Conservation law for distributed entanglement of formation and quantum discord

Abstract

We present a direct relation, based upon a monogamic principle, between entanglement of formation (EOF) and quantum discord (QD), showing how they are distributed in an arbitrary tripartite pure system. By extending it to a paradigmatic situation of a bipartite system coupled to an environment, we demonstrate that the EOF and the QD obey a conservation relation. By means of this relation we show that in the deterministic quantum computer with one pure qubit the protocol has the ability to rearrange the EOF and the QD, which implies that quantum computation can be understood on a different basis as a coherent dynamics where quantum correlations are distributed between the qubits of the computer. Furthermore, for a tripartite mixed state we show that the balance between distributed EOF and QD results in a stronger version of the strong subadditivity of entropy.the pure qubit and the mixed ones [4]. Arguably, the power of the quantum computer is supposed to be related to QD, rather than entanglement [6]. Here, using the conservation relation, we have shown that even in the supposedly entanglement-free quantum computation there is a certain amount of multipartite entanglement between the qubits and the environment, which is responsible for the non-zero QD (See Fig. 1).

Why entanglement of formation is not generally monogamous

“…Since δ ← AB is related to the EOF of the pair AC, E AC , through the relation [8,19] δ ← AB = E AC + S A|C , and the fact that S(A|B) = −S(A|C) for pure states, we obtain…”

“…Given the definitions above we now discuss the condition for a monogamous distribution of EOF through the system. We begin by using the conservation law for distribution of the EOF and QD [8] …”

“…It is not surprising though that both forms of quantum correlation can be related to each other through extended system [7] distribution formulas. For example, it is possible to describe a conservation relation [8] for distribution of the EOF and quantum discord (QD), a measure of quantum correlation; for an arbitrary tripartite pure system, the sum of the QD of a chosen partition, given measurements on the complementary partitions, must be equal to the sum of the pairwise EOF between the chosen partition and the complementary ones. Surprisingly, the sum of the pairwise EOFs appears in a fashion quite similar to the desired expression for the so-called monogamy of entanglement, and the relation obtained can be connected to the way that classical correlations [9,10] are distributed [11][12][13].…”

Unified scheme for correlations using linear relative entropy

“…Analytical results are known only in a few families of two-qubit states [27,28,29,30,45,46,47] (see also [48,49] and references therein). The alternative geometric way, proposed in [35], quantifies the quantum discord as the minimal Hilbert-Schmidt distance between a given state ρ and the closest classical states of the form…”

Quantumness and the role of locality on quantum correlations

“…The measure δ attempts to capture the minimal disturbance suffered by the state under a local non-selective measurement. Also, QD is an essential resource in the performance of many quantum protocols [27][28][29][30][31]. Thus, Potential Discord (PD) should be defined as…”