LOCC entanglement cost of entangled qubit Bell-diagonal states

Problem

What is the LOCC entanglement cost of an entangled qubit Bell-diagonal state? Let \(\mathbf p=(p_I,p_X,p_Y,p_Z)\) be a probability vector, and let \(\lvert\Phi^+\rangle=(\lvert00\rangle+\lvert11\rangle)/\sqrt2\) and \(\lvert\Phi_P\rangle=(I\otimes P)\lvert\Phi^+\rangle\) for \(P\in\{I,X,Y,Z\}\). The state is

\begin{equation} \rho_{\mathbf p} :=\sum_{P\in\{I,X,Y,Z\}}p_P \lvert\Phi_P\rangle\!\langle\Phi_P\rvert, \qquad p_P\geq0, \qquad \sum_{P\in\{I,X,Y,Z\}}p_P=1. \tag{1} \end{equation}

The state in Eq. (1) is entangled exactly when \(p_\star:=\max\{p_I,p_X,p_Y,p_Z\}>1/2\) [ZCH10]. Determine \(E_C(\rho_{\mathbf p})\) for every probability vector satisfying \(p_\star>1/2\), where \(E_C\) is the infimum asymptotic rate of ebits consumed by protocols using local operations and classical communication (LOCC) that prepare \(\rho_{\mathbf p}^{\otimes n}\) with trace-norm error tending to zero.

Source

The question for entangled qubit Bell-diagonal states is derived from the regularized-entanglement-of-formation characterization of LOCC entanglement cost and Wootters’ one-copy formula for two-qubit states [HHT01], [Woo98].

Progress

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Let \(p_\star:=\max\{p_I,p_X,p_Y,p_Z\}\), \(C_{\mathbf p}:=\max\{0,2p_\star-1\}\), and \(h_2(x):=-x\log_2x-(1-x)\log_2(1-x)\), with \(0\log_2 0:=0\). Regularized relative entropy of entanglement, entanglement cost, and Wootters’ one-copy entanglement of formation give the rigorous bounds

\begin{equation} L(p_\star)=E_R^\infty(\rho_{\mathbf p}) \leq E_C(\rho_{\mathbf p}) \leq E_F(\rho_{\mathbf p}) =h_2\!\left(\frac{1+\sqrt{1-C_{\mathbf p}^{2}}}{2}\right), \qquad L(t):= \begin{cases} 0, & 0\leq t\leq\tfrac12,\\ 1-h_2(t), & \tfrac12<t\leq1. \end{cases} \tag{2} \end{equation}

Here \(E_C=E_F^\infty\) [HHT01]; Wootters gives the right endpoint [Woo98]; and additivity of the relative entropy of entanglement gives \(E_R^\infty(\rho_{\mathbf p})=L(p_\star)\) [ZCH10]. By themselves, the bounds in Eq. (2) determine the cost for \(p_\star\leq1/2\) but leave a gap for \(1/2<p_\star<1\). A 2025 semidefinite-programming construction gives a faithful, efficiently computable lower bound on \(E_C(\rho_{\mathbf p})\) for every entangled two-qubit Bell-diagonal state, without determining the exact cost [WJZ25].
If at most two Bell probabilities are nonzero, say \(q\) and \(1-q\), then the entanglement of formation is strongly additive and

\begin{equation} E_C(\rho_{\mathbf p})=E_F(\rho_{\mathbf p}) =h_2\!\left(\frac12+\sqrt{q(1-q)}\right). \tag{3} \end{equation}

Vidal, Dür, and Cirac proved Eq. (3) for a mixture of \(\lvert\Phi^+\rangle\) and \(\lvert\Phi^-\rangle\); local unitaries extend it to any pair of Bell states [VDC02]. This earlier result settled the rank-at-most-two boundary of the family.
On the isotropic line, \(p_X=p_Y=p_Z\), write the probabilities and the known one-copy value as

\begin{equation} p_I=F, \qquad p_X=p_Y=p_Z=\frac{1-F}{3}, \qquad E_F(\rho_{\mathbf p})= \begin{cases} 0, & 0\leq F\leq\tfrac12,\\ h_2\!\left(\dfrac12+\sqrt{F(1-F)}\right), & \tfrac12<F\leq1. \end{cases} \tag{4} \end{equation}

Terhal and Vollbrecht determined this one-copy quantity [TV00]; it is also the qubit specialization of Wootters’ formula. Fang’s subsequent result also identifies this value with \(E_C\) throughout \(0\leq F\leq1\) [Fan26].
Fang’s 2026 preprint reports the complete resolution: Theorem 7, Eq. (40), gives, for every probability vector in Eq. (1),

\begin{equation} E_C(\rho_{\mathbf p})=E_F(\rho_{\mathbf p}) =h_2\!\left(\frac{1+\sqrt{1-C_{\mathbf p}^{2}}}{2}\right). \tag{5} \end{equation}

Its Eq. (33) gives \(E_F(\rho_{\mathbf p}^{\otimes n})\geq nE_F(\rho_{\mathbf p})\) for every integer \(n\geq1\); subadditivity and regularization yield Eq. (5) [Fan26]. The proof uses exact tensorization for primitive reversible qubit semigroups with positive off-diagonal scaling, established in Theorem 3.14 of [DGO+26].

Comment

The Solved designation records the full resolution reported in Fang’s preprint [Fan26], Theorem 7, Eq. (40). Equation (5) covers all qubit Bell-diagonal states, including entangled rank-three and rank-four states. The amplitude-damping Choi-state entanglement-cost problem asks the analogous question for a different two-qubit family.

References

[HHT01]: P. M. Hayden, M. Horodecki, and B. M. Terhal, “The Asymptotic Entanglement Cost of Preparing a Quantum State,” Journal of Physics A: Mathematical and General 34, 6891–6898 (2001).DOI arXiv
[Woo98]: W. K. Wootters, “Entanglement of Formation of an Arbitrary State of Two Qubits,” Physical Review Letters 80, 2245–2248 (1998).DOI arXiv
[ZCH10]: H. Zhu, L. Chen, and M. Hayashi, “Additivity and Non-Additivity of Multipartite Entanglement Measures,” New Journal of Physics 12, 083002 (2010).DOI arXiv
[WJZ25]: X. Wang, M. Jing, and C. Zhu, “Computable and Faithful Lower Bound on Entanglement Cost,” Physical Review Letters 134, 190202 (2025).DOI arXiv
[VDC02]: G. Vidal, W. Dür, and J. I. Cirac, “Entanglement Cost of Bipartite Mixed States,” Physical Review Letters 89, 027901 (2002).DOI arXiv
[TV00]: B. M. Terhal and K. G. H. Vollbrecht, “Entanglement of Formation for Isotropic States,” Physical Review Letters 85, 2625–2628 (2000).DOI arXiv
[Fan26]: K. Fang, “Entanglement cost of quantum depolarization,” preprint (2026).arXiv
[DGO+26]: Y. Dong, L. Gao, F. Ou, P. Yao, and H. Zhou, “Dimension-Free Approximate Tensorization of Quantum Hypercontractivity for Qudit Depolarizing Semigroups,” preprint (2026).DOI arXiv

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