Non-zero total correlation means non-zero quantum correlation

Physics Letters A 378 (2014) 1249–1253

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Physics Letters A www.elsevier.com/locate/pla

Non-zero total correlation means non-zero quantum correlation Bo Li a,b , Lin Chen c,d,e,∗ , Heng Fan f a

Department of Mathematics and Computer, Shangrao Normal University, Shangrao 334001, China Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China c Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada d Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada e Center for Quantum Technologies, National University of Singapore, Singapore f Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China b

a r t i c l e

i n f o

Article history: Received 11 July 2013 Received in revised form 21 February 2014 Accepted 25 February 2014 Available online 4 March 2014 Communicated by P.R. Holland Keywords: Super discord Weak measurement Optimal state discrimination

a b s t r a c t We investigated the super quantum discord based on weak measurements. The super quantum discord is an extension of the standard quantum discord defined by projective measurements and also describes the quantumness of correlations. We provide some equivalent conditions for zero super quantum discord by using quantum discord, classical correlation and mutual information. In particular, we find that the super quantum discord is zero only for product states, which have zero mutual information. This result suggests that non-zero correlations can always be detected using the quantum correlation with weak measurements. As an example, we present the assisted state-discrimination method. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Quantum measurement plays a key role in quantum mechanics and has certain interesting quantum properties that are rarely seen in everyday life. These properties include the collapse of the wave function, the concept of compatible observables and the contextuality phenomenon. To perform a quantum measurement, one must construct a set of orthogonal projection operators corresponding to the observable eigenvector spaces of a Hermitian operator. The possible outcomes of the measurement correspond to the eigenvalues of the Hermitian operator. This type of measurement is the standard von Neumann measurement, or projective measurement [1]. Recently, the formalism was generalized to the positive-operator-valued measure (POVM) [2], which can capture many phenomena beyond what can be probed by projective measurements. However, the measurement of a quantum state inevitably disturbs the quantum system, which, in turn, determines what knowledge can be retrieved regarding the measured system. To exert the least influence on the original quantum state, a measurement can be introduced that induces a partial collapse of the quantum state. This is the so-called weak measurement [3–5]. Quantum states can be retrieved with a non-zero success probability when the in-

*

Corresponding author. E-mail address: [email protected] (L. Chen).

http://dx.doi.org/10.1016/j.physleta.2014.02.036 0375-9601/© 2014 Elsevier B.V. All rights reserved.

teraction between the system and the measurement apparatus is weak [6]. It has been shown that any generic measurement can be decomposed into a sequence of weak measurements [7]. Therefore, weak measurements are universal. Furthermore, the reverse process has drawn considerable attention both theoretically and experimentally [8,9] because of its potential applications in quantum information processing [10]. In addition, weak measurements can amplify extremely tiny signals [11,12]. Searching for quantum correlation in a composite system and identifying its role in quantum information processing is one of the fundamental problems in quantum mechanics. Quantum entanglement is widely regarded as having a crucial role in quantum teleportation and superdense coding [2]. Quantum discord [13–15], which is more stringent than quantum entanglement, can effectively elucidate the role of the quantumness of correlations and is different from the classical correlation. Quantum discord has been proven to be present in deterministic quantum computation with one qubit (DQC1) [16] and can be used as a resource in remote state preparation [17]. Furthermore, the consumed discord bounds the quantum advantage in encoded information [18]. Quantum dissonance (or one-side discord) has been proven to be required for optimal assisted discrimination [19–21]. It is known that quantum entanglement can be described and detected using various methods [22–24]. However, quantum discord can exist when entanglement is absent. The quantum discord vanishes for the so-called classical–classical (CC) state, the classical–quantum (CQ) state and the quantum–classical (QC) state [25–27].

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However, studies indicate that the quantum advantage may exist even for vanishing discord [28]. It should, then, be possible to construct a measure of quantum correlation that always exists, except for product states. A good candidate for this measure is the super quantum discord, which is an extension of the quantum discord based on weak measurements [29]. The super quantum discord is always larger than the normal discord induced by strong (projective) measurements, which suggests that the super quantum discord can capture significantly more correlation information. Furthermore, super discord can result in the improvement of the entropic uncertainty relations [30,31]. Thus, we can ask: what is the criterion by which the super quantum discord exists in a quantum system? Can super discord exist in a quantum information model in which quantum discord and entanglement do not exist? In this article, we provide a necessary and sufficient condition for the vanishing of the super discord in terms of classical correlation, mutual information and normal discord. Our results indicate that the quantum correlation measured by the super quantum discord always exists, except when there is no correlation. Thus, we can confirm the expectation that all correlations can be detected from the perspective of quantum correlation. We further illustrate that super discord can occur in optimal assisted state discrimination on both sides, whereby only one-side quantum discord is present and entanglement is unnecessary. This article is organized as follows. In Section 2, we review the definition and several properties of super discord. In Section 3, we provide a series of necessary and sufficient conditions for the vanishing of the super discord. An illustration of the super discord present on both sides in optimal assisted state discrimination is given in Section 4. Finally, we present our summary in Section 5. 2. The concept and properties of super discord Consider the bipartite state ρ on the space H A ⊗ H B . Let {πk } be one-dimensional von Neumann projectors, and the probability is given by pk = Tr( I ⊗ πk )ρ ( I ⊗ π k ). The completeness of the operators {πk } implies the formula k p k = 1. Next, S (ρ ) := − Tr ρ log ρ is the von Neumann entropy, whereby “log” denotes “log2 ” throughout the article. We refer to ρ A , ρ B as the reduced density operators of ρ . Then, we denote the mutual quantum in+ S (ρ B ) − S (ρ ) and the classical corformation by I (ρ ) := S (ρ A )  relation by C (ρ ) := maxπk I ( k ( I ⊗ πk )ρ ( I ⊗ πk )) [13,14,32]. Both these terms are non-negative because the mutual information is non-negative [2]. The quantum discord for ρ is defined as the difference between the mutual information and the classical correlation as follows [14,25]:

D (ρ ) = I (ρ ) − C (ρ )

= S (ρ B ) − S (ρ ) + min πk



 pk S

( I ⊗ πk )ρ ( I ⊗ πk )

k

.

(1)

It is  known that [26,27] the (“right”) discord is zero if and only if ρ = i p i ρi ⊗ |ϕi ϕi |, where the |ϕi  are o.n. basis. This criterion defines the so-called classical state of the system B. Next, we recall the super quantum discord D w (ρ ) for the twoqubit states ρ introduced in [29], which is defined as follows:

 

D w (ρ ) := min S w A  P B (x) − S ( A | B ), {π0 ,π1 }

(2)

where the conditional entropy S ( A | B ) = S (ρ ) − S (ρ B );

 



S w A  P B (x)

= p (x) S (ρ A | P B (x) ) + p (−x) S (ρ A | P B (−x) ),



p (±x) = Tr I ⊗ P B (±x) ρ I ⊗ P B (±x) , 



P (x) = P (−x) =

1 p (±x)

Tr B

1 − tanh x



2





I ⊗ P B (±x)



π0 +

1 + tanh x 2

1 + tanh x



π0 +

2

(5)

π1 ,

1 − tanh x 2



ρ I ⊗ P B (±x) ,

(6)

π1 ,

(7)

and x ∈ R \ {0} is a parameter that describes the strength of the measurement process. Using Eq. (2), we can write D w (U ⊗ V ρ U † ⊗ V † )  D w (ρ ) with arbitrary unitary U , V . One can similarly obtain D w (U ⊗ V ρ U † ⊗ V † )  D w (ρ ); thus, we can write





D w U ⊗ V ρ U † ⊗ V † = D w (ρ ).

(8)

In other words, the super discord is invariant up to the local unitary. This property is the same as that of the normal discord. Using Eqs. (6) and (7), we can obtain the completeness relation as follows:

π0 + π1 = f P (x)† P (x) + P (−x)† P (−x) = I .

(9)

Using Eqs. (4) and (9), we observe that the probability sum is equal to one as follows:

p (x) + p (−x) = 1.

(10)

Using the concavity of the von Neumann entropy and Eqs. (3) and (5), we can easily obtain I (ρ )  D w (ρ ). By incorporating the theorem of [29], we can obtain

I (ρ )  D w (ρ )  D (ρ )

(11)

for any two-qubit states. However, these three quantities are not quantitatively related to the classical correlation. It follows from Ref. [33] that the difference C (ρ ) − D (ρ ) can be either positive or negative for the two-qubit Bell diagonal states ρ in [33]; see [34]. Nevertheless, we derive the relations among the classical correlation, mutual information, super discord and discord for product states in the next section. 3. Condition for zero super discord Similar to the case of discord, we can ask the following question: what are the states ρ whose super discord is zero? According to Eq. (11) and [26,27], such states ρ must be classical in the system B. However, the converse is not manifestly true; see Theorem 1 below. Thus, we require a preliminary lemma. It is known that the classical correlation is zero for a product state [13]. We can show that the inverse is also true. Lemma 1. Any bipartite state ρ that satisfies C (ρ ) = 0 is a product state, i.e., ρ = ρ A ⊗ ρ B .



pk

ρ A | P B (±x) =



Proof. By definition, the condition C (ρ ) = 0 implies that I ( k ( I ⊗ πk )ρ ( I ⊗ πk )) = 0 holds for any  {πk }. By the subadditivity of the von Neumann entropy, the state k ( I ⊗ πk )ρ ( I ⊗ πk ) is a product state. By tracing out the system A or B, we obtain



( I ⊗ πk )ρ ( I ⊗ πk ) = ρ A ⊗

k





πk ρ B πk

(12)

k

for any {πk }. Let ρ B = i p i |b i b i | be the spectral decomposition,  and we can assume ρ = i j ρi j ⊗ |b i b j |. By choosing πi = |b i b i | in Eq. (12), normalization con we obtain ρii = p i ρ A , ∀i. Using the  dition i p i = 1, we obtain ρ = ρ A ⊗ ρ B + i = j ρi j ⊗ |b i b j |. By substituting this expression for ρ into Eq. (12), we obtain

(3)



(4)

k

  ( I ⊗ πk ) ρi j ⊗ |bi b j | ( I ⊗ πk ) = 0 i = j

(13)

B. Li et al. / Physics Letters A 378 (2014) 1249–1253

for any {πk }.  Because any two summands are orthogonal, we obtain ( I ⊗ πk )( i = j ρi j ⊗ |b i b j |)( I ⊗ πk ) = 0, ∀k. By choosing



1



1

πk = √ |bl  + √ e iα |b j  2

2

1

1



√ bl | + √ e−i α b j | , 2

(14)

2

we find that ρlj e i α + ρ jl e −i α = 0 for any real α . Hence, ρlj = ρ jl = 0 for any j = l. Thus, ρ = ρ A ⊗ ρ B , and the assertion follows, thereby completing the proof. 2 Theorem 1. The following seven statements are equivalent for the twoqubit state ρ : (a) (b) (c) (d) (e) (f) (g)

ρ is a product state; ρ has zero classical correlation; ρ has zero super discord; ρ has zero mutual information; ρ has equal discord and super discord; ρ has equal discord and mutual information; and ρ has equal super discord and mutual information.

Proof. (a) → (b) follows from the definition of classical correlation. (b) → (c) can be proven as follows. Using Lemma 1, we can assume ρ = ρ A ⊗ ρ B . Using Eqs. (4) and (5), we can obtain ρ A | P B (±x) = ρ A . Using Eqs. (3) and (10), we can obtain S w ( A |

{ P B (x)}) = S (ρ A ). Then, Eq. (2) implies that D w (ρ ) = 0; thus, (b) → (c) follows. (c) → (e) follows from D (ρ )  0 and Eq. (11). (e) → (d) can be proven as follows. Let {πi } be the measurement basis that minimizes the super discord in Eq. (2). Using [29, Eq. (11)] and Eq. (1), we can obtain

D w (ρ ) 

1 

 pk S

( I ⊗ πk )ρ ( I ⊗ πk )



pk

k =0

− S ( A | B )  D (ρ ).

(15)

According to the hypothesis, both equalities in Eq. (15) hold. Based on the results in [29] and the concavity of the von Neu( I ⊗π0 )ρ ( I ⊗π0 ) = mann entropy, the first equality holds only if Tr B p 0

( I ⊗π )ρ ( I ⊗π )

1 1 Tr B [2]. Because the operators p1 second equality implies that

D (ρ ) =

1 

 pk S Tr B

( I ⊗ πk )ρ ( I ⊗ πk )



pk

k =0

πk are of rank one, the

− S ( A|B )

  ( I ⊗ π0 )ρ ( I ⊗ π0 ) − S ( A|B ) = S Tr B =S

p0

1 

pk Tr B

( I ⊗ πk )ρ ( I ⊗ πk )

k =0

pk

− S ( A|B )

 



D w (ρ ) = min S w A  P B (x) {π0 ,π1 }

− S ( A|B )

= S (ρ A ) − S ( A | B )  

 max S w A  P B (x) − S ( A | B ). {π0 ,π1 }

(17)



Thus, the quantity S w ( A  { P B (x)}) is constant for any π0 , π1 . The equality in Eq. (17) holds if and only if ρ A | P B (x) = ρ A | P B (−x) = ρ A for any π0 , π1 . Using Eqs. (5), (6) and (9), we can obtain ( I ⊗ πk )ρ ( I ⊗ πk ) ∝ ρ A . This fact and Eq. (1) together imply that D (ρ ) = I (ρ ). Thus, we have proven (g) → (f) → (b) → (a), where the last relation follows from Lemma 1, thereby completing the proof. 2 As a typical example, Theorem 1 implies that the completely mixed state 14 I ⊗ I has zero super discord. This observation has been included as a special case in [29]. Note that the equivalence in Theorem 1 does not hold for states with zero discord because such states may not be product states. The super discord measured via weak measurements has a natural similarity to the Gaussian quantum discord [37] that is restricted to Gaussian measurements in the realm of continuous– variable (CV) systems, where the measurement class is larger than the class that includes all local projective measurements. Recently, quantum discord with non-Gaussian measurements has also been studied [38]. Theorem 1 suggests that weak measurements can reveal significant correlation information, because the super discord is always larger than the quantum discord and vanishes only for product states. Therefore, the super discord is ubiquitous in quantum systems. A natural and interesting question is whether Gaussian measurements or even non-Gaussian measurements can also reveal significantly more quantum correlation information than can projections alone. We propose this as an open question because it may help us to obtain a deeper understanding of how the quantum correlation behaves with regard to different measurement classes. It is widely accepted that the mutual information of a system contains both the classical and quantum correlations. However, they share the same vanishing condition as the super discord. Based on Theorem 1, we can state that the super discord is generally larger than the normal discord, which means that the super discord is quite likely to reveal significantly more quantum correlation information than the quantum discord. Another advantage of the super discord is that its criterion for vanishing does not rely on a specific side, even though we perform a measurement that acts on the “left” or “right” system. The reason for this lack of side dependence is that if the “left” super discord is zero, then the state is a product state according to Theorem 1. Thus, the “right” super discord must also be zero. As an example, in the next section, we illustrate how the super discord present in the optimal assisted discrimination is different from the normal discord. 4. Super discord in optimal state discrimination

= S (ρ A ) − S ( A | B ) = I (ρ ).

1251

(16)

The second equality follows from the formula p 0 + p 1 = 1, and the fourth equality follows from Eq. (9). It follows from Eq. (1) that Eq. (16) holds only if C (ρ ) = 0. Then, Lemma 1 implies that ρ is a product state; thus, (e) → (d). (d) → (f) can be proven as follows. The hypothesis I (ρ ) = 0 implies that ρ is a product state. Hence, the discord is also zero, and the assertion follows. (f) → (g) is a corollary of Eq. (11). (g) → (a) can be proven as follows. Using Eq. (2) and the concavity of the von Neumann entropy, we can obtain

The super discord vanishes only for product states. It exists more universally in quantum information processing than do other quantum correlations, such as entanglement and quantum discord. In other words, certain examples can be found in which the super discord is non-zero, while neither the entanglement nor the quantum discord is non-zero. In this section, we consider a statediscrimination method and provide an example in which the super discord exists more universally than do the entanglement and quantum discord. To achieve this goal, we focus on the case of the minimal error probability, in which the entanglement and one-side quantum discord vanish. We first review the state-discrimination method introduced by Roa, Retamal and Alid-Vaccarezza (RRA scheme) [19]. Consider two

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non-orthogonal states |ψ+  and |ψ−  randomly prepared in one of the a priori probabilities p + or p − , where p + + p − = 1. To discriminate the two states |ψ+  and |ψ− , couple the original system to an auxiliary qubit A using a joint unitary transformation U such that



1 − |α+ |2 |+|0a + α+ |0|1a ,

U |ψ+ |ka = U |ψ− |ka =



1 − |α− |2 |−|0a + α− |0|1a ,

(18)

where |ka is an auxiliary state√ with the orthonormal basis {|0a , |1a }, and |± ≡ (|0 ± |1)/ 2 are the orthonormal states of the system that can be discriminated. The state of the system and the ancilla qubits is given as follows:





ρ|α+ | = p + U |ψ+ ψ+ | ⊗ |ka k| U †



+ p − U |ψ− ψ− | ⊗ |ka k| U † .

(19)

It has been proven that the conclusive recognition between the two non-orthogonal states relies on the existence of entanglement and discord in the general case of the RRA scheme [19]. However, in the optimal case, or that with the maximum recognition probability, only the one-side (“right” side) discord (or dissonance) is non-zero [20]. An interesting question to ask is what types of nonclassical correlations can be regarded as candidates for resources in the measurement method for the optimal case? Using the fact that the super quantum discord is always greater than or equal to the normal quantum discord and the equivalent condition given in Theorem 1, we can suppose that the super quantum discord truly captures the non-classical correlation of the RRA scheme on both sides. In the following discussion, we concentrate on the state (19) in the zero “left” discord case. Because the “right” discord is always present in a quantum system, and the super discord is larger than the normal discord, we can conclude that the “right” super discord must be non-zero. As indicated in [20], in the case in which the “left” discord disappears, the following three criteria must be satisfied: α is a real number, and α  0; p + = p − = 12 ; √ √ and |α+ | = |α− | = |α | = α . This configuration can then be considered to be the optimal assisted state discrimination. For convenience, we set α+ = c to be a real number. Then, the state ρ in (19) is reduced to

ρc =

1 − c2  2



I ⊗ |00| + |00|

√ √  

2c 1 − c 2  |01| + |10| . ⊗ c 2 |11| + 2

(20)

We use the weak measurement P (x) ⊗ I , P (−x) ⊗ I to act on the state ρc in (20), where P (x), P (−x) is given by Eqs. (6) and (7), ψ |, |ψ = cos θ|0 + e i ϕ sin θ|1, and and π0 = |ψψ|, π1 = |ψ  = sin θ|0 − e i ϕ cos θ|1. Then, the weak “left” conditional en|ψ tropy for this state is given as follows:

 



S w B  P A (x)

  = − p (x) λ+ (x) log λ+ (x) + λ− (x) log λ− (x)   − p (−x) λ+ (−x) log λ+ (−x) + λ− (−x) log λ− (−x) ,

where p (x) =

λ± (x) =

(21)

1 (1 − tanh(x) cos(2θ)c 2 ); 2

1 2(1 − tanh(x) cos(2θ)c 2 )



1 − tanh(x) cos(2θ)c 2

 ± 1 − 2c 2 + 2c 4 − 2c 2 tanh(x) cos(2θ)

 

2 1

+ 2c 2 − c 4 tanh(x) cos(2θ) 2 ,

(22)

Fig. 1. Super discord in the optimal case of assisted state discrimination as a function of α+ = c and the strength x of the measurement for 0  α+  1 and 0  x  2.

and λ± (−x) can be similarly defined. After the calculation, we determine that S ( A B ) = S ( A ). Let D w ( B : A ) = D w (ρ ) in Eq. (2) by exchanging systems A and B. Using Eqs. (2) and (21), we can obtain

 



D w ( B : A ) = min S w B  P A (x) {πiA }

 

= min S w B  P A (x) , (23) θ

where D w ( B : A ) is a function of x and c. In Fig. 1, we have plotted a visualization of D w ( B : A ). It can be observed that for all 0 < c < 1, the super discord increases as the strength of the measurement x decreases. When x → +∞, the weak measurement reduces to the strong measurement, and the super discord approaches the normal discord. The discord and entanglement are always zero in this optimal case. Thus, we have demonstrated that the super discord can be regarded as a resource for optimal assisted state discrimination. In summary, we have determined that in the optimal assisted discrimination method, the entanglement and one-side quantum discord are unnecessary. In this approach, we require only the super discord, which is always present between the principal qubit and the ancilla. This discovery reveals the mysterious properties of the non-classical correlation in quantum information processing and that neither the quantum discord nor the entanglement is the fundamental ingredient in the non-classical correlation. Our findings could stimulate additional research concerning the role of non-classical correlation in quantum information processing. 5. Summary and discussions In this article, we obtained several equivalent conditions for zero super discord. It was determined that the vanishing of the super quantum discord is equivalent to the vanishing of the classical correlation and mutual information, etc. Thus, the super quantum discord is a type of quantum correlation that ubiquitously exists in quantum systems. Furthermore, the super quantum discord can be present in certain quantum information processing tasks in which the entanglement is entirely unnecessary and only the one-side quantum discord is non-zero. One fundamental problem in quantum information is to quantify correlations. The quantum discord emerges when separating the total correlations into quantum and classical components. A few studies indicate that all correlations behave as if they were exclusively quantum [28,35,36]. In this article, we confirmed this

B. Li et al. / Physics Letters A 378 (2014) 1249–1253

concept by demonstrating that the super quantum discord vanishes only when the mutual (total) information vanishes. This result extends the regime of the quantumness of correlations to all bipartite quantum states except product states. Based on this conclusion, we can always rely on the quantum correlation in various quantum information protocols because a non-zero total correlation implies a non-zero super quantum discord. Keeping this in mind, we can safely state that quantum correlation exists in tiny signal-amplifying processes in which weak measurements are performed. Acknowledgements This study was conducted while L.C. was visiting the Institute of Physics, CAS, China. He was primarily supported by MITACS and NSERC. The CQT is funded by the Singapore MoE and the NRF as part of the Research Centres of Excellence program. B.L. and H.F. were supported by the 973 program (2010CB922904), NSFC (11175248, 11305105) and NSFJXP (20132BAB212010). References [1] J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932. [2] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, England, 2000. [3] Y. Aharonov, D.Z. Albert, L. Vaidman, Phys. Rev. Lett. 60 (1988) 1351. [4] A.N. Korotkov, A.N. Jordan, Phys. Rev. Lett. 97 (2006) 166805. [5] Q. Sun, M. Al-Amri, M.S. Zubairy, Phys. Rev. A 80 (2009) 033838. [6] M. Ueda, M. Kitagawa, Phys. Rev. Lett. 68 (1992) 3424; M. Koashi, M. Ueda, Phys. Rev. Lett. 82 (1999) 2598. [7] O. Oreshkov, T.A. Brun, Phys. Rev. Lett. 95 (2005) 110409. [8] H.M. Wiseman, Phys. Lett. A 311 (2003) 285. [9] R. Mir, J.S. Lundeen, M.W. Mitchell, A.M. Steinberg, J.L. Garretson, H.M. Wiseman, New J. Phys. 9 (2007) 287.

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