the Schmidt decompositions (B.1) and (B.2) can be used in finite dimensions to test the presence of entanglement in Q.For this, let us recall the well known realignment criterion: if Q is separable then the sum of the corresponding canonical Schmidt numbers τ is not bigger then 1 [59,53].
For other, generalized version of this criterion see [56,60,61,62,63,64].The infinite dimensional applications are also possible and are reported in a separate note [52].
With the help of these expansions, the following formulas for the corresponding reduced density operators (RDM) on the local L 2 -spaces are derived The operator |Q Q| acts in the Hilbert-Schmidt space of operators acting in H as an orthogonal projector.The spaces of operators acting on the space of states E(H) are called often the space of superoperators.From the physical point of view the most important class of superoperators are those which are completely positive and trace preserving [22,23,55].Such superoperators are called quantum channels.From our considerations it follows that any superoperator from He2(He2(H)) can be decomposed similarly to the decompositions (B.5)- (B.8).
C Operator valued (renormalized) map (Q → log(I H + Q)) Several useful properties of the map Q → log(I H + Q) will be collected in this supplement.To start with let us consider non-negative Q ∈ L(H).Using the spectral theorem we can define operator log(I H + Q).
Proposition C.1 The map log(1 H + .)with values in L + 1 (H) is well defined on L + 1 (H) and moreover, for Q ∈ L + 1 (H): Lemma C.2 (C.2) For any Q 0 , Q 1 ∈ L + 1 (H) the strong Frechet directional derivative of the map log(1 H + ..) in the point Q 0 and in the direction to Q 1 is given by the formula: (C.4) Proof.All the formulated here results are valid in the finite dimensional setting.The corresponding infinite dimensional results follows by performing the finite dimensional approximations and then performing the passage (controllable by the L 1 -continuity) to limiting cases.
C.1 Continuation of the proof of Theorem 2.8 The case τ = 1.If τ = 1 then it follows that Q or Q ′ or both one are pure states.Assume that Q, Q ′ are both pure states.Then, there exist two unit vectors

Lemma 3 . 1 Proposition 3 . 2 f 3 . 3
Let as assume that f is a continuous, increasing and convex function on R. If a b then f (a) f (b).It is clear from the very definitions that a m-b iff log(a+1) log(b+1).Let a, b ∈ C ≥ and let as assume that a m-b.Let f be continuous, increasing function and such that the composition f • exp(x) is convex on a suitable domain.Then f (a) f (b).Proof.For fix n we have: (b i ) .(3.44)In particular taking f (x) = x we conclude Corollary Let and let as assume that a, b ∈ C ≥ and a m-b.Then a b.
.77) the proof follows as the proof of Theorem 3.8.named partial trace map is quantum operation in the sense of the previously introduced definition in Section 3.

Theorem 4 . 8
Let H A and H B be a pair of separable Hilbert spaces of an arbitrary dimensions H = H A ⊗ H B and let Q ∈ E(H) and let

Theorem 4 . 11
For any unit vector |Ψ ∈ H there exist • sequence of non-negative numbers τ n (called the Schmidt coefficients of Ψ) and such that ∞ n=1 τ 2 n = 1, 99)and similarly ∆ B (Ψ) :J B † J B : H B → H B (4.100)Some elementary properties of the introduced operators ∆ A and ∆ B are collected in the following proposition.

4. 3
The case of pure states Proposition 4.12 The operators ∆ A and ∆ B have the following properties: RDM1 They both are non-negative and bounded ∆ A 1 = ∆ B 1 = 1.

Ψ
(4.105)In particular case z = 1 the value of the gramian function G of state |Ψ will be called the gramian volume of |Ψ and denoted as G(Ψ).The logarithm of the gramian volume will be called the logarithmic (gramian) volume of |Ψ and denoted as g(Ψ).Using (4.105) it follows that ij |e A i ⊗ |e B j ∈ H. (4.107)Then the gramian volume G(Ψ) has the following properties: 4. Let Φ A(B) be any local quantum operation on the local space H A (resp.H B ). Then (a) G((Φ A ⊗ I B )(Ψ)) ≤ G(Ψ), (b) G((

Corollary B. 5 5 )
Let H = H A ⊗ H B be a bipartite separable Hilbert space and letQ ∈ E(H).Then L 2 -RDM of |Q Q| ∈ L 2 (L 2 (H)) are given by QQ A = Tr L 2 (H B ) (|Q Q|) = and QQ B = Tr L 2 (H A ) (|Q Q|) =The hermitean version of this expansion is given:Corollary B.6 Let H = H A ⊗ H B bea bipartite separable Hilbert space and let Q ∈ E(H).Then HeL 2 -RDM of |Q Q| ∈ He(He(H)) are given by QQ A = Tr L 2 (H B ) (|Q Q|) = and QQ B = Tr L 2 (H A ) (|Q Q|) =
.61) Proposition 4.2 Let H A and H B be a pair of separable Hilbert spaces of an arbitrary dimensions H = H A ⊗ H B and let .67) Theorem 4.4 Let H A and H B be an pair of separable Hilbert spaces of an arbitrary dimensions and H = H A ⊗ H B and let are states on H A .The obtained systems of operators {Q A n } and {Q B n } consist of bounded non-negative self-adjoint, local operators of class L 1 (H A ), respectively of class 1 (H B ) and therefore they are locally measurable.In particular the squares of the Schmidt coefficients τ n i of the Schmidt decompositions of the eigenfunctions of the parent state Q are observable (measurable locally) quantities.Proposition 4.9 Let H = H A ⊗H B be a bipartite separable Hilbert space and let .97)Let us define pair of linear maps JA : H A → H B , resp.J B : H B → H .109) where τ n are the Schmidt numbers of |Ψ .Let U(H) be a multiplicative group of unitary operators acting in the Hilbert space H. Then the logarithmic gramian volume of |Ψ is invariant under the action on of the local unitary groups U(H A ) ⊗ I B , I A ⊗ U(H B ), U(H A ) ⊗ U(H B ) (4.111) 5. Let Φ A(B) be any local quantum operation on the local space H A (resp. Theorem 4.17 Let H = H A ⊗ H B , where H A and H B are separable Hilbert spaces.Then, for any pure state |Ψ ∈ E(H) the renormalized entropy defined as Theorem 4.18 Let U(H) be a multiplicative group of unitary operators acting in the Hilbert space H. Then the renormalized entropy of |Ψ is invariant under the action on of the local unitary groups U(H A ) ⊗ I B , I A ⊗ U(H B ) and also U(H A ) ⊗ U(H B ).Theorem 4.19 Let H = H A ⊗ H B , where H A and H B are separable Hilbert spaces.Then, for any pure state |Ψ ∈ E(H) the renormalized entropy de- A (resp.H B ).
.2) Remarks B.3 Whether the Schmidt numbers of both expansions are identical or not is not clear for us.Also the operators Ω appearing in Theorem B.1 and Theorem B.2 are different in general.In particular, all the operators appearing in (B.2) are hermitean.If all the operators appearing in Eq. (B.2) are non-negative then Q is separable.