Entanglement cost of a qubit Bell-diagonal state
- Field
- Topics
Problem
What is the entanglement cost of a qubit Bell-diagonal state for an arbitrary probability vector \(\mathbf p=(p_I,p_X,p_Y,p_Z)\)? 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
For the state in Eq. (1), determine \(E_C(\rho_{\mathbf p})\) for every \(\mathbf p\) in the probability simplex, where \(E_C\) is the infimum asymptotic rate of ebits consumed by LOCC protocols that prepare \(\rho_{\mathbf p}^{\otimes n}\) with trace-norm error tending to zero.
Source
The problem is implicit in the regularized-entanglement-of-formation characterization of entanglement cost and Wootters’ one-copy formula for two-qubit states [HHT01], [Woo98].
Progress
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]. Thus Eq. (2) solves \(p_\star\leq1/2\) but generally leaves a gap for \(p_\star>1/2\). A recent semidefinite-programming construction gives a faithful, efficiently computable lower bound on \(E_C(\rho_{\mathbf p})\) for every entangled two-qubit Bell-diagonal state, but does not close the gap [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 settles the rank-at-most-two boundary of the family, not generic rank-three or rank-four states.
On the isotropic line singled out in the question, \(p_X=p_Y=p_Z\), 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. For \(1/2<F<1\), no proof is known that it equals the asymptotic LOCC entanglement cost.
Comment
Equation (4) determines \(E_C\) only for \(F\leq1/2\) and \(F=1\); for \(1/2<F<1\) it is an \(E_F\) formula. The open task is to regularize \(E_F\) for entangled rank-three and rank-four Bell-diagonal states. Problem 8 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).DOIarXiv
- [Woo98]
- W. K. Wootters, “Entanglement of Formation of an Arbitrary State of Two Qubits,” Physical Review Letters 80, 2245–2248 (1998).DOIarXiv
- [ZCH10]
- H. Zhu, L. Chen, and M. Hayashi, “Additivity and Non-Additivity of Multipartite Entanglement Measures,” New Journal of Physics 12, 083002 (2010).DOIarXiv
- [WJZ25]
- X. Wang, M. Jing, and C. Zhu, “Computable and Faithful Lower Bound on Entanglement Cost,” Physical Review Letters 134, 190202 (2025).DOIarXiv