Quantum Information and Quantum Computation
Open Problem Zoo
Random Unsolved Solved

Relative entropy of entanglement for two qubits

Unsolved ID op_e2149f4ced34d1a8 Last edited 25 September 2026
Edit
Field
Topics

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

Reports do not certify correctness or automatically change the problem's status. Progress policy.

  • 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].

  • Reported progress: Issue #104 (withdrawn).

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).DOIarXiv
[MI08]
A. Miranowicz and S. Ishizaka, “Closed Formula for the Relative Entropy of Entanglement,” Physical Review A 78, 032310 (2008).DOIarXiv
[FG11]
S. Friedland and G. Gour, “Closed Formula for the Relative Entropy of Entanglement in All Dimensions,” Journal of Mathematical Physics 52, 052201 (2011).DOIarXiv
[KW05]
O. Krüger and R. F. Werner (eds.), “Some Open Problems in Quantum Information Theory,” arXiv:quant-ph/0504166 (2005).DOIarXiv

Cite this problem

Please also cite the primary sources listed under References. Cite this page for the statement, status, and stable identifier.

BibTeX

@incollection{qiqcop_op_e2149f4ced34d1a8,
  title = {Relative entropy of entanglement for two qubits},
  booktitle = {Quantum Information and Quantum Computation Open Problem Zoo (QIQCOP Zoo)},
  year = {2026},
  howpublished = {\url{https://qiqc-op.com/problem/op_e2149f4ced34d1a8/}},
  note = {Stable ID op_e2149f4ced34d1a8; status: Unsolved; accessed 2026-10-08}
}

Plain text

“Relative entropy of entanglement for two qubits,” Quantum Information and Quantum Computation Open Problem Zoo (QIQCOP Zoo), https://qiqc-op.com/problem/op_e2149f4ced34d1a8/, ID op_e2149f4ced34d1a8, accessed 2026-10-08.

Share this problem

Permanent link

Identifiers

op_e2149f4ced34d1a8
01M1HME780JCWZMCJMARNSAXH3