Deformed entropic and information inequalities for X - states of two-qubit and single qudit states
aa r X i v : . [ qu a n t - ph ] O c t DEFORMED ENTROPIC AND INFORMATION INEQUALITIESFOR X - STATES OF TWO-QUBIT AND SINGLE QUDIT STATES V.I. Man’ko and L.A. Markovich ∗ P.N. Lebedev Physical Institute, Russian Academy of SciencesLeninskii Prospect 53, Moscow 119991, Russia Institute of Control Sciences, Russian Academy of SciencesProfsoyuznaya 65, Moscow 117997, Russia ∗ Corresponding author e-mail: kimo1 @ mail.ru
Abstract
The q -deformed entropies of quantum and classical systems are discussed. Standard and q -deformedentropic inequalities for X - states of the two-qubit and a state of single qudit with j = 3 / Keywords: q -deformed entropy, Entropic inequalities, X - states, entanglement, noncomposite systems. The quantum correlations of bipartite qudit systems are characterized, e.g. by entropic inequalitieswritten for von Neuman entropies [1] of the system and its subsystems [2]. The q -deformed entropieswere introduced in [3, 4]. These entropies being the functions of extra parameter contain more detailedinformation on properties of density matrices of the qudit states and the qudit subsystem states. TheTsallis entropy of bipartite qudit system was shown to satisfy the generalized subadditivity condition [5,7].This condition is the inequality available for Tsallis entropy of the bipartite system state and Tsallisentropies of two subsystem states. In approach [8–12] it was shown that the relations for compositesystem state can be extended to be valid for noncomposite systems, e.g. for the single qudit state.These inequalities reflect some quantum correlation properties of degrees of freedom either of subsystems(in the case of bipartite system) or degrees of freedom of the single qudit in the case of noncompositesystem states. One of the important states of the two-qubit systems are X -states. The properties of thisstates were studied for example in [12–14]. The partial case of X -state is the Werner state [15]. Theentanglement properties of the Werner state were studied in detail for example in [16].The aim of our work is to obtain new deformed entropic inequality for X -state of composite (bipartite)and noncomposite (single qudit with j = 3 /
2) quantum systems. We consider both Renyi and Tsallisentropic inequalities.The paper is constructed as follows. In Sec. 2 we review the notion of Renyi and Tsallis entropies forbipartite systems. In Sec. 3 we obtain the new Tsallis entropic inequalities for X -state of noncompositequantum system. The latter entropic inequality is illustrated on the example of the Werner state of thesingle qudit. 1 Renyi and Tsallis entropies
Let us introduce the quantum state in the Hilbert space H defined by the following density matrix ρ = ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ , T r ( ρ ) = 1 , ρ = ρ † , ρ ≥ . (1)If we apply the invertible map of indices 1 ↔ / /
2; 2 ↔ / − /
2; 3 ↔ − / /
2; 4 ↔ − / − / ρ = T r ρ (1 ,
2) and ρ = T r ρ (1 ,
2) which describe the states of the subsystems 1 and2, respectively. Applying another invertible map of indices 1 ↔ /
2, 2 ↔ /
2, 3 ↔ − /
2, 4 ↔ − / j = 3 /
2. Hence it is possible to use the density matrix in the form (1) to describe bothbipartite systems as well as systems without subsystems. This idea to use invertible map of integers1 , , . . . onto the pairs (triples, etc) of integers ( i, k ), j, k = 1 , , . . . to formulate the quantum propertiesof systems without subsystems was applied in [8–12]. That gives us possibility to translate knownproperties of quantum correlations associated with structure of bipartite system like entanglement to thesystem without subsystems, e.g. single qudit.An important measure of entanglement is entropy. The most known is the von Neumann entropy. Itis obtained as S N = − T rρ ln ρ. More flexible are Tsallis and Renyi entropies. The Renyi entropy generalizes the Shannon entropy, theHartley entropy, the min-entropy, and the collision entropy. Both, Tsallis and Renyi entropies, depend onextra parameter q , thus they are called q -entropies. The classical q -entropies for the probability vector,constructed from the diagonal elements of the density matrix (1) −→ p = ( p = ρ , p = ρ , p = ρ , p = ρ ), are S Tq = 11 − q X i =1 p qi − ! , S Rq = 11 − q ln X i =1 p qi ! . (2)When q → S Tq reduces to the von Neumann entropy. Tsallis and Renyi entropies can be written in thefollowing form S Tq = − T rρ ln q ρ, S Rq = 11 − q ln ( T rρ q )where ln q ρ = ( ρ q − − Iq − , if q = 1 , ln ρ, if q = 1 , (3)2or any real q > I is identity matrix. The logarithm (3) is called q -logarithm or deformed logarithm.The relations between Tsallis and Renyi entropies are given by the following formulas S Tq = exp( S Rq (1 − q )) − − q , S Rq = ln(1 + (1 − q ) S Tq )1 − q . (4)If the density matrix (1) describes the bipartite state (two-qubit system), then we can consider twosubsystems on spaces H and H such that H = H ⊗ H . Reduced density matrices ρ , ρ are definedas partial traces of (1). Result matrices are density matrices of the density operators acting on spaces H and H , respectively. Thus the reduced density matrices of the first and the second qubit are definedas ρ = ρ + ρ ρ + ρ ρ + ρ ρ + ρ ! , ρ = ρ + ρ ρ + ρ ρ + ρ ρ + ρ ! . (5)It is well known, that the von Neumann entropy is subadditive. In [5] was proved the subadditivity ofthe Tsallis entropy for q > S Tq ( ρ ) ≤ S Tq ( ρ ) + S Tq ( ρ ) . There are other entropic inequalities, for example strong subadditivity condition [6], which holds forthe von Neumann entropy of three-partite quantum system. The fact that Tsallis entropy is not strongsubadditive was recently proved in [7]. Let us define the q -information as I Tq = S Tq ( ρ ) + S Tq ( ρ ) − S Tq ( ρ ) ≥ . (6)The subadditivity condition for the Tsallis entropy provides the inequality for the Renyi entropyexp( S Rq ( ρ )(1 − q )) + exp( S Rq ( ρ )(1 − q )) − exp( S Rq ( ρ )(1 − q )) < . (7)Since Renyi and Tsallis entropies tend to the von Neumann entropy for q →
1, both inequalities (6) and(7) in this limit give the standard positivity condition of the von Neumann mutual information. X -state Using the invertible mapping 1 ↔ /
2, 2 ↔ /
2, 3 ↔ − /
2, 4 ↔ − /
2, the density matrix (1) can berewritten as ρ / = ρ / , / ρ / , / ρ / , − / ρ / , − / ρ / , / ρ / , / ρ / , − / ρ / , − / ρ − / , / ρ − / , / ρ − / , − / ρ − / , − / ρ − / , / ρ − / , / ρ − / , − / ρ − / , − / . This matrix is a density matrix of the single qudit state with spin j = 3 /
2. Such system has nosubsystems, thus it is impossible to write the reduced density matrices for it. But using the form (1) we3an successfully write them. If ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = 0 then the densitymatrix (1) has the view of X -state density matrix ρ X = ρ ρ ρ ρ ρ ρ ρ ρ = ρ ρ ρ ρ ρ ∗ ρ ρ ∗ ρ , (8)where ρ , ρ , ρ , ρ are positive reals and ρ , ρ are complex quantities. The latter matrix has theunit trace and is nonnegative if ρ ρ ≥ | ρ | , ρ ρ ≥ | ρ | . The reduced density matrices are definedas ρ = ρ + ρ ρ + ρ ! , ρ = ρ + ρ ρ + ρ ! . Hence, the q -information (6) for the X -state of the single qudit is I Tq = 11 − q ( ρ + ρ )(( ρ + ρ ) q − −
1) + ( ρ + ρ )(( ρ + ρ ) q − −
1) (9)+ ( ρ + ρ )(( ρ + ρ ) q − −
1) + ( ρ + ρ )(( ρ + ρ ) q − − − ( ρ + ρ + ρ + ρ )( ρ q − − ! ≥ . As an example of the X -state density matrix of the qudit state with spin j = 3 / ρ W = p p − p − p p p , (10)where parameter is − ≤ p ≤
1. The parameter domain < p ≤ p = marksthe border between the separable and entangled Werner states. One can see general behavior of the q -information against parameter p for different values of deformation parameter q . In the domain of theentangled states the q -information increases with increasing the degree of the state entanglement. Thesensitivity of the q -information to the degree of entanglement depends on the deformation parameter q . To conclude we point out the main results of the work. We applied q -deformed entropies of Renyiand Tsalles as a measure of entanglement for the systems without subsystems. New deformed entropicinequality for the X -state of noncomposite quantum state (qudit with spin j = 3 /
2) was written. As anexample of the X -state the Werner state with one parameter was taken.4 − Figure 1: The q -information I Tq for the Werner state (10) of the qudit with spin j = 3 / p for different values of the deformation parameter q .Despite there are no subsystems in such systems it is possible to introduce analogs of partial traceslike for composite systems using special mapping illustrated in the text. Certainly it is necessary tounderstand that these partial traces for noncomposite systems have not the same meaning as for thecomposite system quantum state. The understanding of physical background of the correlations insidethe system without subsystems is still not well understood and will be developed in future article of theauthors. References [1] J. von Neumann,
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