Circulant States with Vanishing Quantum Discord

Quantum state of composite system contains both classical and quantum correlation. Usually by quantum correlations one means quantum entanglement which provides an essential resource for quantum information processing such as quantum cryptography, dense coding, quantum computing [1, 2]. However, a quantum state of a composed system may contain other types of nonclassical correlation even if it is separable (not entangled). For a recent “catalogue" of nonclassical correlations, see [3]. The most popular measure of such correlations – quantum discord – was introduced by Ollivier and Zurek [4] and independently by Henderson and Vedral [5]. Recently quantum discord has received increasing attention. It was analyzed in the context of broadcasting of quantum states [6, 7]. Interestingly, it turned out that quantum discord might be responsible for the quantum computational efficiency of some quantum computation tasks [8–11]. Moreover, the dynamics of discord, both Markovian and non-Markovian, [12–21] was analyzed. Quantum discord was generalized for continuous variables to study correlations in Gaussian states [22, 23]. A geometric measure for quantum discord was introduced in [24] and analyzed in [25]. Finally, an operational interpretation of quantum discord was provided in [26, 27] – quantum discord received clear information-theoretic operational meaning in terms of entanglement consumption in an extended quantum state merging protocol. Remarkable progress in characterization of set of zero-discord states was done. Interestingly, it was shown [28] to have vanishing volume in the set of all states. Actually, this result holds true for any Hilbert space dimension. It shows that a generic state of composed quantum system does contain non-classical correlation. Necessary and sufficient conditions were provided [24, 29–31], to determine states with vanishing discord. Moreover nonlinear witnesses of discord were introduced [32–34]. For very recent papers analyzing various aspects of quantum discord see also [35–41]. In the present paper we analyze a large class of two-qudit states introduced in [42] called circulant states (see also [43]). Construction of these states is based on a certain decomposition of the total ddimensional Hilbert space into d mutually orthogonal d-dimensional subspaces. A density matrix ρ representing a circulant state is a convex combination of density matrices supported on different subspace. Interestingly, circulant states provide natural generalization of so called X-states of two qubits. In the present paper we address a question when a quantum discord for a circulant state does vanish. The paper is organized as follows: in Section II we recall basis definitions related to quantum discord and formulate necessary and sufficient condition for the vanishing discord. Section III introduces the construction of circulant states in C⊗C and analyzes the case of discord zero. Section IV discusses several well known examples of circulant states invariant under then action of the symmetry group – a subgroup of the unitary group U(d). In Section V we investigate special class of circulant states – so called Bell diagonal states [44, 45] – which play important role in the theory of quantum entanglement. To simplify our discussion we consider only the case when d is prime. Prime dimensions already appeared for example in the discussion of mutually unbiased basis [46]. It turns out that in this case the discussion considerably simplifies. The general case may we analyzed in the same way. However, the corresponding analysis is technically much more involved. We illustrate our discussion with several examples and conclude in the last Section.