Quantum Discord

Quantum discord is defined as the difference between two classically equivalent expressions for mutual information that become inequivalent for quantum states. The total mutual information of a bipartite system AB is I(A:B) = S(rho_A) + S(rho_B) - S(rho_AB), where S denotes von Neumann entropy. The classical correlation J(A:B) is defined by optimizing over all local measurements on subsystem B: J(A:B) = S(rho_A) - min_{Pi_B} S(rho_A | Pi_B), where the minimization is over positive operator-valued measures on B. Quantum discord is then D(A:B) = I(A:B) - J(A:B), the portion of mutual information that cannot be captured by any local classical measurement.

A striking feature of quantum discord is that it can be nonzero even for states that contain no entanglement. Separable mixed states, which can be written as convex mixtures of product states and thus cannot be distilled into Bell pairs, may still carry quantum correlations that no local measurement on one subsystem can fully extract. This means entanglement is a proper subset of quantum correlations, and discord captures the broader class. For pure states, discord reduces to the entanglement entropy, but for mixed states it provides genuinely new information about the quantumness of correlations present in the system.

Computing quantum discord is NP-hard in general because the optimization over measurements on subsystem B involves a search over an infinite-dimensional space of POVMs. Analytical results exist for special families of two-qubit states, including X-states (those whose density matrix is nonzero only along the main diagonal and anti-diagonal), but even there subtle errors appeared in the early literature. This computational difficulty has motivated geometric measures of discord that replace the measurement optimization with a distance to the set of classically correlated states, trading operational meaning for tractability.

The DQC1 (deterministic quantum computation with one clean qubit) model illustrates why discord matters for quantum advantage. DQC1 evaluates the normalized trace of a unitary using one clean qubit coupled to n maximally mixed qubits, achieving an exponential speedup over known classical algorithms even though the circuit generates negligible entanglement. Datta, Shaji, and Caves showed in 2008 that the quantum discord between the clean qubit and the mixed register scales with the size of the speedup, suggesting discord rather than entanglement is the relevant resource. This connection motivates ongoing research into discord as a resource in quantum sensing, thermodynamics, and quantum communication protocols.