Relative entropy of entanglement for two qubits

Unsolved ID op_e2149f4ced34d1a8 Last edited 4 September 2026

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Field: Quantum Resource Theory
Topics: Entanglement measures
Quantum separability
Quantum relative entropy

Problem

Find a closed formula for the relative entropy of entanglement of every two-qubit density operator \(\rho\), including an explicit closest separable state. With \(\operatorname{Sep}(\mathbb{C}^2:\mathbb{C}^2)\) denoting the two-qubit separable states, the quantity is

\begin{equation} E_R(\rho) :=\min_{\sigma\in\operatorname{Sep}(\mathbb{C}^2:\mathbb{C}^2)} D(\rho\Vert\sigma), \qquad D(\rho\Vert\sigma) :=\begin{cases} \operatorname{Tr}\!\left[\rho(\log\rho-\log\sigma)\right], &\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma,\\ +\infty,&\text{otherwise}. \end{cases} \tag{1} \end{equation}

In the finite branch of Eq. (1), the trace is evaluated on \(\operatorname{supp}\rho\), with \(0\log 0:=0\). The formula sought must determine at least one minimizing state \(\sigma_\rho\) for every \(\rho\).

Source

The general two-qubit formula is listed by Krüger and Werner, and Miranowicz and Ishizaka explicitly distinguish this unresolved forward problem from their solved inverse construction [KW05], [MI08].

Progress

For two qubits, separability is equivalent to positivity under partial transpose. Thus Eq. (1) is a convex optimization over the positive-partial-transpose set, but this equivalence does not produce a closed optimizer [HHH96].
Miranowicz and Ishizaka solved the inverse problem: from a boundary separable state \(\sigma\), they parameterized entangled states for which \(\sigma\) is closest. Their construction yields formulas for special families but does not invert to a closed map \(\rho\mapsto\sigma_\rho\) for an arbitrary input [MI08].
Friedland and Gour extended the inverse optimality characterization to general dimensions and proved uniqueness of the closest separable state for full-rank entangled inputs. These structural results still do not evaluate Eq. (1) in closed form for every two-qubit state [FG11].

Comment

The source collection and the full discussion of the inverse construction explicitly identify the forward formula as open [KW05], [MI08]. The unresolved step is a closed determination of \(\sigma_\rho\) from arbitrary input data \(\rho\); special-state formulas and an inverse parameterization do not supply it.

References

[HHH96]: M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of Mixed States: Necessary and Sufficient Conditions,” Physics Letters A 223, 1–8 (1996).DOI arXiv
[MI08]: A. Miranowicz and S. Ishizaka, “Closed Formula for the Relative Entropy of Entanglement,” Physical Review A 78, 032310 (2008).DOI arXiv
[FG11]: S. Friedland and G. Gour, “Closed Formula for the Relative Entropy of Entanglement in All Dimensions,” Journal of Mathematical Physics 52, 052201 (2011).DOI arXiv
[KW05]: O. Krüger and R. F. Werner (eds.), “Some Open Problems in Quantum Information Theory,” arXiv:quant-ph/0504166 (2005).DOI arXiv

Relative entropy of entanglement for two qubits

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