Prime decomposition and correlation measure
of finite
quantum systems
D. Ellinas ^{⋄}^{⋄}⋄ Email: and E. G. Floratos ^{◁}^{◁}◁ Email:
Applied Mathematics and Computers Lab.
Technical University of Crete
GR  73 100 Chania Crete Greece
NCRC Demokritos, Institute of Nuclear Physics
GR  15 310 Ag. Paraskevi Attiki Greece
and
Department of Physics, University of Crete, Crete Greece.
Abstract
Under the name prime decomposition (pd), a unique decomposition of an arbitrary dimensional density matrix into a sum of seperable density matrices with dimensions determined by the coprime factors of is introduced. For a class of density matrices a complete tensor product factorization is achieved. The construction is based on the Chinese Remainder Theorem, and the projective unitary representation of by the discrete Heisenberg group . The pd isomorphism is unitarily implemented and it is shown to be coassociative and to act on as comultiplication. Density matrices with complete pd are interpreted as grouplike elements of . To quantify the distance of from its pd a tracenorm correlation index is introduced and its invariance groups are determined.
03.65 Bz, 42.50 Dv, 89.70.+c
Journal of Physics A: Math. Gen. 32 (1999) L63L69
Quantum correlations, an emblematic notion of quantum theory, remains an open challenge since the early days of Quantum Mechanics [1, 2]. Recent investigations have set important questions concerning classification of various types of quantum correlations and their appropriate quantification. These theoretical activities have parallel developments with, and are partly motivated from recent interesting proposals which engage quantum correlations to such diverge tasks as e.g quantum computation and communication [3, 4], quantum cryptography [5], teleportation [6], and some new frequency standards [7]. Although the classification of quantum correlations is still open to refinements, it appears to include the following cases: for pure states, correlations entail nonlocality and give rise to violation of Bell inequalities [2]. For mixed states, two systems are considered uncorrelated if the composite system density matrix factorizes into a product of reduced density matrices, one for each isolated quantum subsystem viz. , where , are determined by partial tracing. Quantification measures for that case include the von Neumann entropy [8] and other invariant indices [9]. On the other hand classical correlations for quantum subsystems imply seperability of the joint system density matrix, which is analysed into a convex sum for products of pure states viz. , , , [10]. Necessary and sufficient conditions for the existence of such convex decompositions for ’s acting on and , became available recently [11, 12]. Upper bounds for the number of terms in such convex expansions of seperable matrices have also been determined, together with construction algorithms for the cases dim [13] and dim [14]. Beyond these types of classical correlations we encounter inseperable or entangle quantum states. For their characterization and the quantification of their entanglement some general conditions have been presented that good entanglement measures should satisfy [15].
In this Letter we address the problem of the , of a finite but otherwise arbitrary dimensional square density matrix , into a sum of products of elementary density matrices, the number and the respective dimensions of which are determined by the compositeness of the dimension of . This is achieved by means of the so called Chinese Remainder Theorem (CRT) [16], that is based on the prime decomposition of (this also explains the name we have chosen for the decomposition), and by the fact that the discrete Heisenberg group , provides a projective representation of the abelian cyclic group [17]. More concretely, if , is the prime factor decomposition of , where ’s are distinct primes, then the pd of the density matrix involves square matrices , with power prime dimension equal to . Also the number of factors is bounded by the number of coprime factors of . As a measure of the correlation of a given mixed state , with its possible prime or other decomposition, we evaluate the tracenorm distance between the two densities, study its unitary invariant symmetries, and interpret it in terms of the quantum variances between local operators of the subsystems of the decomposition.
We start by considering the matrix realization of the discrete Heisenberg group generated by the operator set of elements , where the matrices
(1) 
satisfy the relations , , , , , with , and , the square indexlattice. By virtue of these relations the following commutators are valid [18], [19], [20]:
(2) 
Moreover due to linear independence, completeness and the orthonormality issued by the relation
(3) 
the same generator set forms a basis of the matrices [17].
Let us consider a dimensional quantum system with Hilbert space . The generators of the finite Heisenberg group provide an operator basis , for the decomposition of the density matrix of , viz.
(4) 
with , We note here that due to the Hermitian conjugation of the basis elements i.e , the Hermiticity of the density matrix implies for its elements the reality conditions . Let us assume to be a composite positive integer with primepower decomposition , where each of the factors is distinct, uniquely determined and relatively prime to each other, i.e when . Then according to the CRT the isomorphism , is valid for the index cyclic groups labelling the operator basis. To proceed we introduce the group isomorphic map , between the cyclic groups. The explicit definition reads: , where , , , , stand for the residues of the division of by .
Next we regard the fact that , provides a projective unitary representation of the additive cyclic group , by means of the map . More explicitly, , with the property . Then the following commuting diagram: , where
Fig. 1
given by the equation , induces the isomorphism of CRT from the indexgroups to the associated Heisenberg groups by the following component version of the above diagram:
Fig. 2
We state the main proposition for the prime decomposition:
Proposition. The isomorphism , determined by the commuting diagram of Fig. 1, via its component version Fig. 2, is a linear map which induces the map of CRT into the Heisenberg group , and provides the unique pd of elements of . Also is implemented by unitary operator in the Hilbert space and possesses the coassociativity property.
Proof. If and , then , with and , the residues of the division of by respectively. According to CRT is an isomorphism the determination of which provides the solution of a system of congruences and , when , i.e when , are pairwise coprime positive integers. Inversely, given the residues, the numbers can be determined in a mixedradix notation by and , where and is the solution of the congruence . Alternatively by means of the Euler function , which counts the positive integers , which are coprime to , the is given by . Then , are expressed in the form and .
We turn now to study the consequences of this decomposition for the generators of . With the notation as before we obtain the relations
(5) 
where and . Analogous relations hold for the generator . By direct computations it is verified that if , and , where . This definition implies that is periodic with respect to the coprime factors of i.e , for . Using the above commutation properties of the generators we write:
(6)  
The isomorphism introduced above is based on the fact the the ’s are commuting for different ’s and their moduli make them to behave as copies (isomorphic images ) of the original , with periodicities ; this is similar to harmonics in Fourier analysis. The following embedding provides the explicit form of the isomorphism:
(7) 
with ; this provides the commutativity of the diagrams.
The prime decomposition of a general density matrix given in eq. (4) with coefficients , can now be evaluated and reads,
(8)  
where .
This suggests that we first map into , according to the previous analysis and then project along the ’s , in order to determine the coefficients of the matrix factors in the prime decomposition.
A special form of pd that contains only a single product term is possible for a special class of density matrices with coefficients , where , an arbitrary real function. If , the tranformed coefficients , due to , factorize as follows:
(9)  
Note that power raising is counted , so above a Frobenious type of map has been used i.e . Therefore we obtain the pd , where . In view of the coassociative property of the , to be established shortly, we see that those matrices that admit such complete factorization behave as grouplike elements under the pd map.
To proceed we study the unitary implementation of the pd of density matrices. We introduce the operator , given by
(10) 
and its conjugate
(11) 
(this as the one used earlier maps numbers to their respective residues; see below). These operators form a conjugate pair that obeys the unitarity condition and . Then it is straightforward to verify that the pd map acting on general density matrix is implemented by the unitary similarity transformation i.e .
Finally we study briefly an important property of the pd map , namely that it becomes a coassociative comultiplication of the Heisenberg group ; this illustrates a connection of CRT with Hopf algebras [21] in the framework of quantum mechanical correlations. Consider the map , , by which a , decomposes into its resedues wrt coprimes . Also consider its dual map , which constructs the solution of the congruences , , according to CRT. Then we check that for three coprime factors of , the following equation is valid on any :
(12) 
This is dual to the relation
(13) 
which holds if we are given three congruences and combine them pairwise in two different ways. This (co)associativity of the CRT maps, in turn is induced into the pd map , where it takes the form
(14) 
As an example we take the system
(15) 
with solution , and obtain
(16) 
Dualizing we recover the relation for the ’s which induces the coassociativity of the pd mapping:
(17) 
for . Closing this proof we note that the integral , with definition , is invariant under the comultiplication , in the sense that
We turn now to the study of the correlation between finite quantum systems. We start with two systems with state vector Hilbert spaces of dimension respectively. Any observable and density matrix is expressed by the elements of the Lie algebra , correspondingly. For the density matrix of system1 e.g ,
(18) 
and similarly for system2. The choice of the operator basis , for the Lie algebra is an important one. For a composite system the density matrix reads [22]
(19)  
where , and , the correlation components. Also by partial tracing we define , . To proceed with the definition of the correlation index we view the space of matrices , as a norm space with HilbertSchmidt (HS) norm,
(20) 
for . This is essentially a Frobenius type matrix norm, which is unitarily invariant i.e , for unitary (the lower index of the norm will be omitted hereafter). Then we propose the following
Definition. The correlation scalar index of two coupled finite quantum systems been in a mixed state is defined as [22]
(21) 
Index provides us with a measure of correlation between the coupled systems in terms of the difference of the factorized partial density matrices from the density of the composite system. It is cast in the form
(22)  
where modulo arithmetic applies in the respective indices.
The index vanishes for product states and by using the reality conditions of the ’s i.e , , we introduce the matrix , and reexpress the index in the form
(23) 
This last expression suggests first, that the index is determined by the trace of the covariance matrix of local observables and , and second that it is invariant under general unitary tranformations of the group , i.e , with . The last inclusion describes the fact that the invariance unitary group of , is in general larger than the local unitary transformations in which case the symmetry group factorizes (c.f [9]).
Extensions to three and more coupled systems is straightforward. For three systems e.g the composite density matrix involves terms of the operator basis where the ’s are embedded in all possible ways in the 3tensor space. Also for the reduced matrices there are various possibilities in this case i.e and , with cyclic permutations of . This gives rise to different correlation indices i.e
(24) 
Closing we should mention that the correlation index can be expressed in terms of the function of the involved density matrices, associated with the group coherent state of dimension . This possibility as will be explained elsewhere [23], is based on the fact the the algebra generators used here in the expansion of the dim density matrices, can be embedded (by means of the polar decomposition of the algebra), into the enveloping algebra . Examples of the finite case together with extensions to infinite dimensional quantum systems will also be reported elsewhere.
We acknowledge support from the Greek Secretariat of Research and Technology under contract 95/1981.
Figure captions.
Fig. 1. Induction of CRT into the Heisenberg group.
Fig. 2. Component version of the induction of fig.1.
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