A family of generalized quantum entropies: definition and properties
We present a quantum version of the generalized -entropies, introduced by Salicrú et al. for the study of classical probability distributions. We establish their basic properties, and show that already known quantum entropies such as von Neumann, and quantum versions of Rényi, Tsallis, and unified entropies, constitute particular classes of the present general quantum Salicrú form. We exhibit that majorization plays a key role in explaining most of their common features. We give a characterization of the quantum -entropies under the action of quantum operations, and study their properties for composite systems. We apply these generalized entropies to the problem of detection of quantum entanglement, and introduce a discussion on possible generalized conditional entropies as well.
Keywords:Quantum entropies Majorization relation Entanglement detection
During the last decades a vast field of research has emerged, centered on the study of the processing, transmission and storage of quantum information JozSch94 (); Sch95 (); NieChu10 (); Ren13 (). In this field, the need of characterizing and determining quantum states stimulated the development of statistical methods that are suitable for their application to the quantum realm OgaHay04 (); Hol11 (); GilGut13 (); YuDua14 (). This entails the use of entropic measures particularly adapted for this task. For this reason, quantum versions of many classical entropic measures started to play an increasingly important role, being von Neumann entropy vNeu27 () the most famous example, with quantum versions of Rényi Ren61 () and Tsallis Tsa88 () entropies as other widely known cases. Many other examples of interest are also available in the literature (see, for instance, CanRos02 (); HuYe06 (); Kan02 ()).
Quantum entropic measures are of use in diverse areas of active research. For example, they find applications as uncertainty measures (as is the case in the study of uncertainty relations MaaUff88 (); Uff90 (); WehWin10 (); ZozBos13 (); ZozBos14 (); ZhaZha15 ()); in entanglement measuring and detection HorHor94 (); AbeRaj01:01 (); TsaLlo01 (); RosCas03 (); BenZyc06 (); Hua13 (); OurHam15 (); as measures of mutual information Yeu97 (); ZhaYeu98 (); Car13 (); GroWal13 (); and they are of great importance in the theory of quantum coding and quantum information transmission JozSch94 (); Sch95 (); WilDat13 (); DatRen13 (); AhlLob01 ().
The alluded quantum entropies are nontrivially related, and while they have many properties in common, they also present important differences. In this context, the study of generalizations of entropic measures constitutes an important tool for studying their general properties. In the theory of classical information measures, Salicrú entropies SalMen93 () are, up to now, the most generalized extension containing the Shannon Sha48 (), Rényi Ren61 () and Tsallis Tsa88 () entropies as particular examples and many others as well HavCha67 (); Dar70 (); Rat91 (); BurRao82 (). But a quantum version of Salicrú entropies has not been studied yet in the literature. We accomplish this task by introducing a natural quantum version of the classical expression. Our construction is shown to be of great generality, and contains the most important examples (von Neumann, and quantum Rényi and Tsallis entropies, for instance) as particular cases.
We show that several important properties of the classical counterpart are preserved, whereas other new properties are specific of the quantum extension. In our proofs, one of the main properties to be used is the Schur concavity, which plays a key role, in connection with the majorization relation BenZyc06 (); LiBus13 () for (ordered) eigenvalues of density matrices. Our generalization provides a formal framework which allows to explain why the different quantum entropic measures share many properties, revealing that the majorization relation plays an important role in their formal structure. At the same time, we give concrete clues for the explanation of the origin of their differences. Furthermore, the appropriate quantum extension of generalized entropies can be of use for defining information-theoretic measures suitable for concrete purposes. Given our generalized framework, conditions can then be imposed in order to obtain families of measures satisfying the desired properties.
The paper is organized as follows. In Sec. 2 we give a brief review of (classical) Salicrú entropies, also known as -entropic forms. Our proposal and results are presented in Sec. 3. In 3.1 we start proposing a quantum version of the -entropies using a natural trace extension of the classical form, followed by the study of its Schur-concavity properties in 3.2. Then, in 3.3, we study further properties related to quantum operations and the measurement process. In 3.4 we discuss the properties of quantum entropic measures for the case of composite systems focusing on additivity, sub and superadditivity properties, whereas applications to entanglement detection are given in 3.5. Sec. 4 contains an analysis of informational quantities that could be derived from the quantum -entropies. Finally, in Sec. 5, we draw some concluding remarks.
2 Brief review of classical -entropies
Let us consider an -dimensional probability vector with . The so-called -entropy is defined as
where the entropic functionals and are such that either: (i) is increasing and is concave, or (ii) is decreasing and is convex. The entropic functional is assumed to be strictly concave/convex, whereas is taken to be strictly monotone, together with and .
We notice that in the original definition SalMen93 (), the strict concavity/convexity and monotony characters were not imposed. These considerations will allow us to determine the case of equality in some inequalities presented here. The assumption is natural in the sense that one can expect the elementary information brought by a zero-probability event to be zero. Also, an appropriate shift in allows to consider only the case , thus not affecting generality, while giving the vanishing of entropy (i.e., no information) for a situation with certainty.
The -entropies (1) provide a generalization of some well-known entropies such as those given by Shannon Sha48 (), Rényi Ren61 (), Havrda–Charvát, Daróczy or Tsallis HavCha67 (); Dar70 (); Tsa88 (), unified Rathie Rat91 () and Kaniadakis Kan02 (), among many others. In Table 1 we list some known entropies and give the entropic functionals and that lead to these quantities. Notice that the entropies given in the table enter in one (or both) of the special families determined by entropic functionals of the form: and concave BurRao82 (), or and ZozBos14 (). Indeed, the so-called -entropy (or trace-form entropy) is defined as
where is concave with , whereas the -entropy is defined as
where is increasing with , and the entropic parameter is nonnegative and . With the additional assumption that is differentiable and , one recovers the Shannon entropy in the limit .
As recalled in Ref. ZozBos14 (), the -entropies share several properties as functions of the probability vector :
is invariant under permutation of the components of . Hereafter, we assume that the components of the probability vectors are written in decreasing order.
: extending the space by adding zero-probability events does not change the value of the entropy (expansibility property).
decreases when some events (probabilities) are merged, that is, . This is a consequence of the Petrović inequality that states that for a concave function vanishing at 0 (and the reverse inequality for convex ) (Kuc09, , Th. 8.7.1), together with the increasing (resp. decreasing) property of .
Other properties relate to the concept of majorization (see e.g. MarOlk11 ()). Given two probability vectors and of length whose components are set in decreasing order, it is said that is majorized by (denoted as ), when for all and . By convention, when the vectors do not have the same dimensionality, the shorter one is considered to be completed by zero entries (notice that this will not affect the value of the -entropy due to the expansibility property). The majorization relation allows to demonstrate some properties for the -entropies:
It is strictly Schur-concave: with equality if and only if . This implies that the more concentrated a probability vector is, the less uncertainty it represents (or, in other words, the less information the outcomes will bring). The Schur-concavity of is consequence of the Karamata theorem Kar32 () that states that if is [strictly] concave (resp. convex), then is [strictly] Schur-concave (resp. Schur-convex) (see (MarOlk11, , Chap. 3, Prop. C.1) or (Bha97, , Th. II.3.1)), together with the [strictly] increasing (resp. decreasing) property of .
Reciprocally, if for all pair of entropic functionals , then . This is an immediate consequence of Karamata theorem Kar32 () (reciprocal part) which states that if for any concave (resp. convex) function one has , then for all and (see also (MarOlk11, , A.3-(iv), p. 14 or Ch. 4, Prop. B.1) or (Bha97, , Th. II.3.1)).
It is bounded:
where stands for the number of nonzero components of the probability vector. The bounds are consequences of the majorization relations valid to any probability vector (see e.g. (MarOlk11, , p. 9, Eqs. (6)-(8)))
together with the Schur-concavity of . From the strict concavity, the bounds are attained if and only if the inequalities in the corresponding majorization relations reduce to equalities.
From the previous discussion we can see immediately that the -entropies fulfill the first three Shannon–Khinchin axioms Khi57 (), which are (in the form given in Ref. Tem16 ()) (i) continuity, (ii) maximality (i.e., it is maximum for the uniform probability vector) and (iii) expansibility. The fourth Shannon–Khinchin axiom, the so-called Shannon additivity, is the rule for composite systems that is valid only for the Shannon entropy (notice that there are other axiomatizations of Shannon entropy, e.g., those given by Shannon in Sha48 () or by Fadeev in Fad56 ()). A relaxation of Shannon additivity axiom, called composability axiom, has been introduced Tsa09 (); Tem16 (); it establishes that the entropy of a composite system should be a function only of the entropies of the subsystems and a set of parameters. The class of entropies that satisfy these axioms (the first three Shannon–Khinchin axioms and the composability one) is wide Tem15 () but nevertheless can be viewed as a subclass of the -entropies.
It has recently been shown that the -entropies can be of use, for instance, in the study of entropic formulations of the quantum mechanics uncertainty principle ZozBos13 (); ZozBos14 (). They have also been applied in the entropic formulation of noise–disturbance uncertainty relations Ras16 (). Our aim is to extend the definition of the -entropies for quantum density operators, and study their properties and potential applications in entanglement detection.
3 Quantum -entropies
3.1 Definition and link with the classical entropies
The von Neumann entropy vNeu27 () can be viewed as the quantum version of the classical Shannon entropy Sha48 (), by replacing the sum operation with a trace. We recall that for an Hermitian operator , with being its eigenvectors in and being the corresponding eigenvalues, one has , and the trace operation is the sum of the corresponding eigenvalues (i.e., , where stands for the trace operation). In a similar way to the classical generalized entropies, we propose the following definition:
Let us consider a quantum system described by a density operator acting on an -dimensional Hilbert space , which is Hermitian, positive (that is, ), and with . We define the quantum -entropy as follows
where the entropic functionals and are such that either: (i) is strictly increasing and is strictly concave, or (ii) is strictly decreasing and is strictly convex. In addition, we impose and .
The link between Eqs. (1) and (5) is the following. Let us consider the density operator written in diagonal form (spectral decomposition) as , with eigenvalues satisfying , and being an orthonormal basis. Then, the quantum -entropy can be computed as
This equation states that the quantum -entropy of a density operator , is nothing but the classical -entropy of the probability vector formed by the eigenvalues of . Notice that despite the link between the quantal and the classical entropies defined from a pair of entropic functionals , we keep a different notation for the entropies ( and , respectively) in order to distinguish their very different meanings.
The most relevant examples of quantum entropies, which are the von Neumann one vNeu27 (), quantum versions of the Rényi, Tsallis, unified and Kaniadakis entropies HuYe06 (); Ras11:06 (); FanCao15 (); OurHam15 (), and the quantum entropies proposed in Refs. CanRos02 (); Sha12 (), are clearly particular cases of our quantum -entropies (5).
In what follows, we give some general properties of the quantum -entropies (the validity of the properties for the von Neumann entropy is already known, see for example Lie75 (); Weh78 (); OhyPet93 (); BenZyc06 (); NieChu10 ()). In our derivations, we often exploit the link (6). With that purpose, hereafter we consider, without loss of generality, that the eigenvalues of a density operator are arranged in a (probability) vector , with components written in decreasing order.
3.2 Schur-concavity, concavity and bounds
One of the main properties of the classical -entropies, namely the Schur-concavity, is preserved in the quantum version of these entropies:
Let and be two density operators, acting on and respectively, and such that . Then
with equality if and only if either , or , for any isometric operator (i.e., ), where stands for the adjoint of . Reciprocally, if Eq. (7) is satisfied for all pair of entropic functionals, then .
Let and be the vectors of eigenvalues of and , respectively, rearranged in decreasing order and adequately completed with zeros to equate their lengths. By definition, means that (see (BenZyc06, , p. 314, Eq. (12.9))). Thus, the Schur-concavity of the quantum -entropy (and the reciprocal property) is inherited from that of the corresponding classical -entropy, due to the link (6). From the strict concavity or convexity of and thus the strict Schur-concavity of the classical -entropies, the equality holds in (7) if and only if , that is equivalent to have either (when ) or (when ).
As a direct consequence, the quantum -entropy is lower and upper bounded, as in the classical case:
The quantum -entropy is lower and upper bounded
where stands for the rank of an operator (the number of nonzero eigenvalues). Moreover, the lower bound is achieved only for pure states, whereas the upper bounds are achieved for a density operator of the form for some orthonormal ensemble , with in the tightest situation and in the other one (in the latter case, necessarily , being the identity operator in ).
Let be the vector formed by the eigenvalues of . Clearly , so that the bounds are immediately obtained from that of the classical -entropy, due to the link (6). Moreover, in the classical case, if and only if , that is is a pure state. On the other hand, the upper bounds are attained if and only if , with or .
The classical -entropies and their quantum versions are generally not concave. We establish here sufficient conditions on the entropic functional to ensure the concavity property of the quantum -entropies. We notice that, with the same sufficient conditions, the classical counterpart is also concave:
If the entropic functional is concave, then the quantum -entropy is concave, that is, for all ,
Let us first recall the Peierls inequality (see (BenZyc06, , p. 300)): if is a convex function and is an Hermitian operator acting on , then for any arbitrary orthonormal basis , the following inequality holds
Consider written in its diagonal form, decreasing and convex. Then
Notice that in the case concave, these two inequalities are reversed. Thus, one finally has
Notice that in the case increasing, the first inequality holds, and the second inequality holds as well, from concavity of (with equality valid when is the identity function). Making use of Def. 2, the proposition is proved in both cases, under the condition that the entropic functional is concave.
Note also that for the class of -entropies, the concavity of is equivalent to that of . Moreover, for the von Neumann and quantum Tsallis entropies the conditions of Proposition 3 are satisfied, and it is well known that these entropies have the concavity property. For quantum Rényi entropies, the concavity property holds for as consequence of Proposition 3, but for the proposition does not apply (see (BenZyc06, , p. 53) for an analysis of concavity in this range for classical Rényi entropies). For the quantum unified entropies, the concavity property holds in the range of parameters and or and as consequence of Proposition 3, which complements the result of Ref. HuYe06 () and improves the result of Ref. Ras11:06 ().
It is interesting to remark that using the concavity property given in Proposition 3, it is possible to define in a natural way, for concave, a (Jensen-like) quantum -divergence between density operators and , as follows:
which is nonnegative and symmetric in its arguments. This is similar to the construction presented in Ref. BurRao82 () for the classical case, and offers an alternative to the quantum version of the usual Csiszár divergence Csi67 (); Sha12 (). It can be shown that for pure sates and the quantum -divergence (11) takes the form
Indeed, the square root of this quantity in the von Neumann case provides a metric for pure states LamPor09 (). Notice that the right-hand side of Eq. (12) is a binary -entropy. Other basic properties and applications of the quantum -divergence are currently under study BosBel16 ().
3.3 Specific properties of the quantum -entropy
We recall that the quantum entropy of a density operator equals the classical entropy of the probability vector formed by its eigenvalues. In other words, considering a density operator as a mixture of orthonormal pure states, its quantum entropy coincides with the classical entropy of the weights of the pure states. This is not true when the density operator is not decomposed in its diagonal form, but as a convex combination of pure states that do not form an orthonormal basis. The quantum -entropy of an arbitrary statistical mixture of pure states, is upper bounded by the classical -entropy of the probability vector formed by the mixture weights:
Let be an arbitrary statistical mixture of pure states , with and . Then, the quantum -entropy is upper bounded as
First, we recall the the Schrödinger mixture theorem (BenZyc06, , Th. 8.2): a density operator in its diagonal form can be written as an arbitrary statistical mixture of pure states , with and , if and only if, there exist a unitary matrix such that
As a corollary, one directly has Nie00 ()
where are the elements of the bistochastic matrix111It is assumed that , otherwise is completed with zeros; when , the remaining terms that do not appear in Eq. (14) are added in order to fulfill the unitary of and is to be understood as completed with zeros (for more details, see the proof of the Schrödinger mixture theorem (BenZyc06, , p. 222-223)). . From the lemma of Hardy, Littlewood and Pólya (BenZyc06, , Lemma 2.1) or (MarOlk11, , Th. A.4), this is equivalent to the majorization relation . Therefore, from (6) and the Schur-concavity of the classical -entropy, we immediately have .
The previous proposition is a natural generalization of a well-known property of von Neumann entropy. One can also show that a related inequality holds:
Let be an arbitrary orthonormal basis of and, for a given density operator acting on , let us denote by the probability vector with elements , that is, the diagonal elements of related to that basis. Then
The decomposing of in the basis has the form where the diagonal terms are . The Schur–Horn theorem (BenZyc06, , Th. 12.4) states that the vector of the diagonal terms of is majorized by the vector of the eigenvalues of . Thus, from the Schur-concavity property of the classical -entropy, we have .
We consider now the effects of transformations. Among them, unitary operators are important since the time evolution of an isolated quantum system is described by a unitary transformation (i.e., implemented via the action of a unitary operator on the state). One may expect that a “good” entropic measure remains unchanged under such a transformation. This property, known to be valid for von Neumann and quantum Rényi entropies MulDup13 () among others, is fulfilled for the quantum -entropies, and even in a slightly stronger form, i.e., for isometries. We recall that an operator is said to be isometric if it is norm preserving. This is equivalent to . On the other hand, an operator is then said to be unitary if it is both isometric and co-isometric, that is, both and are isometric. When (both Hilbert spaces having the same dimension) is isometric, it is necessarily unitary (see e.g. Hal82 ()).
The quantum -entropy is invariant under any isometric transformation where is an isometric operator:
Let us write in its diagonal form, . Clearly, , where with , form an orthonormal basis (due to the fact that is an isometry). Since and have the same eigenvalues, and thus, using Eq. (6), we conclude that they have the same -entropy.
When dealing with a quantum system, it is of interest to estimate the impact of a quantum operation on it. In particular, one may guess that a measurement can only perturb the state and, thus, that the entropy will increase. This is also true for more general quantum operations. Moreover, one may be interested in quantum entropies as signatures of an arrow of time: to this end one can see how the value of an entropic measure changes under the action of a general quantum operation. More concretely, let us consider general quantum operations represented by completely positive and trace-preserving maps , expressed in the Kraus form (with satisfying the completeness relation ). It can be shown that the behavior of entropic measures depends nontrivially on the nature of the quantum operation (see e.g. (BenZyc06, , Sec. 12.6)). For example, a completely positive map increases the von Neumann entropy for every state if and only if it is bistochastic, i.e., if it is also unital ( also satisfies the completeness relation), so that the operation leaves the maximally mixed state invariant. This is no longer true for the case of a stochastic (but not bistochastic) quantum operation. What can be said of the generalized quantum -entropies? This is summarized in the following:
Let be a bistochastic map. Then, the quantum operation can only degrade the information (i.e., increase the -entropy):
with equality if and only if for a unitary operator .
This is a well-known property of von Neumann entropy, when dealing with projective measurements (NieChu10, , Th. 11.9). It turns out to be true for the whole family of -entropies, and in a more general context than projective measurements. However, as we have noticed above, generalized (but not bistochastic) quantum operations can decrease the quantum -entropy. Let us consider the example given in (NieChu10, , Ex. 11.15, p. 515). Let be the density operator of an arbitrary qubit system, with nonvanishing quantum -entropy, and consider the generalized measurement performed by the measurement operators and (a completely positive map, but not unital). Then, the system after this measurement is represented by with vanishing quantum -entropy.
Note that Proposition 5 can be viewed as a consequence of Proposition 7. Indeed, it is straightforward to see that the set of operators defines the bistochastic map . Thus, Proposition 5 can be deduced applying successively Proposition 7 and Proposition 4.
In the light of the previous discussions and results, we can reinterpret Proposition 5 as follows: the quantum -entropy equals the minimum over the set of rank-one projective measurements of the classical -entropy for a given measurement and density operator. Indeed, we can extend the minimization domain to the set of rank-one positive operator valued measurements (POVMs)222 Recall that a POVM is a set of positive definite operators satisfying the resolution of the identity. As a consequence, we can give an alternative (and very natural, from a physical perspective) definition for the -entropies.
Let be the set of all rank-one POVMs. Then
where is the probability vector for the POVM given the density operator , i.e., .
Let us consider an arbitrary rank-one POVM and consider the positive operators . Let us then define
where we have used the fact that is rank-one, so its square-root can be written in the form (with not necessarily normalized), allowing us to introduce the pure states . From the completeness relation satisfied by the POVM, is then a doubly stochastic map. Thus, applying successively Proposition 7 and Proposition 4 we obtain
Since is arbitrary, we thus have
Consider then where is the orthonormal basis that diagonalizes . Thus
which ends the proof.
We notice that the alternative definition of quantum -entropy given in this proposition, can not be extended to any POVM. The following counterexample shows this impossibility. Let us consider the density operator with even, and the POVM formed by the positive operators and , where is an arbitrary orthonormal basis of . Thus, we obtain and consequently from the Schur-concavity and the expansibility of the classical -entropy we have .
3.4 Composite systems I: additivity, sub and superadditivities, and bipartite pure states
We focus now on some properties of the quantum -entropies for bipartite quantum systems represented by density operators acting on a product Hilbert space . Specifically, we are interested in the behavior of the entropy of the composite density operator , with reference to the entropies of the density operators of the subsystems333By definition, the partial trace operation over , , is the unique linear operator such that for all and acting on and , respectively. For instance, let us consider the bases and of and respectively, and the product basis of . Let us denote by the components in the product basis of an operator acting on . Thus, the partial trace over of gives the density operator of the subsystem , , whose components are in the basis ., and .
Now, we give sufficient conditions for the additivity property of quantum -entropies:
Let be an arbitrary product density operator of a composite system , and and the corresponding density operators of the subsystems. If, for and , and satisfy the Cauchy functional equations either of the form (i) and , or of the form (ii) and . Then the -entropy satisfies the additivity property
In case (i), by writing the density operators and in their diagonal forms, it is straightforward to obtain
where we used . Similarly, for case (ii),
The domains where the functional equations have to be satisfied are respectively the domain of definition of and the image of (see Proposition 2).
Note that, on the one hand, in case (i) the functional equation for can be recast as with . Thus, and with are entropic functionals that are solutions of the functional equations (i) 444Notice that the Cauchy equations , and are not necessarily linear, logarithmic or power type respectively without additional assumptions on the domain where they are satisfied and on the class of admissible functions (see e.g. Cau21 (); Kuc09 ()). But, recall that the entropic functionals and are continuous and either increasing and concave, or decreasing and convex.. These solutions lead to von Neumann entropy, which, as it is well known, is additive (see e.g. Lie75 (); Weh78 ()). On the other hand, in case (ii), and with and or and are entropic functional solutions. This is the case for the Rényi entropies, which are also known to be additive (see e.g. MulDup13 ()). In general, however, -entropies are not additive, for instance quantum unified entropies (including quantum Tsallis entropies) do not satisfy this property for all the possible values of the entropic parameters HuYe06 (); Ras11:06 (). For the (not so general) quantum -entropies, we can give necessary and sufficient conditions for the additivity property:
Let be an arbitrary product density operator of a composite system , and and the corresponding density operators of the subsystems. Then, for any the additivity property
holds if and only if for .
The ‘if’ part is a direct consequence of Proposition 9 where and satisfy the Cauchy equations of condition (ii).
Reciprocally, if is additive, we necessarily have that for any pair of arbitrary states. Denoting and and analyzing the image of for any density operator acting on , we necessarily have over the domain specified in the proposition, which ends the proof.
Notice that, if is twice differentiable, one can show that is proportional to the logarithm thus, among the quantum -entropies, only the von Neumann and quantum Rényi entropies are additive.
As we have seen, the -entropies are, in general, nonadditive. However, as suggested in Rag95 (), two types of subadditivity and superadditivity can be of interest. One of them compares the entropy of with the sum of the entropies of the subsystems and (global entropy vs sum of marginal-entropies), and the other one compares the entropy of with that of the product state (global entropy vs product-of-marginals entropy). The general study of subadditivity of the first type, , is difficult, even if one is looking for sufficient conditions to insure this subadditivity. Although it is not valid in general, there are certain cases for which it holds. For example, it holds for the von Neumann entropy Weh78 (), quantum unified entropies for a restricted set of parameters HuYe06 (); Ras11:06 (), and quantum Tsallis entropy with parameter greater than 1 Rag95 (); Aud07 (). On the other hand, it is possible to show that only the von Neumann entropy (or an increasing function of it) satisfies subadditivity of the second type, provided that some smoothness conditions are imposed on . This is summarized in the following:
Let be a density operator of a composite system , and and the corresponding density operators of the subsystems. Assume that is twice differentiable on . The -entropy satisfies
if and only if is an increasing function of the von Neumann entropy, given by .
The proof is based on two steps:
First, an example of a two qutrit diagonal system acting on a Hilbert space is presented, for which it is shown that cannot be subadditive, with the exception of certain functions satisfying a given functional equation.
Next, under the assumptions of the proposition, the functional equation is solved, and it is shown that all the entropic functionals for which we could not conclude on the subadditivity of , can be reduced to the case and increasing.
Step 1. Consider the composite two qutrit systems acting on a Hilbert space , of the form
where is an orthonormal basis for , , the coefficients in the set
and in the interval