Measure of quantum correlations that lies approximately between entanglement and discord

When a quantum system is divided into two local subsystems, measurements on the two subsystems can exhibit correlations beyond those possible in a classical joint probability distribution; these are partially explained by entanglement, and more generally by a wider class of measures such as the quantum discord. In this paper, I introduce a simple thought experiment defining a measure of quantum correlations, which I call the accord, and write the result as a minimax optimization over unitary matrices. I find the exact result for pure states as a simple function of the Schmidt coefficients and provide a complete proof, and I likewise provide and prove the result for several classes of mixed states, notably including all states of two qubits and the experimentally relevant case of a pure state mixed with colorless noise. I demonstrate that for two qubit states the accord provides a tight lower bound on the discord; for Bell diagonal states it is also an upper bound on entanglement.

Figure 1

(a) Various measures of quantum correlations computed for isotropic states of the form given in Eq. (43) with $d=2$. Entanglement, teleportation fidelity, nonlocality, and nonclassicality are captured by concurrence, singlet fraction, violations of the CHSH inequality, and discord, respectively. The accord appears similar to the discord. (b) Accord ($A$) vs concurrence ($C$) for ${10}^6$ randomly generated Bell diagonal states. Evidently these satisfy $C\left(\rho \right)\le A\left(\rho \right)$. (c) Accord vs discord ($D$) for ${10}^6$ randomly generated Bell diagonal states. These satisfy $J\left[A\right(\rho \left)\right]\le D\left(\rho \right)$ where the function $J$ is defined in Eq. (55).

Figure 2

(a) Accord vs concurrence for ${10}^6$ arbitrary two qubit states found by tracing out two sites of random four qubit pure states [71]; a line showing $C\left(\rho \right)=A\left(\rho \right)$ is provided as a guide to the eye. The inset shows the distribution of $\left(\text{accord-concurrence}\right)$, computed as a histogram with ${10}^3$ bins. Surprisingly, it is possible for states to be entangled while having no measurement correlations. (b) Accord vs discord, computed by numerical optimization, for ${10}^5$ arbitrary two qubit states, again found by tracing out sites from four qubit pure states [72]. The solid line shows the conjectured bound $J\left[A\right(\rho \left)\right]\le D\left(\rho \right)$; the bound is never violated. The dashed line shows the relation for pure states. (c) Accord vs discord for ${10}^5$ two qubit states drawn from a different distribution [71].

Figure 3

(a) Tensor network representation of Eq. (A1). Each node is a tensor, and lines indicate tensor contraction. Filled circles are higher-dimensional identity tensors, the diamonds are diagonal matrices the diagonal entries of which are the Schmidt coefficients, and for nondiagonal matrices inward arrows indicate the row index and outward arrows indicate the column index. (b) Derivation of Eq. (21) using the identities from Fig. 4. The circle labeled “c” is a vector the entries of which are the Schmidt coefficients. (c) Derivation of Eq. (22).

Figure 4

Tensor network identities used in deriving Eqs. (21) and (22). The notation used is defined in the caption of Fig. 3.

Figure 5

(a) Tensor network representation of $〈{\psi }_{\text{m}}|\psi 〉$ from the calculation of the singlet fraction. (b) The network immediately simplifies to this one, where $U$ is some unitary matrix.