Summary of professional accomplishments

Transcription

1 Summary of professional accomplishments. Name and surname: Zbigniew Walczak 2. Diplomas and scientific degrees: MSc in theoretical physics Faculty of Mathematics, Physics and Chemistry, University of d, 993 On q-deformation of Hamilton s equations PhD in physics Faculty of Physics and Chemistry, University of d, 998 Non-algebraic approach to the problem of quasi-exact solvability 3. History of employment: Faculty of Physics and Chemistry, University of d, Faculty of Physics and Applied Informatics, University of d, since Achievements pursuant to art. 6 (2) of the Act of 4 March 2003 on academic degrees and titles and on arts degrees and titles (Journal of Laws no. 65, item 595, as amended): For achievement as grounds for the habilitation procedure I present a monothematic cycle of five publications [, 2, 3, 4, 5] entitled Information-theoretic approach to the problem of classical and quantum correlations present in mixed quantum states Z. Walczak, Total correlations and mutual information, Physics Letters A 373 (2009), 88. Z. Walczak, Comment on Quantum correlation without classical correlations, Physical Review Letters 04 (200), Z. Walczak, Information-theoretic approach to the problem of detection of genuine multipartite classical correlations, Physics Letters A 374 (200), M. Okrasa, Z. Walczak, Quantum discord and multipartite correlations, Europhysics Letters 96 (20), M. Okrasa, Z. Walczak, On two-qubit states ordering with quantum discords, Europhysics Letters 98 (202),

2 Introduction In quantum information theory, the problem of characterization of correlations present in a quantum system has been intensively studied both theoretically and experimentally for over two decades (for review, see [6, 7, 8]). For almost twenty years the Werner paradigm [9] based on the entanglement-separability dichotomy dominated in this field of research. Within this paradigm the only kind of quantum correlations is quantum entanglement [6]. However, it has become clear gradually that the Werner paradigm is too narrow and needs reconsideration because, contrary to common intuition, separable quantum states can have non-classical correlations beyond quantum entanglement (see, e.g. [0,, 2, 3, 4, 5, 6]). The first step in this direction has been made by Ollivier and Zurek [7], who introduced the concept of quantum discord as a measure of non-classical correlations present in bipartite quantum states. Independently of Ollivier and Zurek, a similar way of shifting the Werner paradigm has been proposed by Henderson and Vedral [8] who studied the problem of coexistence of classical and quantum correlations present in bipartite quantum states. The Ollivier Zurek paradigm has received increased attention after the recent discovery [9, 20] that quantum discord can be responsible for the remarkable computational efficiency of Knill Laflamme algorithm [0]. In the context of unitary evolution of composite quantum systems, it was shown that if two subsystems are initially in a zero-discord state, then a subsystem dynamics is completely positive [2, 22]. It was also discovered that a random quantum state possesses in general strictly positive discord and an arbitrarily small perturbation of a zero-discord state generates quantum discord [23]. Moreover, it was shown that quantum states with non-zero quantum discord cannot be locally broadcast [24]. It is worth emphasizing that in the case of pure entangled states quantum correlations are limited to quantum entanglement 2, however in the case of mixed entangled states there is no simple relation between quantum entanglement and quantum discord 3 [25, 26, 27, 28, 29]. Moreover, it was shown that the dynamics of quantum discord is essentially different than the dynamics of quantum entanglement [30, 3, 32, 33, 34, 35, 36]. Recently, operational interpretation of quantum discord has been provided [37, 38], significant progress has been also made toward understanding of connections between quantum discord and quantum entanglement irreversibility [39], connections between quantum discord and distillable quantum entanglement [40], and connections between quantum discord and distributed quantum entanglement The paper [7] was cited 29 times until 2008 and 505 times since 2009 (according to Web of Science). 2 In this case quantum discord is equal to quantum entanglement. 3 In general quantum discord may be smaller than quantum entanglement. 2

3 [4, 42, 43]. Moreover, it was shown that quantum discord, much like quantum entanglement, is not monogamic [44, 45] and similarly to quantum entanglement, the concept of quantum discord witness can be introduced [46, 47]. Unfortunately, in general evaluation of quantum discord requires the use of complicated optimization procedure, thus the analytical expressions for quantum discord are known only for two-qubit Bell-diagonal states [25], for seven-parameter two-qubit X-states [26], for two-mode Gaussian states [48, 49], for a class of two-qubit states with parallel non-zero Bloch vectors [50], two-qubit Werner states [5] and for two-qubit isotropic states [5]. That is why Dakić, Vedral and Brukner [52] introduced the concept of geometric quantum discord as an alternative to quantum discord measure of quantum correlations present in bipartite quantum states. Since the geometric quantum discord involves a simpler optimization procedure than quantum discord, the analytical expression for geometric quantum discord was obtained for arbitrary two-qubit quantum states [52] as well as for arbitrary qubit-qudit quantum states [53, 54]. As with quantum discord, geometric quantum discord became a subject of intensive study [8]. In particular, it was checked whether geometric quantum discord may be responsible for the remarkable computational efficiency of Knill Laflamme algorithm [0]. Moreover, the dynamics of geometric quantum discord [55, 56, 57, 58, 59, 60] and relations between geometric quantum discord and other measures of non-classical correlations [6, 27, 62, 63, 64] were investigated. The aim of presented cycle of publications [, 2, 3, 4, 5] is to analyze the information-theoretic aspects of classical and quantum correlations present in bipartite and multipartite quantum states. 2 Quantum correlations without classical correlations In [65] it was shown that, for odd n 3, the n-qubit state ρ 2...n = ( W W + 2 W W ), where W = n ( ) and W = n ( ) has genuine n-partite quantum correlations, understood as quantum entanglement, and moreover all covariances Cov(X,..., X n ) = (X X ) (X n X n ) for traceless observables X,..., X n vanish for this state. According to the authors of [65], if for some choice of local observables X,..., X n the covariance Cov(X,..., X n ) is non-zero, then a n-qubit state has genuine n-partite classical correlations. Moreover, if for all choices of local observables the covariance is zero, then a n-qubit state has no genuine n-partite classical correlations. Using the latter claim as a criterion of non-existence of genuine n-partite classical correlations in a n-qubit state, they came to an incorrect conclusion that the state ρ 2...n has only 3

4 genuine n-partite quantum correlations. It is well-known that if the measurement outcomes of observables X,..., X n are independent, then the covariance Cov(X,..., X n ) = 0. However, the converse is not necessarily true the covariance can be zero without implying independence of the measurement outcomes. This means that the criterion of non-existence of genuine n- partite classical correlations in a n-qubit state, used by the authors of [65], is necessary but not sufficient. In the paper [2] I showed that the state ρ 2...n has genuine n-partite classical correlations, despite the fact that all covariances Cov(X,..., X n ) vanish. Let us assume that X = X 2 = = X n = X n = σ z. The probability that the measurement outcome of X i is ± is given by p(x i = ±) = Tr[ 2 (I ± σ z)ρ i ] = 2, where ρ i = I is the state of the i-th qubit. From the other hand, the joint 2 probability that the measurement outcome of X, X 2,..., X n, X n is,,...,, is given by p(x =, X 2 =,..., X n =, X n = ) = Tr[2 n (I + σ z ) n ρ 2...n ] = Tr[ 0 0 n ρ 2...n ] = 0. Therefore, the measurement outcomes of traceless observables X = σ z, X 2 = σ z,, X n = σ z, X n = σ z are not independent, despite the fact that Cov(X,..., X n ) = 0. This means that the n-qubit state ρ 2...n has not only genuine n-partite quantum correlations, contrary to what was suggested in [65]. In this context, the work [66] is worth mentioning where the axiomatic approach to the problem of coexistence classical and quantum correlations in multipartite quantum states was presented. Within this approach, it was shown that the vanishing of covariance cannot be treated as an indicator of the absence of genuine multipartite classical correlations, which shows the advantage of the choice of axioms presented in [66]. 3 Classical correlations in multipartite quantum states In the paper [3] I formulated, in the language of information theory, the reliable criterion of non-existance of classical correlations in multipartite quantum states. Let us note that from the perspective of the authors of [65] classical correlations present in the n-qubit state ρ 2...n should be related to observables X,..., X n and therefore the correlations between random variables corresponding to the measurement outcomes of these observables are the classical correlations present in the state ρ 2...n a similar approach to the problem was presented in [24]. For simplicity, but without loss of generality, in the paper [3] I considered a system consisting of three spins /2 in the state ρ ABC. Moreover, I assumed that spins A, B and C are measured along directions a, b and c a measurement of spin along a given direction n corresponds to the von Neumann measurement of the observable S n = n σ, where σ = (σ x, σ y, σ z ). 4

5 Then I showed that the binary random variables A, B and C with alphabets a = {, }, b = {, } and c = {, }, and probability mass functions p A = [p(a)], p B = [p(b)] and p C = [p(c)] where p(a) = Tr[ 2 (I + as a)ρ A ], p(b) = Tr[ 2 (I + bs b)ρ B ] and p(c) = Tr[ 2 (I + cs c)ρ C ] correspond to the measurements of S a, S b and S c in the state ρ ABC. In classical information theory the dependence or redundancy between given two random variables A and B is measured by mutual information I(A : B) (eq. (8) in [3]), because I(A : B) 0 with equality iff A and B are independent [67]. Therefore, the vanishing of the mutual information between any two random variables can be used as the information-theoretic criterion of their independence in other words, if I(A : B) > 0, then A and B are correlated. It would seem that if three random variables are pairwise independent then they have to be independent, but in the paper [3] I showed by the following example that the pairwise independence of three random variables does not necessarily imply their independence. Let us consider three spins /2 in the state ϱ ABC = ( ). In this case, the measurement outcomes of traceless observables S a, S b and S c, and thereby the random variables A, B and C, are pairwise independent for all directions a, b and c, because ϱ AB = ϱ AC = ϱ BC = 2 I 2 I. However, they are not independent, because for instance if a = b = c = (0, 0, ), then p(a =, b =, c = ) = 4 p(a = )p(b = )p(c = ) = 8. In the paper [3] I also showed that the information-theoretic criterion of independence of two random variables can be extended in a natural way to the case of three random variables, if we take into account that the mutual information is only a special case of the relative entropy (eq. (9) in [3]) the relative entropy of the joint probability mass function p AB = [p(a, b)] with respect to the product of the marginal probability mass functions p A p B, i.e. I(A : B) = D(p AB p A p B ). In the later part of the paper [3] I explained why from the information-theoretic point of view pairwise independence of three random variables does not necessarily imply their independence, i.e. why vanishing of mutual informations I(A : B), I(A : C) and I(B : C) is a necessary but not sufficient condition for independence of random variables A, B and C. Moreover, I explained why from the information-theoretic point of view vanishing of mutual informations I(C : A, B), I(B : A, C) and I(A : B, C) is a necessary and sufficient condition for independence of random variables A, B and C. Therefore, we conclude that (i) state ρ ABC can have genuine classical correlations despite the fact that it has no bipartite classical correlations, (ii) state ρ ABC is a product state, i.e. ρ ABC = ρ A ρ B ρ C iff for all possible directions a, b and c the measurement outcomes of observables S a, S b and S c are uncorrelated across any 5

6 bipartite cut: A : BC, B : AC or C : AB, (iii) state ρ ABC has no genuine classical correlations iff for all possible directions a, b and c the measurement outcomes of observables S a, S b and S c are uncorrelated across a bipartite cut. It is worth noticing here that the similar results can be obtained within the axiomatic approach [66], which is fundamentally different from the informationtheoretic one presented in [3]. Moreover, it is worth pointing out that the results from the paper [3] can be generalized to the multipartite case. 4 Mutual information as a correlation measure In the paper [] I argued that quantum mutual information cannot be considered as a measure of total correlations present in bipartite quantum states, because there exist states for which it does not cover all aspects of total correlations. In the paper [] I showed that there exist classically correlated quantum states for which quantum mutual information is not a proper indicator of strength of correlations present in those states. As an example, I considered the following two-qubit state ρ AB = α ( α) where α (0, ) (eq. (5) in []). This state cannot have quantum correlations because it is separable [9] and the qubit A (B) is in the state ρ A(B) = α ( α) which is a mixture of orthogonal states 0 and. This means that the state ρ AB has only classical correlations [7]. Let us note that classical correlations present in the state ρ AB, understood as correlations between two orthogonal states of qubits A and B, are simply correlations between binary random variables A and B corresponding to the von Neumann measurements of two observables M A = a a and M B = b b, or measurements of the state of the qubits in the computational basis if the measurement outcome of M A (M B ) is a i (b i ), then the qubit A (B) is in the state i, from the other hand a i (b i ) are the values of random variable A(B) with the probability mass function p A(B) = [p A(B) i ], where p A(B) i is the probability that the measurement of M A (M B ) gives a i (b i ) []. A M A ρ AB M B B Next, I showed that if the measurement of M A is performed before the measurement of M B, then the measurement outcomes, and thereby random variables A and B, are perfectly correlated, as illustrated in the diagram below, where p B A j i denotes the conditional probability that the measurement outcome of M B is b j provided that the measurement outcome of M A was a i (see eqs. (6) and (7) in []). From the other hand, I showed that in this case mutual information I(A : B) (eq. (0) in []) can be arbitrarily small (as illustrated in the figure below), despite the fact 6

7 p A i α α A p B A j i B a 0 b 0 a b p B j α α that random variables A and B are perfectly correlated, therefore quantum mutual information (eq. (9) in []), which in this case is equal to mutual information I(A : B), does not give the correct answer to the question how strong the correlations present in the state ρ AB = α ( α) are. I A:B Α This is because mutual information I(A : B) is the average information gain about the measurement outcome of M B (M A ) due to the knowledge of the measurement outcome of M A (M B ), in other words mutual information tell us how many perfectly correlated bits can be, in average, carried by one pair of the measurement outcomes. From the information theory point of view mutual information can be arbitrarily small, despite the fact that random variables are perfectly correlated. It follows directly from the fact that the average information gain about the measurement outcome of M A (M B ) is given by Shannon entropy H(A) (H(B)), which can be arbitrarily small and from the other hand I(A : B) Min(H(A), H(B)). Therefore, a natural question arises: what is an information-theoretic measure of the strength of correlations between random variables A and B? For a pair of random variables A and B with identical probability mass functions Cover and Thomas [67] proposed the following measure of the strength of correlations C(A, B) = I(A : B)/H(A) 4 (eq. () in []). Let us note that in the case under consideration, p A = p B and I(A : B) = H(A) [], therefore C(A, B) =, i.e. A and B are perfectly correlated for all α (0, ), as it was shown above. In the later part of the paper [] I generalized the Cover Thomas measure of correlations to the case when the probability mass functions are not identical. To do this I considered the following two-qutrit state ρ AB = (eq. (2) in []) which does not have quantum correlations, because it 3 4 The Cover Thomas measure of correlations has the following properties (i) C(A, B) = C(B, A), (ii) 0 C(A, B), (iii) C(A, B) = 0 iff A and B are independent, and (iv) C(A, B) = iff A and B are perfectly correlated. 7

8 is separable [9] and qutrits A and B are in the states ρ A = and 3 3 ρ B = being a mixture of orthogonal states Let us note that classical correlations present in the state ρ AB are simply the correlations between ternary random variables A and B corresponding to the von Neumann measurements of two observables M A = a a + a and M B = b b + b Next I showed that in this case p A p B (see eqs. (3) in []) and the correlations strength between random variables A and B depends on the temporal order of the measurements of observables M A and M B, as illustrated in the following diagrams. p A i A p B A j i B p B j p B i B p A B j i A p A j a 0 b 0 a b 2 a 2 2 b b 0 a 0 b a b 2 a In particular, I showed that if the measurement of M A is performed before the measurement of M B, then the correlations strength between random variables A and B is measured by I(A : B)/H(B), but if the measurement of M B is performed before the measurement of M A, then the correlations strength is measured by I(A : B)/H(A) in the case under consideration we have I(A : B)/H(B) and I(A : B)/H(A) = (see eqs. (8) and (22) in []). Taking into account the fact that the correlations strength between random variables A and B can depend on the temporal order of measurements of observables M A and M B, I proposed the following measure of the correlations strength between arbitrary random variables C(A, B) = I(A : B)/Min(H(A), H(B)) (eq. (23) in []) and I showed that this measure has the same properties as the Cover Thomas measure of correlations. Therefore, C(A, B) is a measure of strength of correlations present in a classically correlated quantum state ρ AB [24], if we assume that classical correlations present in this state are simply the correlations between the outcomes of the von Neumann measurements of the state of subsystems A and B, in the computational basis. It is worth noting here that C(ρ AB ) = max MA,M B C(A, B) is a measure of strength of classical correlations present in a bipartite quantum state ρ AB, where random variables A and B correspond to the von Neumann measurements described by complete sets of one-dimensional orthogonal projectors {Π A i } and {Π B i }, i.e. the von Neumann measurements of observables M A = i a iπ A i and M B = i b iπ B i. 8

9 5 Quantum discord and multipartite correlations In the paper [4] I provided the systematic analysis of quantum and classical correlations present in both bipartite and multipartite quantum states in the framework of the Ollivier Zurek paradigm 5 [7]. The starting point was to prove that according to what was suggested in [68] quantum discord D A(B) (ρ AB ) 6 is equal to the minimal amount of correlations that are lost during the non-selective von Neumann measurement M A(B) {Π performed on i } the subsystem A(B), i.e. D A (ρ AB ) = inf [I(ρ A(B) {Π i } AB) I(M (ρ A(B) {Π i } AB))], where M {Π A i }(ρ AB ) = i (ΠA i I)ρ AB (Π A i I) and M {Π B i }(ρ AB ) = i (I ΠB i )ρ AB (I Π B i ) (eq. (7) in [4]); the measurement M for which infimum is attained is called the { ΠA(B) i } optimal non-selective von Neumann measurement. In this context a natural question arises: what type of correlations are lost during the optimal non-selective von Neumann measurement? In the paper [4] I showed that the optimal non-selective von Neumann measurement M { ΠA i } does not cause the loss of classical correlations present in the state ρ AB, but it causes only the loss of quantum correlations, because D A (M { ΠA i } (ρ AB)) = 0 and C A (M { ΠA i } (ρ AB)) = C A (ρ AB ) 7 (eq. (2) in [4]), where M { ΠA i } (ρ AB) = i pa Π i A i (eq. (3) in [4]) is the post-measurement state of the system. ρ B i Here another question arises: is it possible that the zero-discord state may still have quantum correlations? At first glance the question might seem rhetorical, but in the paper [4] I brought to attention the fact, previously ignored in this context, that the state M { ΠA i } (ρ AB) = i pa i Π A i ρ B i can have quantum correlations, despite the fact that D A (M { ΠA i } (ρ AB)) = 0, because according to classification of bipartite quantum states [24], if the states ρ B i commute, then the state M { ΠA i } (ρ AB) has only classical correlations, otherwise the state M { ΠA i } (ρ AB) has classical and quantum correlations. Therefore, another natural question arises: what amount of quantum correlations remain after the optimal non-selective von Neumann measurement M { ΠA i }? First, let us note that the optimal non-selective von Neumann measurement M { ΠB j } does not lead to the loss of classical correlations present in the state M { ΠA i } (ρ AB), 5 Within the Ollivier Zurek paradigm total correlations present in a bipartite state ρ AB are measured by quantum mutual information I(ρ AB ), but quantum discord D A(B) (ρ AB ) and C A(B) (ρ AB ) = I(ρ AB ) D A(B) (ρ AB ) are non-symmetric measures of quantum and classical correlations, respectively. 6 D A(B) (ρ AB ) = inf A(B) {Π i } [I(ρ AB) J A(B) {Π i } (ρ AB)], where I(ρ AB ) is quantum mutual information of ρ AB, and J A(B) {Π i } (ρ AB) is quantum mutual information induced by the von Neumann measurement, described by a set of one-dimensional orthogonal projectors {Π A(B) i }, performed on the subsystem A(B) (notation form [4]). 7 C A(B) (ρ AB ) = sup A(B) {Π i } J {Π A(B) i } (ρ AB) is the Henderson Vedral measure of classical correlations [8] (notation from [4]). 9

10 but only to the loss of quantum ones because D B (M { ΠB j } (M { Π A i }(ρ AB))) = 0 and C B (M { ΠB j } (M { Π A i }(ρ AB))) = C B (M { ΠA i } (ρ AB)) (eq. (6) in [4]). Moreover, according to classification of bipartite quantum states [24], the state of the system after the measurement M { ΠB j }8 does not have quantum correlations, but only classical ones, that were present in the state ρ AB. Thus, we conclude that the state M { ΠA i } (ρ AB) had as many quantum correlations as were lost during the optimal non-selective von Neumann measurement M { ΠB j }, i.e. D B(M { ΠA i } (ρ AB)). Therefore, within the Ollivier Zurek paradigm a measure of the overall quantum correlations present in a state ρ AB can be introduced in natural way as the sum of quantum correlations which are lost during the optimal non-selective von Neumann measurements M { ΠA i } and M { Π B j }, Q(ρ AB) = D A (ρ AB )+D B (M { ΠA i } (ρ AB)) (eq. (7) in [4]), because the subsequent optimal measurement lead only to the loss of all quantum correlations present in the state ρ AB, leaving classical correlations unaffected. Let us note that the measure of the overall quantum correlations Q(ρ AB ) can be rewritten in the following form Q(ρ AB ) = I(ρ AB ) C B (M { ΠA i } (ρ AB)) (eq. (8) in [4]), which means that within the Ollivier Zurek paradigm C(ρ AB ) = C B (M { ΠA i } (ρ AB)) (eq. (9) in [4]) is a measure of the overall classical correlations present in the state ρ AB. In the paper [4] I showed that in general quantum discord D A(B) (ρ AB ) is not greater than the overall quantum correlations Q(ρ AB ), while the Henderson Vedral measure of classical correlations C A(B) (ρ AB ) is not less than the overall classical correlations C(ρ AB ). Moreover, I proved that both Q(ρ AB ) and C(ρ AB ) are symmetric measures of correlations (see eqs. (2) and (22) in [4]). It is worth emphasizing that using the above approach to the problem of correlations one can easily explain the results presented in [33, 69]. In the later part of the paper [4] I introduced in a natural way the concept of quantum discord D Ak (ρ A ) 9 as a measure of quantum correlations present in m-partite quantum state ρ A, which allows one to deliver the systematic analysis of classical and quantum correlations present in multipartite quantum states. First, let us note that quantum discord D Ak (ρ A ) is equal to the minimal amount of correlations which are lost during the non-selective von Neumann measurement M A, i.e. D {Π k i } A k (ρ A ) = inf A [I(ρ {Π k i } A) I(M A (ρ {Π k i } A)], where M A (ρ {Π k i } A) = i (I ΠA k i I)ρ A (I Π A k i I). Moreover, the optimal non-selective von Neumann measurement M { ΠA k does not cause the loss of classical correlations i } present in the state ρ A, but only the loss of quantum correlations [4]. 8 M { ΠB j } (M { Π A}(ρ AB)) = ij pab ij Π A i Π B j, where pab ij = Tr[( Π A i Π B j )ρ AB] is the probability i of getting the results i and j, respectively (eq. (5) in [4]). 9 D Ak (ρ A ) = inf A [I(ρ {Π k i } A) J A (ρ {Π k i } A)], where I(ρ A ) is quantum mutual information of ρ A (A denotes A A 2... A m A m ), and J A (ρ {Π k i } A) is quantum mutual information induced by the von Neumann measurement, described by a set of one-dimensional orthogonal projectors {Π A k i }, performed on the subsystem A k. 0

11 Next, I showed that the subsequent optimal non-selective von Neumann measurements M,..., M { ΠA i } { Π A m i } lead to the corresponding loss of quantum correlations D A (ρ A ), D A2 (M { ΠA i } (ρ A)), which may be present in the states D A3 (M { ΠA 2 i 2 } (M { Π A i } (ρ A))),. D Am (M { ΠA m i m } (... (M { Π A i } (ρ A)))) ρ A, M { ΠA i } (ρ A), M { ΠA 2 i 2 } (M { Π A i } (ρ A)), leaving classical correlations unaffected [4].. M { ΠA m i m } (... (M { Π A i } (ρ A))), Therefore, we see that within the Ollivier Zurek paradigm Q(ρ A ) = D A (ρ A ) + D A2 (M { ΠA i } (ρ A)) + D A3 (M { ΠA 2 i 2 } (M { Π A i } (ρ A))) + + D Am (M { ΠA m i m } (... (M { Π A i } (ρ A)))) is a measure of the overall quantum correlations present in m-partite quantum state ρ A (eq. (35) in [4]). Let us note that the measure of the overall quantum correlations Q(ρ A ) can be rewritten in the following form Q(ρ A ) = I(ρ A ) C Am (M { ΠA m i m } (... (M { Π A i } (ρ A)))) (eq. (36) in [4]), which means that within the Ollivier Zurek paradigm C(ρ A ) = C Am (M { ΠA m i m } (... (M { Π A i } (ρ A)))) is a measure of the overall classical correlations present in the state ρ A (eq. (37) in [4]). In the paper [4] I showed that in general quantum discord D Ak (ρ A ) is not greater than the overall quantum correlations Q(ρ A ) the m-partite zero-quantum state D Ai (ρ A ) can have quantum correlations, while the Henderson Vedral measure of classical correlations C Ak (ρ A ) 0 is not less than the overall classical correlations C(ρ A ). Moreover, I proved that Q(ρ A ) and C(ρ A ) are symmetric measures of correlations (see eqs. (38) and (39) in [4]). 0 C Ak (ρ A ) = sup {Π A k i J } {Π A (ρ k i } A).

12 6 Quantum states ordering with quantum discords In the paper [5] I studied the problem of two-qubit Bell-diagonal states ordering with quantum discord and geometric quantum discord. Let us note that in general there may exist bipartite quantum states ρ AB and ρ AB for which the states ordering with respect to quantum discord D A (ρ AB ) differs from that given by geometric quantum discord D G A (ρ AB). In other words, in general there may exist quantum states ρ AB and ρ AB for which the following condition D A (ρ AB ) ( )D A (ρ AB) D G A(ρ AB ) ( )D G A(ρ AB) is not satisfied (eq. () in [5]), i.e. the states ordering with quantum discords is violated. Recently, two-qubit quantum states for which the states ordering with quantum discord is violated were discovered [57], which shows that the lack of the unique ordering of states with quantum entanglement measures [70, 7, 72, 73, 74, 75, 76, 77, 78, 79, 80] goes beyond entanglement. Therefore, the problem of finding bipartite quantum states for which the states ordering with quantum discords is preserved naturally arises. It turned out that the general solution of this problem, i.e. the finding of all quantum states for which the states ordering is preserved, is very difficult even in the case of two-qubit Bell-diagonal states 2 [8, 25] for which the analytical expression for both quantum discord D A (ρ AB ) [25] (eq. (6a) in [5]) and geometric quantum discord D G A (ρ AB) [52] (eq. (6b) in [5]) is known. In the paper [5] I showed that if ρ AB and ρ AB belong to one of twelve two-parameter families of states (corresponding to twelve triangles in the first figure on the next page) c i = 0, c j 0.5, c k + c j, c l = 0, 0.5 c m, c n c l, where i j k and l m n, then the states ordering with quantum discords is preserved (eqs. (5) in [5]). In particular, I showed that if ρ AB and ρ AB belong to the two-parameter family of states: c = 0, c 2 0.5, c 3 + c 2, then the states ordering is preserved, because for given c 2 both quantum discords are convex functions of c 3 with D G A (ρ AB) = inf χab ρ AB χ AB 2 wherethe infimum is over all zero-discord states, D A (χ AB ) = 0, and is the Hilbert Schmidt norm, A = Tr(A A) (notation form [5]). 2 The Bell-diagonal states have the following form ρ AB = 4 (I I + 3 i= c i σ i σ i ), where σ i are the Pauli spin matrices, and coefficients c i R fulfill the following conditions 0 4 ( c c 2 c 3 ), 0 4 ( c + c 2 + c 3 ), 0 4 ( + c c 2 + c 3 ), 0 4 ( + c + c 2 c 3 ). These inequalities describe a tetrahedron with vertices (,, ), (,, ), (,, ) and (,, ) corresponding to the Bell states ψ + = 2 ( ), ψ = 2 ( 0 0 ), ϕ + = 2 ( 00 + ), ϕ = 2 ( 00 ) (see the first figure on the next page). 2

13 .0 c c c A, A G c c local minimum at the same point and D A (ρ AB ) DA G(ρ AB) (D A black line, DA G grey line, see figure above) [5]. Let us note that in a similar way, it can be shown that the ordering of states is preserved for the rest of families of states [5]. Moreover, in the paper [5] I showed that if ρ AB and ρ AB do not belong to one of twelve two-parameter families of states, then one can find both the states for which the states ordering is violated and the states for which it is preserved. For example, in the case of one-parameter family of states: c 0, c 2 = c, c 3 = the states ordering is preserved (the left figure below), however in the case of one-parameter family of states: c = 0.5, c 2 = 0.5, 0 < c 3 the states ordering is violated (the right figure below). The fact that, in particular, one can find the states for which the states ordering is violated despite the fact that they belong to different two-parameter families of 3

14 G A, A.0 A, A G c c 3 states listed above, deserves special attention this is why finding of the general solution of the problem of the two-qubit Bell-diagonal states ordering with quantum discords is a challenging issue [5]. For example, in the case of states ρ AB and ρ AB with c = 0., c 2 = 0, c 3 = 0.75 and c = 0., c 2 = 0, c 3 = 0.9, respectively the states ordering is violated, because D A (ρ AB ) < D A (ρ AB ) and DG A (ρ AB) = DA G(ρ AB ) despite the fact that ρ AB belongs to the family of states: c 2 = 0, c 3 0.5, c + c 3 while ρ AB belong to the family of states: c 2 = 0, 0.5 c 3, c c 3 [5]. In the paper [5] I also noticed that a numerical comparison between quantum discord and geometric quantum discord for general two-qubit states [62, 6] leads one to the conclusion that there exist other states, except the Bell-diagonal states and those considered in [57], violating the states ordering with quantum discords. 7 Summary In the presented cycle of papers [, 2, 3, 4, 5] I have developed the information-theoretic approach to the problem of classical and quantum correlations present in both bipartite and multipartite quantum states. In particular, I have found answers for a number of crucial questions that lead to better understanding of the problem of classical and quantum correlations in the framework of both the Werner and the Ollivier Zurek paradigms. Does the n-qubit quantum state ρ 2...n [65] have only genuine n-partite correlations, from the Werner paradigm point of view? In the paper [2] I have explained that the criterion of non-existence of genuine n-partite classical correlations in a n-qubit state that bases on the vanishing of all covariances Cov(X,..., X n ) is the necessary but not sufficient criterion. Moreover, I have shown explicitly that the state ρ 2...n has genuine n-partite classical correlations, despite the fact that all covariances vanish. Is it possible to formulate a reliable information-theoretic criterion of nonexistence of genuine n-partite classical correlations in multipartite quantum states, in the framework of the Werner paradigm? In the paper [], for simplicity but without loss of generality, I have proposed 4

15 such a criterion for tripartite quantum states. In particular, I have explained why tripartite classical correlations can be present in a tripartite quantum state despite the fact that this state has no bipartite classical correlations and I have given an explicit example of such a state. Does quantum mutual information cover all aspects of total correlations present in bipartite quantum states? In the paper [] I have shown that there exist classically correlated quantum states for which quantum mutual information is not a proper indicator of strength of correlations present in those states. In particular, I have found classically correlated quantum states for which mutual information can be arbitrarily small, despite the fact that total correlations present in those states are maximally strong. Moreover, I have proposed an information theoretic measure of the classical correlations strength in classically correlated quantum state, which can be generalized to the case of arbitrary bipartite quantum states. How many classical and quantum correlations are present in bipartite quantum states from the Ollivier Zurek paradigm point of view? In the paper [4] I have shown explicitly that quantum discord D A(B) (ρ AB ) is equal to the amount of correlations that are lost during the optimal nonselective von Neumann measurement performed on the subsystem A(B). I have also shown that the optimal non-selective von Neumann measurement does not cause the loss of classical correlations present in a bipartite quantum state. Moreover, I have introduced, in a natural way, a measure of the overall quantum correlations present in bipartite quantum states and I have shown that quantum discord is not greater than the overall quantum correlations, which means that vanishing of quantum discord does not necessarily imply the vanishing of quantum correlations. Then, I have introduced a measure of the overall classical correlations present in bipartite quantum states and I have shown that the Henderson Vedral measure of classical correlations is not less than the overall classical correlations. How many classical and quantum correlations are present in multipartite quantum states from the Ollivier Zurek paradigm point of view? In the paper [4] I have generalized, in a natural way, the notion of quantum discord to the case of multipartite quantum states. That allowed me to introduce, in a natural way, a measure of the overall quantum correlations present in multipartite quantum states 3, as well as a measure of the overall classical correlations present in those states. 3 Few months later Rulli and Sarandy introduced the notion of global measure of quantum correlations present in multipartite quantum states [82]. 5

16 Does the lack of the unique ordering of states with quantum entanglement measures goes beyond entanglement? In the paper [5] I have studied the problem of two-qubit Bell-diagonal states ordering with quantum discord and geometric quantum discord. In particular, I have identified twelve two-parameter families of states for which the states ordering with quantum discords is preserved as long as the states belong to the same family of states, and I have shown that in the case of states belonging to different families of states the state ordering may be violated, which is counterintuitive and explains why a general solution of the problem of two-qubit states ordering with quantum discords is a very challenging issue. Despite the fact that works included into presented habilitation cycle [, 2, 3, 4, 5] were published during last few years they were cited 8 times, according to Web of Science. It is worth mentioning that results from [2, 4, 5] were presented in a large review paper [8] concerning the problem of correlations present in bipartite and multipartite quantum systems. The paper [5] was featured on the front cover of Europhysics Letters (January March 203), moreover works [2, 5] were selected to the prestigious journal Virtual Journal of Quantum Information. 5. Other scientific accomplishments Before PhD In [83] the q-deformed phase space was described and the q-deformed Hamilton s equations of motion were derived form Hamilton s principle of least action. Papers [84, 85, 86] were devoted to non-algebraic approach to the problem of quasiexact solvability. In [84] it was shown that the Bender Dunne spectral orthogonal polynomials can be constructed for all quasi-exact solvable models. Moreover, a simple way for building the weight functions associated with these polynomials was presented and it was shown that Bender-Dunne polynomials are orthogonal polynomials of a discrete variable, which takes its values in a fnite set of the exactly calculable energy levels of a given quasi-exactly solvable model. In [85] it was shown that certain one-dimensional quasi-exactly solvable models naturally appear in the form of dublets connected by a discrete transformation, which was called anti-isospectral transformation, because it changes the sign of all energy levels belonging to the exactly calculable part of the spectrum of models forming a given dublet. Moreover, it was shown that the duality property enables one to find an upper bound for the part of spectrum occupied by exactly calculable energy levels for the model under consideration. Furthermore, it was shown that the duality property implies the existence of polynomials of the same degree, which form an orthogonal system on two 6

17 different integrals with the same weight function these polynomials are a natural generalization of the standard orthogonal polynomials connected to exactly solvable systems. After PhD In [86] it was shown that for the time-dependent sextic anharmonic oscillator (the simplest example of the time-dependent quasi-exactly solvable model) the Bethe ansatz equations describe a classical complex dynamical system of Calogero Moser type, which means that there is an exact correspondence between the quasi-exactly solvable quantum models and the classical dynamical models. Moreover, it was shown that it is possible to establish the similar correspondence between the quasi-exactly solvable quantum models and the classical matrix models by using Olshanetsky-Perelomov projection method. Papers [87, 88, 89, 90] were devoted to the problem of destruction of states in quantum mechanics. In [87] a description of the destruction of distinguishable as well as identical particles on the grounds of quantum mechanics rather than quantum field theory was proposed. In particular, several kinds of maps called supertraces were defined and used to describe the destruction procedure. Moreover, it was shown that the concept of destruction of states can be treated as a supplement to the von Neumann Lders measurement. In [88] a few illustrative examples were used to show that the destruction of states procedure can be helpful in description of the Einstein Podolsky Rosen type experiments. In [89] it was shown, by finding its Kraus representation, that destruction of states procedure for one qudit is a quantum operation. In [90] the destruction of quantum states during the spin measurement of a particle involving localization of a particle in a detector in a system of two distinguishable particles of spin-/2 was considered. Papers [9, 92] were devoted to the non-relativistic spin correlation function. In [9] the Einstein Podolsky Rosen type experiment was considered and the spin correlation function was calculated, separately for a pair of distinguishable and identical particles with arbitrary spin, taking into account the relative motion of two observers, finite size of detectors and assuming that the spin measurements are not simultaneous. It was the first such detailed analysis of the non-relativistic spin correlation function in the Einstein Podolsky Rosen type experiment. The paper [92] was devoted to finding the non-relativistic spin correlation function in the Einstein Podolsky Rosen type experiment for two distinguishable spin-/2 particles in the triplet state. In papers [93, 94, 95] it was shown that Bogoliubov transformations, widely used in quantum field theory, can be also useful in quantum information theory. Namely, it was shown that the problem of the choice of tensor product decomposition in the system of two fermions, in the presence of a certain superselection rule, can be analyzed with the 7

18 help of Bogoliubov transformations of creation and annihilation operators. Moreover, it was shown that the Wootters concurrence is not the proper entanglement measure in this case, and the explicit formula for the entanglement of formation was found. This formula shows that the entanglement of a given state depends on the tensor product decomposition of a Hilbert space. It was also proved that the set of separable states is narrower than in the two-qubit case, and there exist states which are separable with respect to all tensor product decompositions of the Hilbert space. Papers [96, 97, 98] were devoted to the problem of time evolution of the unstable particles system particularly with attention to the entanglement evolution of K 0 K0 system in the context of testing the Bell CHSH inequalities. In [96] a description of the decaying particles within the framework of quantum mechanics of open systems was found, taking into account the superselection rule prohibiting the superposition of the particle and vacuum. Within this approach the evolution of the system is given by a family of completely positive trace-preserving maps forming a one-parameter dynamical semigroup. Moreover, the Kraus representation for the evolution of the system was given. It was also shown that there are some restrictions on the possible strength of decoherence. In [97] the time evolution of the system of two non-interacting unstable particles, distinguishable as well as identical ones, in arbitrary reference frame was found, having only the Kraus operators governing the evolution of its components in the rest frame. The EinsteinPodolskyRosen quantum correlation function for K 0 K0 system in the singlet state taking into account CP-violation and decoherence was found. It was shown that in this case the statistics of the particles plays no role. In [98] the evolution of entanglement in an ensemble of bipartite unstable quantum systems was considered. In particular, the quantum entanglement evolution for K 0 K0 systems in the singlet state with CP symmetry violation and decoherence was studied. In [99] it was shown that the spin reduced density matrix for two non-relativistic particles with arbitrary spin is covariant under Galilean transformations. Moreover, it was shown explicitly that the quantum entanglement of the system is the same for all Galilean observers. In [00] it was shown explicitly that the hypothesis of tachyonic neutrinos leads to the same oscillations effect as if they were usual massive particles. In [0] the detailed analysis of the improved purification protocol presented in [02] was given. In particular, it was shown that this protocol is not recursive, and even if it was, it would not be as effective as the authors of [02] claim. Moreover, the error free version of the purification protocol was presented [0]. 8

19 References [] Z. Walczak, Total correlations and mutual information, Phys. Lett. A 373 (2009), 88. [2] Z. Walczak, Comment on Quantum correlation without classical correlations, Phys. Rev. Lett. 04 (200), [3] Z. Walczak, Information-theoretic approach to the problem of detection of genuine multipartite classical correlations, Phys. Lett. A 374 (200), [4] M. Okrasa, Z. Walczak, Quantum discord and multipartite correlations, EPL 96 (20), [5] M. Okrasa, Z. Walczak, On two-qubit states ordering with quantum discords, EPL 98 (202), [6] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 8 (2009), 865. [7] O. Gühne, G. Tóth, Entanglement detection, Phys. Rep. 474 (2009),. [8] K. Modi, A. Brodutch, H. Cable, T. Paterek, V. Vedral, The classical-quantum boundary for correlations: Discord and related measures, Rev. Mod. Phys. 84 (202), 655. [9] R. F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, Phys. Rev. A 40 (989), [0] E. Knill, R. Laflamme, Power of one bit of quantum information, Phys. Rev. Lett. 8 (998), [] S. L. Braunstein, C. M. Caves, R. Jozsa, N. Linden, S. Popescu, R. Schack, Separability of very noisy mixed states and implications for NMR quantum computing, Phys. Rev. Lett. 83 (999), 054. [2] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, W. K. Wootters, Quantum nonlocality without entanglement, Phys. Rev. A 59 (999), 070. [3] D. A. Meyer, Sophisticated quantum search without entanglement, Phys. Rev. Lett. 85 (2000), 204. [4] E. Biham, G. Brassard, D. Kenigsberg, T. Mor, Quantum computing without entanglement, Theor. Comput. Sci. 320 (2004), 5. 9

20 [5] A. Datta, S. T. Flammia, C. M. Caves, Entanglement and the power of one qubit, Phys. Rev. A 72 (2005), [6] A. Datta, G. Vidal, Role of entanglement and correlations in mixed-state quantum computation, Phys. Rev. A 75 (2007), [7] H. Ollivier, W. H. Zurek, Quantum discord: A measure of the quantumness of correlations, Phys. Rev. Lett. 88 (200), [8] L. Henderson, V. Vedral, Classical, quantum and total correlations, J. Phys. A 34 (200), [9] A. Datta, A. Shaji, C. M. Caves, Quantum discord and the power of one qubit, Phys. Rev. Lett. 00 (2008), [20] A. Datta, S. Gharibian, Signatures of nonclassicality in mixed-state quantum computation, Phys. Rev. A 79 (2009), [2] C. A. Rodríguez-Rosario, K. Modi, A. Kuah, A. Shaji, E. C. G. Sudarshan, Completely positive maps and classical correlations, J. Phys. A 4 (2008), [22] A. Shabani, D. A. Lidar, Vanishing quantum discord is necessary and sufficient for completely positive maps, Phys. Rev. Lett. 02 (2009), [23] A. Ferraro, L. Aolita, D. Cavalcanti, F. M. Cucchietti, A. Acín, Almost all quantum states have nonclassical correlations, Phys. Rev. A 8 (200), [24] M. Piani, P. Horodecki, R. Horodecki, No-local-broadcasting theorem for multipartite quantum correlations, Phys. Rev. Lett. 00 (2008), [25] S. Luo, Quantum discord for two-qubit systems, Phys. Rev. A 77 (2008), [26] M. Ali, A. R. P. Rau, G. Alber, Quantum discord for two-qubit X states, Phys. Rev. A 8 (200), [27] A. Al-Qasimi, D. F. V. James, Comparison of the attempts of quantum discord and quantum entanglement to capture quantum correlations, Phys. Rev. A 83 (20), [28] F. F. Fanchini, M. C. de Oliveira, L. K. Castelano, M. F. Cornelio, Why the entanglement of formation is not generally monogamic, Phys. Rev. A 87 (203), [29] S. Campbell, Predominance of entanglement of formation over quantum discord under quantum channels, Quant. Inf. Proc. 2 (203),

21 [30] T. Werlang, S. Souza, F. F. Fanchini, C. J. Villas Boas, Robustness of quantum discord to sudden death, Phys. Rev. A 80 (2009), [3] J. Maziero, L. C. Céleri, R. M. Serra, V. Vedral, Classical and quantum correlations under decoherence, Phys. Rev. A 80 (2009), [32] F. F. Fanchini, T. Werlang, C. A. Brasil, L. G. E. Arruda, A. O. Caldeira, Non- Markovian dynamics of quantum discord, Phys. Rev. A 8 (200), [33] J. Maziero, T. Werlang, F. F. Fanchini, L. C. Céleri, R. M. Serra, Systemreservoir dynamics of quantum and classical correlations, Phys. Rev. A 8 (200), [34] B. Wang, Z.-Y. Xu, Z.-Q. Chen, M. Feng, Non-Markovian effect on the quantum discord, Phys. Rev. A 8 (200), 040. [35] X. Hu, Y. Gu, Q. Gong, G. Guo, Necessary and sufficient condition for Markovian-dissipative-dynamics-induced quantum discord, Phys. Rev. A 84 (20), [36] R. Lo Franco, B. Bellomo, E. Andersson, G. Compagno, Revival of quantum correlations without system-environment back-action, Phys. Rev. A 85 (202), [37] V. Madhok, A. Datta, Interpreting quantum discord through quantum state merging, Phys. Rev. A 83 (20), [38] D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, A. Winter, Operational interpretations of quantum discord, Phys. Rev. A 83 (20), [39] M. F. Cornelio, M. C. de Oliveira, F. F. Fanchini, Entanglement irreversibility from quantum discord and quantum deficit, Phys. Rev. Lett. 07 (20), [40] A. Streltsov, H. Kampermann, D. Bruss, Linking quantum discord to entanglement in a measurement, Phys. Rev. Lett. 06 (20), [4] F. F. Fanchini, M. F. Cornelio, M. C. de Oliveira, A. O. Caldeira, Conservation law for distributed entanglement of formation and quantum discord, Phys. Rev. A 84 (20), [42] F. F. Fanchini, L. K. Castelano, M. F. Cornelio, M. C. de Oliveira, Locally inaccessible information as a fundamental ingredient to quantum information, New J. Phys. 4 (202),