Relative Entropy of Discord

• Relative entropy of discord is a quantum correlation measure that quantifies the minimal information loss induced by optimal local dephasing in composite quantum systems.
• It exhibits key properties such as nonnegativity, local unitary invariance, and a consistent hierarchy with mutual information and entanglement measures.
• Its extensions to multipartite systems and generalized entropies enable finer probes for phase transitions and tighter constraints on entanglement distribution in resource theories.

The relative entropy of discord is a quantifier of quantum correlations in composite quantum systems, defined via the distance (quantified by quantum relative entropy) to the nearest "classical" or "classical-quantum" state under local measurements. This measure extends the concept of quantum discord by providing a geometric interpretation within the quantum information theory landscape and enjoys broad operational and axiomatic justification. Its formulations, properties, and physical significance have been rigorously developed, especially in the context of entanglement distribution and resource theories.

1. Formal Definition and Mathematical Structure

Given a density matrix $\rho_{XY}$ on a bipartite Hilbert space $\mathcal{H}_X \otimes \mathcal{H}_Y$, the (Umegaki) quantum relative entropy between states $\sigma$ and $\tau$ is defined as

$$S(\sigma\|\tau) = \mathrm{Tr}[\sigma \log \sigma] - \mathrm{Tr}[\sigma \log \tau].$$

The relative entropy of discord $D_{X|Y}(\rho_{XY})$ is defined by minimizing the relative entropy between $\rho_{XY}$ and the set of "quantum-classical" (QC) states:

$$D_{X|Y}(\rho_{XY}) = \min_{\chi_{X|Y}} S(\rho_{XY}\|\chi_{X|Y}),$$

where $$\chi_{X|Y} = \sum_j p_j \chi^j_X \otimes |j\rangle\langle j|_Y$$ with $\{|j\rangle_Y\}$ an orthonormal basis of $\mathcal{H}_X \otimes \mathcal{H}_Y$0 and $\mathcal{H}_X \otimes \mathcal{H}_Y$1 density operators on $\mathcal{H}_X \otimes \mathcal{H}_Y$2.

Operationally, this minimization admits the equivalent form:

$\mathcal{H}_X \otimes \mathcal{H}_Y$3

where $\mathcal{H}_X \otimes \mathcal{H}_Y$4 denotes a complete projective measurement (dephasing) on $\mathcal{H}_X \otimes \mathcal{H}_Y$5. Thus, $\mathcal{H}_X \otimes \mathcal{H}_Y$6 quantifies the least entropy increase—i.e., informational loss—caused by dephasing $\mathcal{H}_X \otimes \mathcal{H}_Y$7 in an optimally chosen basis. For symmetric discord, one can define minimization over both subsystems, and in multipartite settings global generalizations follow directly (Rulli et al., 2011).

2. Core Properties and Operational Interpretation

The relative entropy of discord exhibits the following properties:

• Nonnegativity and Faithfulness: $\mathcal{H}_X \otimes \mathcal{H}_Y$8 for all $\mathcal{H}_X \otimes \mathcal{H}_Y$9, attaining zero if and only if $\sigma$0 is quantum-classical on $\sigma$1.
• Invariance under Local Unitaries: $\sigma$2.
• Monotonicity: $\sigma$3 for any completely positive trace-preserving (CPTP) map $\sigma$4 on $\sigma$5.
• Relationship to Mutual Information and Entanglement: For all $\sigma$6,

$\sigma$7

where $\sigma$8 is the quantum mutual information and $\sigma$9 is the relative entropy of entanglement (Chuan et al., 2012).

Operational interpretations include the connection to the extractable work in thermodynamic settings (difference between quantum and classical demons), and the resource-theoretic characterization of "quantumness" as the nonclassical advantage in information processing (Costa et al., 2012).

3. Zero-Discord States and Structure Theorems

Zero relative entropy of discord identifies the set of quantum-classical states. For one-sided discord vanishing ($\tau$0), there exists a projective measurement on $\tau$1 such that $\tau$2 can be reconstructed from the post-measurement state. More generally, symmetric zero-discord states are diagonal in some product basis, and this structure can be derived using Petz's monotonicity theorem and recovery maps (Zhang et al., 2012).

The formulation yields:

• For one-sided zero-discord: $\tau$3.
• For symmetric zero-discord: diagonal in a product basis across all parties. These structures generalize via commutation conditions, yielding "lazy" states and other nontrivial classes.

One-sided relative-entropy discord is exactly the conditional mutual information in a suitable dilation, connecting the measure to the strong subadditivity equality structure and state redistribution protocols (Zhang et al., 2012).

4. Multipartite and Symmetric Generalizations

Global quantum discord (GQD), as formulated in (Rulli et al., 2011), extends the relative entropy of discord to multipartite systems by minimizing the difference between the relative entropy of the state and the relative entropy after local dephasing on all subsystems:

$\tau$4

with $\tau$5 the global dephasing map. GQD is symmetric, nonnegative, and local-unitary invariant. For $\tau$6, this reduces to the standard (symmetric) bipartite relative-entropy discord. GQD can signal critical behavior in quantum many-body systems where pairwise or entanglement measures fail, e.g., in infinite-order transitions (Rulli et al., 2011).

5. Entanglement Distribution and Resource Implications

The relative entropy of discord provides tight constraints on entanglement distribution processes. In a tripartite system $\tau$7-$\tau$8-$\tau$9 where $$S(\sigma\|\tau) = \mathrm{Tr}[\sigma \log \sigma] - \mathrm{Tr}[\sigma \log \tau].$$0 is sent from Alice to Bob, the increase in relative entropy of entanglement is bounded:

$$S(\sigma\|\tau) = \mathrm{Tr}[\sigma \log \sigma] - \mathrm{Tr}[\sigma \log \tau].$$1

This bound means that the maximal entanglement that can be distributed is quantified by the discord between $$S(\sigma\|\tau) = \mathrm{Tr}[\sigma \log \sigma] - \mathrm{Tr}[\sigma \log \tau].$$2 (the carrier) and the other parties. Notably, nonzero discord is necessary for entanglement distribution; if $$S(\sigma\|\tau) = \mathrm{Tr}[\sigma \log \sigma] - \mathrm{Tr}[\sigma \log \tau].$$3, entanglement cannot increase via LOCC (Chuan et al., 2012).

This framework exposes cases where unentangled carriers can still facilitate entanglement distribution, with the entanglement gain tightly bounded by the communicated discord. The bound is tight in paradigmatic protocols, including those based on three-qubit separable-carrier and Werner-mixture states.

6. Generalizations: Tsallis, Rényi, and Sandwiched Relative Entropies

Moving beyond Umegaki (von Neumann) relative entropy, generalized relative entropy of discord definitions employ alternate divergences, enabling finer control and new operational interpretations:

• Tsallis and Rényi Discord: For parameter $$S(\sigma\|\tau) = \mathrm{Tr}[\sigma \log \sigma] - \mathrm{Tr}[\sigma \log \tau].$$4, the Tsallis relative entropy is $$S(\sigma\|\tau) = \mathrm{Tr}[\sigma \log \sigma] - \mathrm{Tr}[\sigma \log \tau].$$5.
• Sandwiched Relative Entropy: For $$S(\sigma\|\tau) = \mathrm{Tr}[\sigma \log \sigma] - \mathrm{Tr}[\sigma \log \tau].$$6, $$S(\sigma\|\tau) = \mathrm{Tr}[\sigma \log \sigma] - \mathrm{Tr}[\sigma \log \tau].$$7 and $$S(\sigma\|\tau) = \mathrm{Tr}[\sigma \log \sigma] - \mathrm{Tr}[\sigma \log \tau].$$8 provide further one-parameter families of discord measures (Misra et al., 2014, Vershynina, 2019).

The standard relative entropy of discord is recovered in the appropriate limit ($$S(\sigma\|\tau) = \mathrm{Tr}[\sigma \log \sigma] - \mathrm{Tr}[\sigma \log \tau].$$9). These generalizations satisfy core properties (nonnegativity, contractivity, local-unitary invariance) for suitable parameter ranges, and for selected state families provide improved sensitivity as order parameters in detecting phase transitions. For instance, the sandwiched-Rényi discord exhibits sharply diverging derivatives at the critical point in the transverse-field Ising chain, outperforming standard discord and entanglement measures in finite-size scaling (Misra et al., 2014).

7. Applications, Examples, and Special Cases

Representative Calculations:

• For symmetric "X"-states (two-qubit and qubit-qutrit), the minimization in the relative-entropy discord simplifies to dephasing in the computational basis. Analytically, $D_{X|Y}(\rho_{XY})$0, with $D_{X|Y}(\rho_{XY})$1 the dephased (classical-classical) state (Mahdian et al., 2013).
• For pure bipartite states, the discord coincides with entanglement entropy; in multipartite settings, it reduces to shared mutual information post optimized local measurements (Rulli et al., 2011).
• Discord based on correlated coherence also leads to a relative-entropy measure coinciding with the original one-sided quantum discord and satisfies resource monotonicity properties under incoherent operations (Guo et al., 2016).

Physical Relevance:

The relative entropy of discord thus captures the minimal nonclassical correlation necessary for entanglement distribution, the loss of extractable information due to "classicalization," and defines a hierarchy bridging mutual information and entanglement. Its generalizations further allow for parameter-tunable probes of quantum correlations in complex, many-body, or thermodynamic settings.

Summary Table: Core Discord Types

Name	Distance Function	Class of Reference States
Relative-entropy discord	$D_{X|Y}(\rho_{XY})$2	Classical-Quantum (CQ)/CC
Tsallis/Rényi discord	$D_{X|Y}(\rho_{XY})$3	Classical-Quantum (CQ)/CC
Sandwiched-entropy discord	$D_{X|Y}(\rho_{XY})$4	Classical-Quantum (CQ)/CC

For all, the minimization is over the set of classical states (product of local orthonormal projectors), with the operational content controlled by the choice of the divergence.

References: (Chuan et al., 2012, Mahdian et al., 2013, Zhang et al., 2012, Guo et al., 2016, Rulli et al., 2011, Misra et al., 2014, Vershynina, 2019, Costa et al., 2012).