OPEN Quantum correlation exists in any
SUBJECT AREAS: non-product state APPLIED MATHEMATICS Yu Guo1 & Shengjun Wu2,3 QUANTUM INFORMATION
1School of Mathematics and Computer Science, Shanxi Datong University, Datong, Shanxi 037009, China, 2Kuang Yaming 3 Received Honors School, Nanjing University, Nanjing, Jiangsu 210093, China, Department of Modern Physics and the Collaborative 29 September 2014 Innovation Center for Quantum Information and Quantum Frontiers, University of Science and Technology of China, Hefei, Anhui 230026, China. Accepted 5 November 2014 Simultaneous existence of correlation in complementary bases is a fundamental feature of quantum Published correlation, and we show that this characteristic is present in any non-product bipartite state. We propose a 1 December 2014 measure via mutually unbiased bases to study this feature of quantum correlation, and compare it with other measures of quantum correlation for several families of bipartite states.
Correspondence and uantum systems can be correlated in ways inaccessible to classical objects. This quantum feature of requests for materials correlations not only is the key to our understanding of quantum world, but also is essential for the should be addressed to Q powerful applications of quantum information and quantum computation1–19. In order to characterize Y.G. (guoyu3@aliyun. the correlation in quantum state, many approaches have been proposed to reveal different aspects of quantum 6–10 com) correlation, such as the various measures of entanglement and the various measures of discord and related measures17–21, etc. It is believed that some aspects of quantum correlation could still exist without the presence of entanglement and these aspects could be revealed via local measurements with respect to some basis of a local system. The simultaneous existence of complementary correlations in different bases is revealed very early by the Bell’s inequalities22. Bell’s inequalities quantify quantum correlation via expectation values of local complementary observables. In Ref. 23, the feature of genuine quantum correlation is revealed by defining measures based on invariance under a basis change: for a bipartite quantum state, the classical correlation is the maximal correlation present in a certain optimum basis, while the quantum correlation is characterized as a series of residual correlations in the bases mutually unbiased (MU) to the optimum basis. In this paper, we use the fact that the essential feature of the quantum correlation is that it can be present in any two mutually unbiased bases (MUBs) simultaneously. Thus, one of the two bases is not necessarily the optimum basis to reveal the maximal classical correlation in this paper. With respect to the measure proposed here, we shall show that only the product states do not contain quantum correlation. A product state contains neither any quantum correlation nor any classical correlation; while any non-product bipartite state contains correlation that is fundamentally quantum! We shall also reveal interesting properties of this measure by comparing this measure to other measures of quantum correlation for several families of bipartite states. The MUBs constitute now a basic ingredient in many applications of quantum information processing: quantum state tomography24, quantum cryptography25, discrete Wigner function26, quantum teleportation27, 28 29 quantum error correction codes , and the mean king’s problem . Two orthonormal bases {jyiæ} and {jwjæ}of a d-dimensional Hilbert space H are said to be mutually unbiased if and only if DE ~ 1ffiffiffi ƒ ƒ yi wj p , V 1 i, j d: ð1Þ d In a d-dimensional Hilbert space, there exist at least 3 MUBs (when d is a power of a prime number, a full set of d 1 1 MUBs exists, more details can be found in Ref. 30). We recall the quantity defined in Ref. 23. Let Hab 5 Ha fl Hb with dim Ha 5 da and dim Hb 5 db be the state space of the bipartite system A1B shared by Alice and Bob. Let {jiæ} and jj9æ be the orthonormal bases of Ha and Hb respectively. Alice selects a basis {jiæ}ofHa and performs a measurement projecting her system onto the basis x r æ r states. The Holevo quantity { abj{ji }}ÈÉ of ab with respectX to Alice’sX localÀÁ projective measurement onto the basis {jiæÆij}, is defined as xrfgjfgjii ~x p ; rb :S p rb { p S rb . A basis {jiæ} that achieves the ab i i i i i i i i maximum (denoted as C1(rab)) of the Holevo quantity is called a C1-basis of rab. There could exist many C1-
SCIENTIFIC REPORTS | 4 : 7179 | DOI: 10.1038/srep07179 1 www.nature.com/scientificreports
bases for a state rab, and the set of these bases is denoted as Crab . Let The nullity of C2. Now we show that any bipartite quantum state a VPa be the set of all bases that are mutually unbiased to P , contains nonzero correlation simultaneously in two mutually a unbiased bases unless it is a product state, this result is stated as P [ Crab . The quantity of quantum correlation in Ref. 23, denoted by Q (r ), is defined as the following theorem, while the proof is left to the Method. 2 ab ÈÉ ~ a Q2ðÞrab : max max xrab P : ð2Þ Theorem. C ðÞr ~0 if and only if r is a product state. Pa [ C ~ a 2 ab ab rab P [ VPa In a sense, this theorem implies that, any non-product bipartite state contains genuine quantum correlation, and C reveals the In other words, Q2 is defined as the Holevo quantity of Bob’s access- 2 ible information about Alice’s results, maximized over Alice’s amount of quantum correlation in the state. In addition, we know projective measurements in the bases that are mutually unbiased to that C2 is different from the quantity Q2 in Ref. 23 since Q2(rcq) 5 0 for any classical-quantum state rcq while C2~0 only for product a C1-basis Crab , and further maximized over all possible C1-bases (if states. The difference between the measure C2 and other measures not unique). Thus, Q2 is actually the maximum correlation present of quantum correlation shall be discussed below for several families simultaneously in any set of two MUBs of which one is a C1-basis. of bipartite states in more details. Now, we shall calculate the quantity for several families of bipartite Results states, and see how our measure in terms of MUBs is well justified as a Our approach - Correlation measure based on MUBs. We now measure of quantum correlation. present our approach in a more general way. Example 1 - Pure statesX. Forp affiffiffiffi bipartite pure state with the Schmidt Definition. Let D denote the set of all two-MUB sets, i.e., decomposition jiy ~ l jia jib , C ~Q ~SðÞr ~SðÞr ~ X i i i i 2 2 a b D~fgfgfgji,fgji : fgji fgji : ð Þ i1 j2 i1 is MU to j2 3 {l log l . It can be easily checked that C coincides with the i i 2 i 2 We define entropy of either reduced state for any pure state, which is also the ÈÉÈÉ ÈÉ a a usual measure of entanglement in a pure state. C2ðÞrab : max min xrab P1 ,xrab P2 : ð4Þ Pa,Pa [ D ðÞ1 2 Example 2 - Werner states. Next, we consider the Werner states of a d fl d dimensional system5, The quantity C2 represents the maximal amount of correlation that is present simultaneously in any two MUBs. Similar to the other usual ~ 1 { rw ðÞI aP , ð6Þ measures of quantum correlation, C2 is local unitary invariant, that is, ddðÞ{a ~ 6 {6 { C2ðÞrab C2 Ua UbrabUa U for any unitary operators Ua and 2 b where 21 # a # 1, I is theX identity operator in the d -dimensional U acting on H and H respectively. d b a b Hilbert space, and P~ jii hjj 6jij hji is the operator that i,j~1 C2 versus Q2. Although both C2 and Q2 represent quantum exchanges A and B. For a local measurement with respect to basis correlation (here the symbol C2 is associated with Correlation in 1 states {jeiæ}ofHa, with probability pi~ , Alice will obtain the k-th two MUBs) instead of classical correlation (represented by C1), d basis state je æ, and Bob will be left with the state they are actually quite different. As C2 is defined as the maximum k X 1 correlation present simultaneously in any two MUBs while Q2 is the b r ~ ðÞI{ajie’k hje’k , where jie’k ~ akjjij’ with akj 5 maximum correlation present simultaneously in two MUBs of which k d{a j Æe jjæ. It is straightforward to show that one is a C1-basis, it is obvious that k ÈÉ { § B d 1 a C2ðÞrab Q2ðÞrab ð5Þ C ðÞr ~x p ; r ~log z log ðÞ1{a ~Q ðÞr :ð7Þ 2 w i i 2 d{a d{a 2 2 w for any rab. In a sense, C2 is more essential than Q2 since the The entanglement0 0 of formation Ef for the Werner11 states is given as maximum in the former one is taken over arbitrarily two MUBs. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Thus, C may reveal more quantum correlation than Q . 2 2 2 @1 @ da{1 AA From the following Theorem and Examples, one knows that there Ef ðÞr ~h 1z 1{ max 0, , with h(x) ; 2x w 2 d{a do exist states such that C2wQ2 (Example 4) and C2~Q2 (Examples 31 1,2,3). A clear illustration of the difference between C2 and Q2 is also log2 x 2 (1 2 x) log2(1 2 x) . The three different measures of given in Fig. 3. We know that Q2 does not exceed quantum discord D quantum correlation, i.e., C2, the quantum discord D and the 23 for all the known examples . However, one can easily find states such entanglement of formation Ef, are illustrated in Fig. 1 for that C2 exceeds quantum discord D (See Fig. 3). comparison. From this figure, we see that the curve for
Figure 1 | Measures of quantum correlation for the Werner states as functions of a when d 5 2 (left) and d 5 3 (right). The red curve represents our measure C2, the green curve represents the quantum discord D and the blue curve represents the entanglement of formation Ef.
SCIENTIFIC REPORTS | 4 : 7179 | DOI: 10.1038/srep07179 2 www.nature.com/scientificreports
Figure 2 | Measures of quantum correlation for the isotropic states as functions of b when d 5 2 (left) and d 5 3 (right). The red curve represents our measure C2, the blue curve represents the quantum discord D and the green curve represents the entanglement of formation Ef. pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi entanglement of formation intersects the other two curves; thus, E 1 2 f where c~ bz ðÞd{1 ðÞ1{b . The quantum discord of the can be larger or smaller than C2. d isotropic state is34 Example 3 - Isotropic states. For the d fl d isotropic states {b {b ~ z 1 1 ÀÁÀÁ DðÞr b log2 b log2 1 z dz1 d2{1 r~ ðÞ1{b Iz d2b{1 P , b [ ½ 0,1 , ð8Þ 2{ ð12Þ d 1 1zdb 1{b{ 1 zdb X { log d : z 1 z 2 2{ where P1 5 jW1æÆW1j, jjW ~ pffiffiffi jii jii’ is the maximally d 1 d 1 d i d d The three different measures of quantum correlation, i.e., C2, the entangled pure state in C 6C . Let {jekæÆekj} be an arbitrarily quantum discord D and the entanglement of formation Ef, are given projective measurement on Alice’s part. Bob’s state after illustrated in Fig. 2 for comparison. From this figure, we see that after Alice gets the k-th measurement result is the curve for entanglement of formation intersects the other two 1 ÀÁÀÁ curves; thus, E can be larger or smaller than C . rb~ dðÞ1{b Iz d2b{1 jie’ hje’ , ð9Þ f 2 k d2{1 k k X Example 4 - A family of two-qubit states. As the last example, we where jie’ ~ a jij’ with a 5 Æe jjæ. As the eigenvalues of rb k j kj kj k k consider a family of two-qubit states that are Bell-diagonal states. This family of states admit the form does not depend on the basis for Alice’s measurement, one can easily ! show that X3 1 s ~ I 6I z r s 6s : ð13Þ dbz1 dbz1 ab 4 2 2 j j j C ðÞr ~Q ðÞr ~log dz log j~1 2 2 2 dz1 2 dz1 ð10Þ d{db d{db We rearrange the three numbers {r1, r2, r3} according to their z log : absolute values and denote the rearranged set as { r , r , r } such z 2 2{ 1 2 3 d 1 d 1 that jj r §jj r §jj r . Next we show that 1 2 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The entanglement of formation E for the isotropic states is given 0 ÀÁ1 f z 2z 2 as32,33 1 r1 r2 =2 C ðÞs ~1{h@ A: ð14Þ 8 2 ab 2 > 0, bƒ 1 , <> d z { { 1 4ðÞd{1 hðÞc ðÞ1 c log ðÞd 1 , vbv 2 , Without loss of generality, we prove (14) only for the case r1§r2§r3. Ef ðÞr ~ 2 d d ð11Þ > { { A projective measurement performed on qubit A can be written as :> ðÞb 1 d log ðÞd 1 4ðÞd{1 2 zlog d, ƒbƒ1, 1 d{2 2 d2 Pa ~ ðÞI +~n:~s , parameterized by the unit vector ~n. When Alice + 2 2