Minkovskii-type inequality for arbitrary density matrix of composite and noncomposite systems
aa r X i v : . [ qu a n t - ph ] M a y Minkovskii-type inequality for arbitrary density matrix of composite andnoncomposite systems
V. N. Chernega, O. V. Man’ko, V. I. Man’koP.N. Lebedev Physical Institute, Russian Academy of SciencesLeninskii Prospect, 53, Moscow 119991, RussiaEmail: [email protected]
Abstract
New kind of matrix inequality known for bipartite system density matrix is obtained forarbitrary density matrix of composite or noncomposite qudit systems including the singlequdit state. The examples of two qubit system and qudit with j = 3 / Key words:
Hermitian matrix, bipartite quantum system, entropic and information inequalities
PACS : 42.50.-p,03.65 Bz
The quantum correlations for multipartite qudit systems are partially characterized by some in-equalities for density matrices of the systems. For example, the system of two qubits has thedensity matrix which in case of separable state of the system obeys the Bell inequality [1, 2]. Theviolation of the inequality provides the characteristics of the entanglement in the two qubit systemwhich is related to the degree of correlations between qubits and correlations can be associatedwith value of Cirelson bound [3]. On the other hand there exist the entropic and informationinequalities, e.g. subadditivity condition which is inequality for von Neumann entropies of thebipartite system and its two subsystem states [4]. For three-partite systems there exists strongsubadditivity condition which is the inequality for the von Neumann entropies of the compositesystem and its subsystems [5, 6]. The subadditivity condition are also valid for Shannon entropies[7] of the bipartite and three-partite systems, respectively. The nonnegativity of the Shannonmutual information and quantum mutual information and conditional classical and quantum mu-tual informations follow from the subadditivity and strong subadditivity conditions. Recently theportrait qubit and qudit map was introduced to study the entanglement phenomenon [8, 9]. Thismethod is appropriate to study quantum correlations in framework of tomographic probabilityrepresentation of quantum states [10, 11, 12, 13, 14]. In this representation which is valid forboth discrete and continious variables [10] the spin states (qudit states) are identified with fairtomographic probability distributions [15, 16, 17, 18]. In view of this the standard formulas forclassical probability distributions like entropies can be easily compared with corresponding quan-tum ones [19, 20]. Using the approach based on the portrait method which in fact is the positivemap approach it was shown [8, 9, 21, 22, 23, 24] that the entropic inequalities valid for compos-ite systems can be extended to the arbitrary systems including systems without subsystems. In125, 26, 27] some inequalities associated with positive operators acting in the Hilbert space whichhas the structure of tensor product of Hilbert spaces were studied. The aim of our work is toobtain new matrix inequalities for density matrices of the qudit states of both composite andnoncomposite quantum systems which do not depend on the tensor product structure of Hilbertspace. We follow the approach of [8, 9, 21, 22, 23, 24] based on the portrait positive map method.The paper is organized as follows. In Sec. 2 we formulate new inequality for the Hermitianmatrix which corresponds to operator inequality for bipartite quantum system given in [25]. InSec. 3 we consider example of this inequality for 4 × j = 3 /
2. The conclusion and perspectives are givenin Sec. 4. N × N - Hermitian matrix We present now the new inequality for density N × N matrix, i.e. ρ + = ρ, Tr ρ = 1 , ρ ≥
0. Let N = nm where n and m are integers. Let us present the matrix ρ in block form ρ = a a · · · a n a a · · · a n · · · · · · · · · · · · a n a n · · · a nn . (1)Here the blocks a jk ( j, k = 1 , , . . . , n ) are the m × m matrices. For arbitrary real number p weintroduce the matrix ρ p and present it in block form ρ p = a ( p ) a ( p ) · · · a n ( p ) a ( p ) a ( p ) · · · a n ( p ) · · · · · · · · · · · · a n ( p ) a n ( p ) · · · a nn ( p ) . (2)The blocks a jk ( p ) are the m × m matrices which depend on the parameter p . New inequalitywhich is valid for arbitrary N × N - matrix ρ reads Tr n X j =1 a jj p /p ≤ Tr Tr a ( p ) Tr a ( p ) · · · Tr a n ( p )Tr a ( p ) Tr a ( p ) · · · Tr a n ( p ) · · · · · · · · · · · · Tr a n ( p ) Tr a n ( p ) · · · Tr a nn ( p ) /p . (3)This inequality is valid for p ≥
1. In this inequality the n × n -matrix in right-hand side has matrixelements Tr a jk ( p ). If 0 ≤ p ≤ N = nm we use the integer N ′ = N + s such that N ′ = nm and consider the density N ′ × N ′ matrix ρ ′ of the form ρ ′ = ρ
00 0 ! = a a · · · a n a a · · · a n · · · · · · · · · · · · a n a n · · · a nn . (4)2ere a jk are blocks which provide the representation of density N ′ × N ′ matrix ρ ′ . Then theinequality (3) takes place for the blocks associated with the matrix ρ ′ by (4). The new inequality(3) is obtained on the base of inequality obtained in [25, 26, 27] for the density operator of abipartite quantum system. But the new inequality (3) is valid for arbitrary density matrices ofmultipartite qudit systems including the single qudit density matrix. If the density matrix ρ isdiagonal matrix the inequality (3) provide the inequalities for arbitrary probability vectors.In fact, let us denote the diagonal elements of the matrix ρ as( a jj ) α = P jα , α = 1 , , . . . m. The nonnegative numbers P , P , . . . , P m , P , P , . . . , P m , . . . , P n , P n , . . . , P nm can be con-sidered as components of a probability N -vector ~P . The inequality (3) written in terms of theprobability vector reads m X α =1 n X j =1 P jα p /p ≤ n X j =1 " m X α =1 ( P jα ) p /p , p ≥ . (5)The reverse inequality holds for 0 < p ≤
1. If N = nm we use N ′ = N + s = nm . The inequality(5) for the probability N ′ -vector holds and the last s components of the probability N ′ -vector areequal to zero. In fact we have the inequality (5) for arbitrary number of nonnegative numberswhich are not necessarily associated with a probability distribution. It is worthy to point out thatthe inequality (3) we obtained for density N × N matrix ρ is valid for any density matrix obtainedfrom this one by all the permutations of indices 1 , , . . . , N → p , p , . . . , N p . The same statementis true for the inequality (5). More generally, the density matrix Φ( ρ ) obtained from the initialmatrix ρ by means of arbitrary positive map ρ → Φ( ρ ) satisfies the inequality (3). It is clear thatone can use different decompositions of the integer N = nm = n ′ m ′ . It means that there existdifferent inequalities for the same density matrix ρ corresponding to different product form of thenumbers N and N ′ .For N = nm we can extend the inequality (3) to the case of arbitrary Hermitian N × N matrix A . Let A = A † and the matrix A has the block form corresponding to decomposition N = nmA = a a · · · a n a a · · · a n · · · · · · · · · · · · a n a n · · · a nn , (6)Let x be the minimal eigenvalue of the matrix A . For arbitrary x ≥ x such that x + x ≥ A ( x ) = A + x N . (7)The matrix A ( x ) has the block form A ( x ) = a + x m a · · · a n a a + x m · · · a n · · · · · · · · · · · · a n a n · · · a nn + x m . (8)3hen the matrix ( A ( x )) p can be presented in block form( A ( x )) p == a ( x, p ) a ( x, p ) · · · a n ( x, p ) a ( x, p ) a ( x, p ) · · · a n ( x, p ) · · · · · · · · · · · · a n ( x, p ) a n ( x, p ) · · · a nn ( x, p ) . (9)For N = nm the new inequality which holds for arbitrary Hermitian N × N matrix A reads Tr n X j =1 a jj ( x ) p /p ≤ Tr Tr a ( x, p ) Tr a ( x, p ) · · · Tr a n ( x, p )Tr a ( x, p ) Tr a ( x, p ) · · · Tr a n ( x, p ) · · · · · · · · · · · · Tr a n ( x, p ) Tr a n ( x, p ) · · · Tr a nn ( x, p ) /p . (10)This inequality holds for p ≥
1. The inequality reverses for 0 ≤ p ≤
1. In case N = nm we use N ′ = N + s = nm and the new inequality for the Hermitian matrix A ′ = A
00 0 ! (11)presented in the block form (6) can be given by the above inequality (10). Using the diagonalHermitian matrix A one can write down inequality for arbitrary finite set of N = nm real numbers P , P , . . . , P m , P , P , . . . , P m , . . . , P n , P n , . . . , P nm . It has the form of inequality for twofunctions P ( x, p ) and P ( x, p ), i.e. P ( x, p ) ≤ P ( x, p ) , p ≥ , P ( x, p ) ≥ P ( x, p ) , < p ≤ . (12)In this inequality P ( x, p ) = m X α =1 nx + n X j =1 P jα p /p , P ( x, p ) = n X j =1 (" m X α =1 P jα ! + mx p ) /p (13)For reals such that P jα ≥ P nj =1 P mα =1 P jα = 1 the inequality (12) can be interpreted as theinequality for probability vector which holds for arbitrary x ≥ J ( p ) = Tr Tr a ( p ) Tr a ( p ) · · · Tr a n ( p )Tr a ( p ) Tr a ( p ) · · · Tr a n ( p ) · · · · · · · · · · · · Tr a n ( p ) Tr a n ( p ) · · · Tr a nn ( p ) /p − Tr n X j =1 a jj p /p ≥ , p ≥ . (14)For bipartite system the function J ( p ) is additional characteristics of correlations to the mutualinformation given by the subadditivity condition terms.4 The inequalities for Hermitian × matrices Let us illustrate the inequalities on example of 4 × × A hasthe 2 × a = ρ ρ ρ ρ ! , a = ρ ρ ρ ρ ! , a = ρ ρ ρ ρ ! , a = ρ ρ ρ ρ ! . (15)The matrix A ( x ) reads A ( x ) = ρ + x ρ ρ ρ ρ ρ + x ρ ρ ρ ρ ρ + x ρ ρ ρ ρ ρ + x . (16)The matrix ( A ( x )) p has the block form( A ( x )) p = a ( x, p ) a ( x, p ) a ( x, p ) a ( x, p ) ! . (17)The inequality (10) has the form " Tr ρ + ρ + 2 x ρ + ρ ρ + ρ ρ + ρ + 2 x ! p /p ≤ Tr Tr a ( x, p ) Tr a ( x, p )Tr a ( x, p ) Tr a ( x, p ) ! /p , p ≥ . (18)If the Hermitian matrix A is nonnegative and Tr A = 1 it can be interpreted as a density matrixeither of two-qubit state or the state of qudit with j = 3 / × p , p , p , p the inequality reads ( x = 0)[( p + p ) p + ( p + p ) p ] /p ≤ ( p p + p p ) /p + ( p p + p p ) /p , p ≥ . (19)One can check that for p = 2 the above inequality is equivalent to inequality a + b ≥ ab . Thefunction J ( p ) for this case reads J ( p ) = ( p p + p p ) /p + ( p p + p p ) /p − h ( p + p ) + ( p + p ) p i /p ≥ , p ≥ . (20)The mutual information for this case has the form I = p ln p + p ln p + p ln p + p ln p − ( p + p ) ln ( p + p ) − ( p + p ) ln ( p + p ) − ( p + p ) ln ( p + p ) − ( p + p ) ln ( p + p ) ≥ onclusion To conclude we point out the main results of our work. We obtained for arbitrary system ofqudits, including single qudit case, the inequalities for the system-state density matrix which isequivalent to known inequalities in case of bipartite quantum system. We obtained new simpleinequality for arbitrary Hermitian N × N -matrix. The inequality can be used to study the groundstate energy property for the Hermitian Hamiltonian and the compatibility of this inequality withentropic and information inequalities which can be obtained for the Hamiltonian. These problemswill be discussed in a future publication. References [1] J. S. Bell, ”On the Einstein Podolsky Rosen paradox”,
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