RELATING QUANTUM DISCORD WITH THE QUANTUM DENSE CODING CAPACITY
Xin Wang", Liang Qiua* Song Li", Chi Zhang", Bin Yeh
" School of Sciences, China University of Mining and Technology 221116, Xuzhou, China
b School of Information and Electrical Engineering, China University of Mining and Technology 221116, Xuzhou, China
Received August 2, 2014
We establish the relations between quantum discord and the quantum dense coding capacity in (/? + l)-particle quantum states. A necessary condition for the vanishing discord monogamy score is given. We also find that the loss of quantum dense coding capacity due to decoherence is bounded below by the sum of quantum discord. When these results are restricted to three-particle quantum states, some complementarity relations are obtained.
DOI: 10.7868/S0044451015010022
1. INTRODUCTION
The protocols of quantum dense coding fl], quantum tclcportation [2], and quantum key distribution [3] are viewed as the beginning of discoveries of quantum communication strategies. These protocols can be effectively used to transmit classical or quantum information in a way that cannot be realized with their classical counterparts. Thus, they have created a very-substantial change in the attitude to modern communication schemes. Such protocols are initially introduced for a single sender and a single receiver, and have been realized experimentally in several physical systems such as photons, trapped ions, atoms in optical lattices, nuclear magnetic resonance, etc. [4 9]. However, fruitful applications and commercialization of these protocols require the implementations of these protocols in a multipartite scenario [10]. For example, quantum dense coding, which is used to transmit classical information, has already been introduced in multipartite systems [11, 12].
Quantum correlations occupy a central position in the quest for understanding and harnessing the power of quantum mechanics. Previously, entanglement has been successfully employed to interpret several phenomena that cannot be understood within classical
E-mail: lqiu'fflcumt .edu.cn
physics [13]. It has also been identified as the vital element for the success of quantum communication protocols [10] and the essential ingredient of quantum computational tasks [14]. Therefore, entanglement is regarded as a unique quantum mechanics trait and considered synonymous with quantum correlations. Conversely, several recent studies have found that separable (i.e., not entangled) states may retain some signatures of quantunmess with potential applications to quantum technology [15 20]. Quantum discord [21, 22] is one of these signatures. The dynamics of quantum discord [23 31] and its physical meaning [32, 33] are extensively studied. Experiments on quantum discord are also implemented [34, 35]. Recently, generalization of quantum discord to multipartite systems has received much attention [36, 37, 38].
To answer the question of whether quantum discord is merely a mathematical construct or has a definable physical role in information processing, the link between quantum discord and actual quantum tasks has been investigated [32,33,39 42]. An operational meaning of geometric quantum discord is given in terms of tclcportation fidelity [40]. For three-qubit pure states, a complementarity relation is established between the capacity of multiport classical information transmission via quantum states and multiparty quantum correlation measures [41]. Inspired by the question, we relate quantum discord to the quantum dense coding capacity in this paper. Moreover, the understanding of quan-
turn discord of multipart it o systems, i.e., systems of more than two particles, is still limited, due to their structural complexity. Therefore, we consider the relation between quantum discord and quantum dense coding capacity for (n + l)-particle quantum states. In the scenario of a single sender and n receivers, we establish a necessary condition of a vanishing discord monogamy score based on the quantum dense coding capacity. Furthermore, in the same and the contrary-scenarios, the relations between quantum discord and the loss of quantum dense coding capacity due to de-coherence are given. The contrary scenario means that there are n senders and only a single receiver.
The paper is organized as follows. We begin with reviews of quantum dense coding capacity and the definition of quantum discord in Sec. 2. In Sec. 3, we give a necessary condition for the vanishing discord monogamy score. In Sec. 4, we establish the relation between quantum discord and the loss of quantum dense coding capacity due to decoherence. We present a conclusion in Sec. 5.
2. QUANTUM DENSE CODING CAPACITY AND QUANTUM DISCORD
Quantum dense coding is a quantum communication protocol by which classical information can be transmitted beyond the classical capacity of a quantum channel. The quantum channel together with a shared quantum state is the available resources for the transmission. Let the sender, called Alice, and the receiver, called Bob, share a bipartite quantum state pab- The amount of classical information that the sender can send to the receiver is given by f11,12,43 47]
C(A,B) =C(pab) = log-2 (1a+S(pb)-S(pAB), (1)
where (1a is the dimension of Alice's Hilbert space, Pb = Tva(pab)< and S(p) = ^Trplog2p is the von Neumann entropy of its argument. The conditional entropy S(pa\b) = S(pab) — S(pb) in the equation can have any sign. In the case where the conditional entropy is negative, the sender can transmit classical information beyond the "classical limit", i.e., log2iiyi, bits to the receiver. For example, when a maximally entangled state is shared between Alice and Bob, the capacity C(A,B) reaches the maximal value. On the contrary, in the case where the conditional entropy is positive, the sender must use a noiseless quantum channel without using a shared quantum state, which is usually referred to as the "classical protocol", to transfer log2 (1a bits of classical information.
We now pass to a brief review of the definition of quantum discord. Quantum discord, defined as the minimum difference between two expressions of mutual information extended from a classical to a quantum system, is introduced to characterize all the nonclassi-cal correlations presented in a bipartite system [21, 22]. The von Neumann mutual information 1 for a bipartite system is given as
1(A, B) = 1(pab) = S(pA) + S(pB) - S(pab). (2)
The mutual information is used to quantify the total correlations.
Conditioned on a complete set of von Neumann measurement Ilf (or, more generally, positive-operator valued measures (POVMs)) performed on subsystem B, the alternative version of quantum mutual information is
J(A,B) = J (pab) = S(pa) - 5'{nf }(Pa\b) =
= S(pa) — min PiS(p.ijj). (3) {nf}
In the equation, the probability of outcome i is pi = Tr „,(/ S O hf pab i a O nf),
and the corresponding post-measurement state for the subsystem ,4 is
pA\i = Tr»!/ s O nf pab I a O n f )//>,
with i a being the identity operator on the Hilbert space of subsystem ,4. Generally, ¿T(A, B) is used to measure the classical correlations in bipartite systems.
Even though the two definitions of mutual information are equivalent for classical systems, their quantum generalizations 1 and J do not coincide in general, and quantum discord is defined as their discrepancy
T>(A,D) = V(PAB) = Apab) - J {PAB). (4)
Quantum discord measures the quantum nature of correlations between two subsystems, and it is always nonnegative. Moreover, quantum discord is in general asymmetric with respect to ,4 and B.
In the subsequent sections, we use S(A,B) to denote S(pab)< and similarly for other quantities.
3. NECESSARY CONDITION FOR THE VANISHING DISCORD MONOGAMY SCORE
In multipartite quantum states, the sharing of quantum correlations among subsystems is often constrained by the concept of monogamy. More precisely,
a bipartito quantum correlation measure Q is said to be monogamous for a (n+ l)-particlo state Pab1b2...b„ if
q(pa\b1b2...b„ ) > qípabí ) +
+ q(pab2 ) + •••+ q(pab„ )• (5)
Here, ,4 is used as the "nodal observer",
q(pabt) = q(r£i'b2...b„ (pab1b2...b„ ))
denotes the quantum correlation (with respect to the measure Q) between the subsystems ,4 and Di, and similarly for others, and Q(pa\b1b2...b„ ) measures quantum correlation of the state in the A\BiB-2 ... Bn bipartite split. When entanglement is quantified by concurrence, such a relation is indeed satisfied, which indicates that two parties cannot have a large amount of entanglement shared with the third party if they are highly entangled [48 51]. As regards quantum discord, Bai et al. [52] proved that the monogamy relation is only satisfied for three-qubit pure states.
The concept of quantum monogamy score, which is independent of whether the given bipartite quantum correlation measure is monogamous, is defined as
ÓQ = Q{A\BÍB2...B„)-Q{A,BÍ)-
-Q(A,B2)-...-Q(A,Bn).
For quantum discord, the discord monogamy score is given as
&D = ViA^B-2 ...Bn)~ T>(A, B1) -
-■D(A,B2)-...-T>(A,Bn). (6)
Now, we present a condition for a vanishing discord monogamy score based on the quantum dense coding capacity. We consider a pure or mixed (n+ l)-particle state Pab1b2...b„ in which the particles can have arbitrary dimensions; a necessary condition for the discord monogamy score to vanish is
■D(A\B1B2...Bn) + J(A,B1) +
+ J(A,B2) + ... + J(A,Bn) < < C(.4, B1) + C(A, B2) + ... + C(.4, B„). (7)
The condition can be obtained easily. Based on the definition of quantum discord
V{A,Bi) =X(A,Di) - J(A,D¡) =
= S(A) + S(Di) - S(A, Bi) - ¿T(A,B¡),
a vanishing discord monogamy score implies that
■D(A\B1B2...Bn) + J(A,B1) +
+ J(A,B2) + ... + J(A,Bn) = = S (A) + S(B!) - S(A,B1) + S (A) + S(B2) -- S(A, B2) + ...+ S (A) + S(Bn ) - S(A, Dn), (8)
where we note that S (A) < log2 (1a- and substitute it into the above equation. From the expression for quantum dense coding capacit
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