Quantum entanglement and entropy

Transcription

1 PHYSICAL REVIEW A, VOLUME 64, 0210 Quantu entangleent and entropy Filippo Giraldi 1,2, and Paolo Grigolini 1,2, 1 Dipartiento di Fisica dell Università di Pisa and INFM, Piazza Torricelli 2, Pisa, Italy 2 Istituto di Biofisica CNR, Area della Ricerca di Pisa, Via Alfieri 1, San Cataldo Ghezzano-Pisa, Italy Center for Nonlinear Science, University of North Texas, P.O. Box 11427, Denton, Texas Received 0 January 2001; published 20 August 2001 Entangleent is the fundaental quantu property behind the now popular field of quantu transport of inforation. This quantu property is incopatible with the separation of a single syste into two uncorrelated subsystes. Consequently, it does not require the use of an additive for of entropy. We discuss the proble of the choice of the ost convenient entropy indicator, focusing our attention on a syste of two qubits, and on a special set, denoted by I. This set contains both the axially and partially entangled states that are described by density atrices diagonal in the Bell basis set. We select this set for the ain purpose of aking our work of analysis ore straightforward. As a atter of fact, we find that in general the conventional von Neuann entropy is not a onotonic function of the entangleent strength. This eans that the von Neuann entropy is not a reliable indicator of the departure fro the condition of axiu entangleent. We study the behavior of a for of nonadditive entropy, ade popular by the 1988 work by Tsallis J. Stat. Phys. 52, We show that in the set I, iplying the key condition of nonvanishing entangleent, this nonadditive entropy indicator turns out to be a strictly onotonic function of the strength of the entangleent, if entropy indexes q larger than a critical value Q are adopted. We argue that this ight be a consequence of the nonadditive nature of the Tsallis entropy, iplying that the world is quantu and that uncorrelated subsystes do not exist. DOI: /PhysRevA PACS nubers: 0.67.a, 0.65.Ta, y, 05.0.d I. INTRODUCTION Entangleent is the fundaental quantu property behind the interesting process of quantu teleportation proposed soe years ago by Bennett et al. 1. For this reason it is iportant to quantify entangleent 2. It was also found that the fidelity of the quantu teleportation is always larger than that of any classical counication protocol, even in the noisy environent. However, for the teleportation to take place perfectly, it is necessary for the sender and the recipient 1 to share a axially entangled state. This generates the need for special purification protocols. On the other hand, a statistical analysis of these protocols shares deep siilarities with the second principle of therodynaics 4 6. In spite of the plausible conjecture that there exists a deep connection between quantu teleportation and therodynaics 4 6, the entangleent is expressed by eans of an entropic structure, the conventional von Neuann entropy, only in the case of pure states. In general the definitions of entangleent dictated by the purification protocol are not directly related to entropy indicators. Cerf and Adai 7 showed that the conditional von Neuann entropy can becoe negative, thereby pointing out the nonordinary inforation aspects of quantu entangleent. However, the reason for the lack of a direct connection between quantu entangleent of ixed states and the entropy indicator is probably that, as shown in this paper, the von Neuann entropy is not a reliable indicator of entangleent 7. The inadequacy of the Shannon inforation, and consequently of the von Neuann entropy as an appropriate quantu generalization of Shannon entropy 8, was already pointed out by Brukner and Zeilinger 9,10. It is notable that according to these authors a new kind of entropy indicator is ade necessary by the following deep difference between quantu and classical inforation. In classical easureent the Shannon inforation is a natural easure of our ignorance about the properties of a syste, whose existence is independent of easureent. Quantu easureent, conversely, cannot be claied to reveal the properties of a syste that existed before the easureent was ade. In this paper we focus on the earlier entioned quantu property, essential for teleportation of inforation: entangleent. Entangleent iplies that a syste cannot be divided into two uncorrelated subsystes; this, in turn, akes useless the ordinary request for an additive for of entropy. Thus in this paper we explore and discuss the possible benefits steing fro the adoption of the nonadditive entropy indicator advocated years ago by Tsallis 11. The choice of the Tsallis for of nonadditive entropy for systes that cannot be divided in uncorrelated subsystes the nonextensive case was ade copelling by recent work of Abe 12. This is so because this author proved that the three axios aking the Shannon entropy a unique for of the extensive condition can be properly generalized so as to ake the Tsallis entropy a unique for of the nonextensive case. The Tsallis entropy is applied to a large nuber of physical conditions, characterized by the existence of extended correlation 1. The use of this for of nonadditive indicator in the field of quantu teleportation, to the best of our knowledge, was discussed in only a few papers Reference 14 claied to the realization of a greater sensitivity to the occurrence of a dephasing process, resulting in the annihilation of any for of entangleent. References pointed out the efficiency of Tsallis entropy for the detection of the breakdown of local realis. Reference 16 aied at proving that the Jaynes principle 18,19, applied to /2001/64/021010/$ The Aerican Physical Society

2 FILIPPO GIRALDI AND PAOLO GRIGOLINI PHYSICAL REVIEW A the nonextensive entropy, naturally yields an entangled state. The present paper, although based on an adoption of the Tsallis nonextensive entropy as in Ref. 16, adopts a quite different perspective, which does not rest on the adoption of the Jaynes principle. The outline of the paper is as follows. Section II is devoted to a concise illustration of the ain properties of the nonextensive entropy at work in this paper. Section III, devoted to the entangleent of foration, shows that the von Neuann entropy, in general, is not a onotonic function of entangleent, while the nonextensive entropy is a onotonically increasing function of the entangleent for suitably large values of the entropy index. In Sec. IV we study the entangleent of a de-phasing process, and we extend the onotonic properties of nonextensive entropy to a condition ore general than that of Sec. III. Section IV is devoted to concluding rearks. II. NONEXTENSIVE ENTROPY The entropy indicator applied in this paper has the for S q Tr q q1. This for was originally proposed by Tsallis 11 for the purpose of establishing the ost convenient therodynaic perspective for fractal processes. It is worth rearking that this is a generalization of the conventional Gibbs-Shannon entropy indicator, whose explicit for is recovered fro Eq. 1 in the liiting case q 1. This entropy indicator does not fit the additivity condition, naely, the requireent that the entropy of a syste AB, consisting of two statistically independent subsystes A and B, be the su of the entropies of these two subsystes. In fact, the definition of Eq. 1 yields, in this case, the equality S q ABS q AS q B1qS q AS q B, aking it evident that the additive property is recovered only in the case q1, which, as noted earlier, akes the nonextensive entropy of Eq. 1 becoe equivalent to the usual Shannon entropy. According to Tsallis and to advocates of this nonextensive entropy indicator for a review, see Ref. 1, the violation of the additive condition turns into a benefit when this entropy indicator is used to study cases where the ideal condition of statistical independence is prevented by the nature of the processes under study 1. Notable exaples are processes with long-range correlation 1. We believe that quantu entangleent, which is the basic property for teleportation 1, is probably the ost evident exaple of a condition incopatible with the existence of uncorrelated subsystes. For this reason, the additivity condition can be safely renounced, and the adoption of an entropy index q 1 ight turn out to be beneficial. This paper is devoted to discussing to what extent this conjecture proves to be correct. 1 2 III. TSALLIS ENTROPY AT WORK: ENTANGLEMENT OF FORMATION In this section we derive the central result of this paper. We show that in the case of initial and final states, both with nonvanishing entangleent, and described by a statistical density atrix diagonal in the Bell basis, the Tsallis entropy decreases upon an increase of the entangleent of foration. The results of this section refer to the entangleent of foration 20 of a syste of two qubits. Thus, to ake this paper as self contained as possible, here we give a concise illustration of this key easure of entangleent. The ain point is that the entangleent property is defined without any abiguity for pure states 1. The entangleent of foration extends to the statistical case the ordinary definition as follows. Let us denote by the statistical density atrix for the ixed state of a space S (1) 1/2 S (2) 1/2 of two spin-1/2 particles. The entangleent of foration 21, denoted by the sybol E F throughout, is defined as the iniu average entangleent of every enseble of pure states that represents, E F in i P i i i i P i E i ), where E( i ) denotes the entangleent of a pure state, which is defined according to the usual prescription 1 by the expression with E)Tr A log 2 A Tr B log 2 B, 4 A Tr 2, B Tr 1. 5 Wootters, in an enlightening paper 20, derived an explicit forula for the entangleent of foration of any arbitrary ixed state of a syste of two qubits. This forula reads where E F h 11C2 2, 6 hxx log 2 x1xlog 2 1x. The quantity C(), referred to by Wootters as concurrence, is defined by Cax0, 1 2, where, 1, 2, and are the square roots of eigenvalues of the atrix, set in a decreasing order, with being the axiu eigenvalue. The atrix denotes a spinflipped state: y y * y y. The values of the entangleent of foration range fro 0 to 1. Furtherore, E F is a onotonically increasing function of

3 QUANTUM ENTANGLEMENT AND ENTROPY PHYSICAL REVIEW A C. The values of the concurrence C, in turn, range fro 0 to 1, as the values of E F do. Consequently, the concurrence itself can be considered as a proper easure of the entangleent of foration. Note that the entangleent of foration is the only kind of entangleent studied in the present paper. For siplicity, we shall often refer to it siply as entangleent. A. On I, the working basis set for a syste of two qubits The ost general expression 22 of a ixed state of the (1) space S 1/2 (2) S1/2 is i1 i1 t ij i j, r i i 11 i1 s i i 10 where r i,s i, and t ij are real paraeters. Due to the excessive nuber of involved paraeters, the direct use of this expression would ake the calculation too coplicated. For this reason we decided to liit our investigation to a set defined by I:C2P 10, 11 where P denotes the greatest eigenvalue of the density atrix describing the quantu state. On behalf of future developents, we reark that the condition P 1/2 akes this axiu value unique. The adoption of the set I, asour working set, does not rule out the possibility of considering physical conditions of interest for the field of quantu teleportation. In fact, the set I contains states with positive entangleent of foration E F ()0 that are described by density atrices corresponding to the solution of the equation These density atrices, recently used by Bennett et al. 21 also see Ref. 2, becoe diagonal when expressed in the Bell basis set. The set I contains entangled Werner states 24 as well as axially entangled states. We also note that any ixed state can be brought into a diagonal for in the Bell basis set by rando bilateral rotations 21. This iplies that, in spite of the siplification ade, our discussion is still rather general. The proble under discussion here is the significance of the Tsallis entropy as a easure of entangleent. The working set of states that we select, I, does not conflict with the possibility of discussing this issue with reference to one of the relevant physical conditions recently exained by investigators in this field of research 21,2. In fact, we shall study the change of the Tsallis entropy indicator upon the entangleent change of a given pair of particles. This change ight be the result of a purification process such as that studied by Bennett and co-workers 21,2. These authors discussed the purification of a pair of Werner states, showing how to increase the entangleent of foration of one pair at the price of decreasing that of another. Here we discuss the proble of detecting this increased entangleent with the Tsallis entropy. We found it to be relatively easy to establish this iportant result within the set I. However, we cannot rule out the possibility that this property is shared by any other set of states, resulting in the concurrence being an increasing function of P. In all these cases we ight find the sae property of the entropy being proportional to the inverse of the entangleent. It is worth anticipating an aspect of fundaental iportance. The plausible reason why, as we shall see, the inverse of the Tsallis entropy indicator is a successful easure of entangleent, in the set I, is the following fundaental fact: The nonextensive nature of Tsallis entropy would ake sense only in a world where it were ipossible to create uncorrelated systes, and consequently, a vanishing entangleent. In fact, the Tsallis entropy is not the su of the entropies of the parts, not even when these parts are uncorrelated. The condition that we set, of nonvanishing entangleent, as stated earlier, is equivalent to ruling out the occurrence of a splitting of the syste into two uncorrelated parts. Therefore, it corresponds to an ideal condition for the application of the Tsallis entropy. We shall coe back to this iportant issue in Sec. IV. B. Paraetrization of the eigenvalues of the statistical density atrix Here we adopt a perspective of therodynaic kind, inspired by the lines proposed by Plenio and Vedral 4. We assue that a purification protocol yields an entangleent change E F. We ai at establishing a change of the nonextensive entropy corresponding to the sae therodynaic transforation. To solve this delicate proble we iagine the statistical density atrix, belonging to the set I, tobe a function of a real paraeter, belonging to an interval 1, 2, which can be thought of as playing the sae role as that of variables like pressure, teperature, and volue in the state transforations in ordinary therodynaics. We assue that the initial and final conditions correspond to ( 1 ) and ( 2 ), respectively, so that E F E F ( 2 ) E F ( 1 ). We set the dependence of on in such a way as to fulfill the constraint iposed by the nor conservation Tr 1, and, aong the infinitely large nuber of possibilities fitting these conditions, we select the for ost convenient for the purpose of evaluating S q S q ( 2 ) S q ( 1 ). Let P be the largest of the four eigenvalues of the statistical density atrix, and let us denote the other three eigenvalues by P 1,P 2, and P. These eigenvalues are assued to be functions of the paraeter with the following condition: in the whole interval 1, 2, we have P The derivatives dp / and d /, with j ranging fro 1 to, are assued to be finite. The nor-conservation constraint enforces the condition dp dp 1 dp 2 dp

4 FILIPPO GIRALDI AND PAOLO GRIGOLINI PHYSICAL REVIEW A As pointed out earlier, concurrence is equivalent to the entangleent of foration and, in the set I, concurrence is a strictly increasing onotonic function of P. Thus the change of P will be either positive or negative according to whether the entangleent change is positive or negative. We shall assue that in the whole interval 1, 2 the derivative dp / is either always positive or always negative. As stressed earlier, this convenient choice is legitiized by the fact that, as in ordinary therodynaics, we are considering a state transforation, consequently both the entangleent and entropy changes depend only on the initial and final states, and are independent of the paths used to connect these two states. C. Failure of the von Neuann entropy Here we show that the von Neuann entropy, naely, the quantu version of the Shannon inforation, turns out to be inadequate as an entangleent indicator. This is so because, as we see, the von Neuann entropy can either increase or decrease, corresponding to the entangleent change E F, regardless of whether the entangleent change is positive or negative. This conclusion agrees with the rearks in the recent work by Brukner and Zeilinger 9. We note that to prove the inadequacy of the von Neuann entropy as an indicator of entangleent, it is enough to find a case where the sign of the entropy change is not strictly deterined by that of the entangleent change. Thus let us consider the special physical condition corresponding to dp 0 15 in the whole interval 1, 2. The von Neuann entropy, corresponding to q1, and consequently denoted as S 1, reads S 1 P ln P ln, and its dependence upon the paraeter is given by ds 1 dp ln P P 1 dp 2 ln P 2 P Let us consider, for exaple, the case dp /0. As noted earlier, in this case when the entangleent of the ixed state is a strictly decreasing function of, the entangleent change E F ust be negative. However, using Eqs. 15 and 14 we see that the sign of ds 1 / depends on the special values selected for the paraeters P,P 1,P 2 and for the corresponding derivatives. Thus it is possible to realize either the inequality dp dp 2 ln P 1 P 2 ln P P 1, 18 which would lead to an entropy increase, or the opposite inequality, which would yield an entropy decrease. This proves that the sign of the von Neuann entropy change is not correlated to that of the entangleent, and that consequently the von Neuann entropy is not an adequate entangleent indicator. D. Search of critical entropy indexes as a function of the transforation of the statistical density atrix eigenvalues The theoretical developents of this subsection show that the failure of the von Neuann entropy is a consequence of the fact that the von Neuann entropy eans q1, and this entropy index is saller than, or equal to, a critical value Q( 1, 2 ), where 1 and 2 denote initial and final states, respectively. In Sec. III E we shall show, in fact, that for q Q( 1, 2 ) the nonextensive entropy becoes a onotonic function of the entangleent strength, inversely proportional to it: the condition E F 0 yields S q 0, while the condition E F 0 yields S q 0. First, we show that the property that we plan to prove is a plausible property. For this purpose, let us study the derivative of the nonextensive entropy. This quantity reads q1 ds q qp q1 dp 1 dp 1 P q1 d. 19 We reind the reader that P is the largest eigenvalue of, thereby iplying the property 0 P By using these inequalities and the assuption, ade in Sec. III B, that the derivatives d / and dp / are finite, we obtain q1 dp 1 d li q 1 P This siple result iplies that great enough values of the entropy index yield the iportant property sgn de F sgn dp sgn ds q, 22 which, in turn, eans that increasing entangleent in the transforation yields a decreasing entropy, and vice versa. After proving the onotonic dependence of the nonextensive entropy on the entangleent strength in the asyptotic liit, we now illustrate a recipe necessary to deterine the critical entropic index, naely, the value of q beyond which the onotonic dependence of entropy on entangleent is insured. More precisely, we provide the recipe for two critical indexes, rather than one, according to whether we consider the case of entangleent decrease case a or increase case b. These two critical entropic indexes are denoted by Q ( 1, 2 ) and Q ( 1, 2 ), respectively. This eans that S q 0ifE0, and S q 0ifE0, provided that q Q ( 1, 2 ) and qq ( 1, 2 ), respectively

5 QUANTUM ENTANGLEMENT AND ENTROPY PHYSICAL REVIEW A The critical values Q ( 1, 2 ) and Q ( 1, 2 ) are defined as follows: Q 1, 2 Q 1, 2 sup [ 1, 2 ] sup [ 1, 2 ] q, q. The auxiliary functions q () and q () are defined by q ax1, 1, 2,, q ax1, 1, 2, The functions j (), with the subscript j running fro 1 to, are functions of the interval 1, 2 given by j 1 ln P 1 ln dp 1 d 27 if the conditions ()0 and d /0 apply. If these conditions do not apply, we set j ()1. The functions j () are given by j 1 ln P 1 ln dp 1 d 28 if the conditions ()0 and d /0 apply. If these conditions do not apply, we set j ()1. The proof of this iportant recipe is given in the Appendix. Note that we have not discussed the proble of the possible divergence of Q ( 1, 2 )orq ( 1, 2 ). We shall coe back to this issue in Sec. III E, where we shall consider, without losing any generality, a special paraetrization of the eigenvalues within which, as we shall see, the critical index Q( 1, 2 ) will be proved to be finite. E. Search of a critical entropy index as a function of initial and final states Now let us see how to use the earlier results to ake predictions in the case where the transforation and the ensuing entangleent change are described only by the initial and final states 1 and 2, with the density atrices belonging to the set I defined by Eq. 11. The ain idea is to build (1) up auxiliary states B and (2) B, equivalent to 1 and 2, respectively, as far as their entangleent and entropy are concerned, but fulfilling the condition of being connected the one to the other by one of the transforations described in Sec. III C. This akes these states copatible with earlier (1) prescriptions, and thus with earlier results. The states B and (2) B are defined as follows. Let P (1),P (1) 1,P (1) (1) 2, and P denote the eigenvalues of the density atrix 1, while P (2),P (2) 1,P (2) 2, and P (2) denote the eigenvalues of the density atrix 2 ; we define the auxiliary states (1) B and (2) B by the expressions (1) B P (1) e e P (1) j e j e j 29 and (2) B P (2) e e P (2) j e j e j, 0 where the set e,e j,,2, is the Bell basis set 21, no atter what the order is. It is easy to check that these quantu states have the following properties: i they belong to the set I, ii E F ( 1 )E F ( B (1) ) and E F ( 2 )E F ( B (2) ), and iii S q ( 1 )S q ( B (1) ) and S q ( 2 )S q ( B (2) ). Now let us introduce the transforation B (1), B (2), defined by (1) B, (2) B (1) B P e e e j e j, 1 where the evolutions of P () and () are given by and P P (1) P (2) P (1) (1) (2) (1), 2 with j running fro 1 to, respectively, and belonging to the interval 0,1. The transforation has the required properties: a it keeps the state (1) B, (2) B ( (1) B ) within the set I for every value of belonging to the interval 0,1, b 0 (1) B, (2) B ( (1) B ) (1) B, and c 1 (1) B, (2) B ( (1) B ) (2) B. Note that the functions P (),P 1 (),P 2 (), and P () are eigenvalues of the quantu states (1) B, (2) B ( (1) B ), and are defined in the interval 0,1. They fulfill the properties of Eq. 1, the paraeter conditions of Sec. III B, the relation dp /0 in the case P (1) P (2), and the relation dp /0 in the case P (1) P (2). This akes it possible for us to use Q ( 1, 2 ) of Eq. 2 and Q ( 1, 2 ) of Eq. 24, and the relations on which these quantities rest as well, to derive Q( 1, 2 ). This is done as follows. We write the explicit fors that Q ( 1, 2 ) and Q ( 1, 2 ) gain when 1 0 and 2 1. Applying the transforations of Eqs. 2 and to the prescriptions of Eqs. 2 28, we obtain the following expressions: Q 0,1 sup 0,1 ax1, 1, 2,. 4 Here the function j (), with, 2, and, is defined as follows. If the constraints (2) (1) and (1) (2) (1) 0, 5 with,, hold true, we set ln P (1) (2) j P (2) P j (1) 1 ln P (1) P (2) P (1) P (1) j P (2) j P (1) j

6 FILIPPO GIRALDI AND PAOLO GRIGOLINI PHYSICAL REVIEW A If the constraints of Eq. 5 do not apply, we set j () 1. It is evident that the functions j () are the counterparts of j () in the particular case of the transforation described by Eqs. 2 and. Note that this atheatical definition ust be interpreted as follows. First we consider a given value of belonging to the interval 0,1. Then we ake the index j run fro 1 to, we select indexes j fulfilling the conditions (2) () (1) () and (1) ( (2) (1) )0, and calculate j () using the above definitions. Finally, we take the axiu of the values of a set whose coponents are given by j () and by 1, and we ake explore all possible values of the interval 0,1. Thus we obtain an infinite set of axia, fro which we select the supreu. The resulting nuber defines the critical index of the left hand side of Eq. 4. The resulting critical index is finite. To prove this iportant property we proceed as follows. We note that the ter that could ake Q (0,1) diverge is P (1) P (2) P (1) (1) (2) (1). We denote this ter by (). The special condition resulting in the divergence of the critical index would be given by 1. We observe that () is either an increasing decreasing or constant function of depending on whether the quantity P (1) j P (2) P (1) P (2) j is positive negative or equal to 0. So the iniu value of () is(0)p (1) /P (1) j in the case of /0, and (1)P (2) (2) / in the case of /0. In the reaining case /0, the two inia obtain the sae value. Fro these properties we obtain the inequality Q 0,1 ax,2,,2, 1 ln in P (1) ln (2) (1) proving that Q (0,1) is finite. As for Q (0,1), we obtain P (1) P (2), (1),P P (2) 1 j 7 Q 0,1 sup ax1, 1, 2,. 8 0,1 If the constraints P (1) j P (2) j and P (1) j P (2) j P (1) j 0 9 hold true, we set j 1 ln P (1) (2) j P (2) P (1) ln P (1) P (2) P (1) P (1) j P (2) j P (1) j. 40 If the constraints of Eq. 9 do not apply, we set j () 1. The criterion adopted to define this critical index is the sae as that illustrated earlier to properly define the critical index of Eq. 4. Thus we can prove that Q (0,1) is finite, adopting a procedure analogous to that used for Q (0,1). In this case we arrive at the inequality Q 0,1 ax,2,,2, 1 ln in P (1) ln (1) (2) P (2) P (1), (1),P P (2) 1 j 41 which shows in fact that also Q (0,1) is finite. At this stage we can finally define the critical value Q( 1, 2 ). This is given by Q 1, 2 axq 0,1,Q 0,1. 42 On the basis of the theoretical treatent described earlier, we conclude that Q( 1, 2 )isfinite and that for any initial and final states 1 and 2, respectively, belonging to the set I, with different entangleent, E F ( 1 )E F ( 2 ), the corresponding entropy change S q is positive or negative, according to whether E0 ore0. Note that we found Q( 1, 2 )1. As a consequence of the pseudoadditivity of Eq. 2, the adoption of a value of the entropy index larger than the unity akes the entropy of the whole syste saller than the su of the entropies of the two parts. However, in this paper we never ake a direct use of this property, since, as stressed earlier, our treatent is valid only in the case of nonvanishing entangleent, which rules out the possibility of realizing the factorized condition behind Eq. 2. F. Fro the nonextensive entropy to the entangleent of foration As a purpose of this subsection, we try to prove a property that is the reverse of that discussed in Sec. III E. Ideally, the reverse of the property of Sec. III E should be expressed as follows. Let us focus our attention on a transforation fro an initial state 1 to a final state 2, both belonging to the set I. Let us consider a case where this transforation causes the nonextensive entropy S q to increase decrease. Then the entangleent decreases increases if an entropy index q larger than the critical value Q is adopted. Unfortunately, we cannot prove this property in this attractive for, but only under weaker conditions. This is so because a transforation resulting in an entropy change does note necessarily iply an entangleent change. We note that the entangleent, expressed in the set I, is a function of the eigenvalue P only, while the nonextensive entropy is a function of all four eigenvalues. Thus the entropy can change without iplying a corresponding entangleent change. The sae difficulty is shared by the nonextensive entropy. However, upon an increase of the entropy index q the dependence of the nonextensive entropy on the other three eigenvalues becoes weaker and weaker. In the case of enough great values of the

7 QUANTUM ENTANGLEMENT AND ENTROPY PHYSICAL REVIEW A entropy index q the nonextensive entropy becoes virtually independent of the other three eigenvalues. This is the reason why in Sec. III E we could find a way to ake the nonextensive entropy becoe a onotonic function of the entangleent. We want to reark that in general the entropy critical index is not the sae as that used in Sec. III E. We believe that one of the benefits resulting fro the adoption of the set I, and of very large entropy indices as well, is that the argin of entangleent dependence on the entropy is significantly reduced. Nevertheless, we are forced to ake a weaker request for the reverse of the property discussed in Sec. III E. We shall show, in fact, that if the entropy increases decreases, and the entangleent changes, then the entangleent decreases increases, for entropy indices q larger than a critical value Q (S), not necessarily equal to Q. The conditions ephasized by the adoption of italics ake the property weaker than we would wish. Even in this case we have to assue the entropy index to be larger than a critical value. We denote this critical value with the sybol Q (S) because, as entioned earlier, we cannot prove that it is identical to the critical entropy index Q of an Sec. III E. In the case of an entropy increase, by expressing the nonextensive entropy as a function of its four eigenvalues, we obtain P (2) q P (1) 1 P (1) q (2) q P (1). 4 Since the two eigenstates have different entangleents, we have P (2) /P (1) 1. The inequality of Eq. 4 ust hold true in the case of entropy indices arbitrarily larger than Q (S), and consequently ust hold true also for values uch larger than the unity. As a consequence we reach the conclusion that P (2) P (1), an inequality that in the set I is equivalent to E F 0. The opposite conclusion would be reached in the case of a negative S q. In spite of earlier restrictions, we can use the obtained results to illustrate one of the ost interesting findings of this work. This is as follows. Let us consider a generic subset I, of set I, fulfilling only the request of containing a finite nuber of states, with different entangleents. Then we can conclude that these entangleents are equivalent to the inverse of the nonextensive entropy, provided that entropy indices q are larger than a given value Q I, which is given by the following forula: Q I axq i, j, i, j I,i j. 44 In the set I for entropy indices larger than the critical value the ordering in the direction of increasing decreasing entangleent is equivalent to ordering in the direction of decreasing increasing entropy. A significant consequence of this is that entropy iniization yields a axially entangled state and entropy axiization yields a inially entangled state. An attractive, albeit heuristic way, of illustrating the sae conclusions is given by the forula E q (eff) 2 arctan i1 4 P i ki P k 4S 2 4 ki P k 2 S q 1, 45 which establishes a direct connection between entangleent and entropy. The quantity E (eff) q is equivalent to the entangleent, in the sense that it increases or decreases upon an increase or decrease of the entangleent strength. Furtherore, it is equal to 1 when the entangleent is 1, and tends to vanish with the entangleent easure tending to zero. The key ingredient of this heuristic forula is the ter 4 arctan, and the factor i1 (P i ki P k )(4S 2 () 4 ki P 2 k ). Without arctan, the condition P 1 would generate divergencies. Furtherore with P (1/2) the inverse of the entropy would tend to a iniu which would be different fro 0, which is the right value. With the factor 4 i (P i ki P k )4S 2 ()4 ki P 2 k, we dispose of the divergencies and we succeed in ensuring that the quantity E (eff) q tends to vanish with the entangleent tending to zero. Note that this ad hoc factor is nothing but the square of the concurrence. In principle, one could express the concurrence in ters of S 2, but this would not afford the attractive condition of the entangleent being a onotonically increasing function of the inverse of the nonextensive entropy. In conclusion, we find that entangleent increase iplies entropy decrease, and vice versa. This property ust be copared with the results of the work of Abe and Rajagopal 15. These authors adopted the principle of entropy axiization under suitable constraints to infer a plausible for of physical state, and concluded that the entangled states are the iportant result of this axiization process. Here we adopt a different perspective, based on the fact that the definition of entangleent of foration is already inspired to statistical echanics 20. Within this perspective the state of axiu entangleent corresponds to the iniu aount of inforation necessary to describe the state. Within this sae perspective, the aount of inforation necessary to describe the state becoes increasingly larger upon reducing the entangleent strength. Fro an intuitive point of view, the occurrence of decoherence, is judged by any authors 25 to be the key condition to derive classical fro quantu physics, iplies a significant entropy increase. However, decoherence, as a for of real wave-function collapse 26, iplies the breakdown, in the long-tie liit, of the entangleent condition, and, as a consequence, the breakdown of the theory itself of the present paper. The result of this paper has to be considered within this perspective. As it appears fro the literature on this new and exciting subject, the therodynaic significance of the processes of quantu teleportation is a very delicate and difficult issue. We are inclined to believe that the adoption of a nonextensive for of entropy ight be of soe relevance, under specific restrictions. The first is that real wave-function collapses are ignored, and the second is that, in a world doinated by quantu entangleent, the condition of axiu entangleent is perceived

8 FILIPPO GIRALDI AND PAOLO GRIGOLINI PHYSICAL REVIEW A as that requiring the iniu aount of inforation. In other words, increasing entangleent eans saller, rather than larger, entropy values. IV. TSALLIS ENTROPY AT WORK: DEPHASING PROCESSES IN THE BELL BASIS SET Before ending this paper, it is convenient to illustrate another interesting result that does not require a restriction to the set I. This has to do with an iportant result obtained by Bennett et al. 21. These authors studied entangleent changes as a function of a dephasing process. More precisely, they focused their attention on the transforation D B 1 U 4 i U i, i0 46 which brings the initial condition described by the density atrix, expressed in the Bell basis, into the diagonal for D B ij ij ij, 47 where the operators U i, i0,1,2, are I, B x B x, B y B y, and B z B z, respectively, and B i is the bilateral rotation of /2 around the ith axis of the space S (1) 1/2 S (2) 1/2. This bilateral rotation was defined by these authors 21 as B i 1 2 I 22 i 1i I 22 i 2i. 48 Note that the atrix D B of Eq. 47 is the diagonal of the statistical density atrix expressed in the Bell basis, and that it results fro a rando application of four local unitary transforations, so that oving fro the initial state (1) to the state described by D B (1) the entangleent cannot increase 27. Consequently, we have E F (1) E F D B (1). 49 We shall analyze these theoretical results by eans of nonextensive entropy. The first analysis is ade by focusing our attention on the natural values n1 of the entropy index q. In this special case the nonextensive entropy reads as follows: S n 1TrUU n Tr n n1 n1. Let us define the auxiliary function g n x xxn n We note that this is a concave function. On the other hand, several years ago Wehrl 28 note this in that case we can write Tr g n D B Tr g n. 52 We note that S n () Tr g n () and S n (D B ) Tr g n (D B ). Consequently, we can write S n D B S n. 5 The dephasing process akes it possible for us to generalize the results of Sec. III. Let us consider a transforation fro an initial state described by a generic density atrix. As to the final state, we set the condition that it belongs to the set I. Let p (1) be the axiu of the diagonal eleents of the initial state (1), expressed in the Bell basis set. Let us also suppose that p (1) is larger than the axiu eigenvalue of the density atrix (2), referring to the final state. This condition is expressed by the relation p (1) ax (1) ii P (2) 1 2. i1,2,,4 As a consequence of this relation, we have E F (1) E F D B (1) E F (2) This is a transforation with a decreasing entangleent. On the basis of the results of Sec. III and of Eq. 5, we are in a position to find values of the entropy index q such that the nonextensive entropy of the final state is larger than that of the initial state. This is done as follows. We ove fro the initial condition (1) to D B (1), through the dephasing process earlier described. As we have seen, with the adoption of natural values, larger than the unity, for the entropy indices, the entropy does not decrease. This eans that S n D B (1) S n (1). 56 According to our assuptions, D B (1) and (2) belong to the set I. Thus we know, on the basis of the results of Sec. III, that there exists a critical value of the entropy index, Q(D B (1), (2) ), beyond which the nonextensive entropy increases. If we choose critical values of the entropy index that are natural nubers larger than NQD B (1), (2), 57 we conclude that the nonextensive entropy increases. As earlier anticipated, this has the effect of aking ore general the results of Sec. III. V. CONCLUSIONS This paper shows that in the set I, enforcing the iportant condition of a nonvanishing entangleent, the Tsallis entropy is a onotonic and decreasing function of the increasing entangleent. The entangleent is, in turn, a onotonic and decreasing function of the increasing entropy under the key restriction of transforations yielding an entangleent change. This conclusion was reached adopting a perspective taking the warning of a recent paper by Horodecki et al. 27 into account. As a atter of fact, these authors showed that the principle of entropy axiization yields fake entangleent, and consequently becoes questionable. We share the conviction of these authors and adopt in fact an approach that does not rest on the Jaynes principle 18,19. Thus we establish a coparison between entangleent and nonexten

9 QUANTUM ENTANGLEMENT AND ENTROPY PHYSICAL REVIEW A dp 1 P q1 d 0. A1 sive entropy without invoking the Jaynes principle. We do not need to axiize entropy after iniizing entangleent, as in Ref. 27, and the onotonic dependence of entropy on entangleent is a natural consequence of the adoption of suitably large entropy indices. This eans that we share the view of Rajagopal and Abe 15 that a nonextensive for of entropy can prove to be a convenient tool to study quantu teleportation. In this sense, this paper contributes to deepening our understanding of the significance of the Tsallis entropy. This entropy indicator does not split into the su of two independent contributions, when applied to a syste consisting of two uncorrelated subsystes. This suggests that this kind of entropy ight be a proper theoretical tool only when applied to cases where a repartition into two uncorrelated systes is ipossible. Quantu-echanical systes, in principle, are significant exaples where this condition applies, if environental decoherence, or other kind of decoherence processes, is ignored. In this condition the Tsallis entropy, according to the ain result of this paper, sees to work properly, provided that the warning of Ref. 27 is taken into account. This is where our procedure departs fro the point of view of Rajagopal and Abe 15. Their approach was still based on the Jaynes principle, suppleented by the choice of a suitable additional constraint, concerning the fluctuations around the average, as well as the ordinary constraint on the ean value also see Ref. 29. This procedure yields convincing, although nongeneral conclusions. Our approach, which unfortunately shares the lack of generality of Ref. 15, is based on a different perspective, aiing at identifying the inverse of entangleent with the non-extensive entropy. We think the alternative perspective adopted in the present paper ight contribute, as Refs. 15,29 do, to a better understanding of the therodynaic nature of entangleent. We are afraid that the nonextensive entropy ight becoe inefficient when we leave the physical condition where the no-cloning theore and the principle of no-increasing entangleent, recently found by Horodecki and Horodecki 0, is broken. According to these authors the occurrence of real wave-function collapses, incopatible with the restriction of adopting unitary transforations, provokes a breakdown of this equivalence. In our opinion, the occurrence of real wave-function collapses is incopatible with the restriction of working on the set I, which enforces the condition of a nonvanishing entangleent. Thus we expect that in this case the theory of this paper, and with it the nonextensive entropy, does not work. To explore the uncertain border between quantu and classical echanics we probably need to adopt a still ore advanced perspective. This critical value q () fulfilling the condition of Eq. A1 is also not unique. We therefore adopt a criterion to estiate one of the possible critical values. This will iply that the resulting Q( 1, 2 ) is not unique, but, as shown below, we shall be able to find at least one of the values fulfilling the earlier entioned properties of Q( 1, 2 ). The choice that we adopt to find one of the possible q () sisas follows. We set the inequality dp 1 P q1 d 1 A2 for every value of the subscript j running fro 1 to. We assue that this property holds true for any qq (). This set of conditions, after easy algebra, yields q ax1, 1, 2,. A As for the definition of j (), with the subscript j running fro 1 to, we ust distinguish two cases. The first is the case when the constraints hold true. In this case, we set 0 and d /0 A4 j 1 ln P 1 ln dp 1 d. A5 If the constraints of Eq. A4 do not apply, i.e., either ()0 ord /0 applies, we set j ()1, in accordance with the inequality A2 which is true for every value of the entropic index q different fro unity. The conclusion of this procedure is that we built up the auxiliary function q () in such a way that for any qq () the condition yields d E F0 d S q0. A6 A7 On the basis of this result, the function Q ( 1, 2 ), defined in Eq. 2, APPENDIX This appendix is devoted to proving the crucial properties of the critical values defined by Eqs Let us consider case a first. As a consequence of de F /0, fro the liit of Eq. 21 we naturally obtain that an auxiliary function, q () in Eq. 25, exists such that qq () yields Q 1, 2 sup [ 1, 2 ] q, A8 has the properties described in Sec. III D. In fact, using Eq. A6 and the ensuing inequality for ds q /, we iediately conclude that for any q fulfilling the inequality q Q ( 1, 2 ) the condition E F 0 yields S q 0. This is

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