Entanglement of formation of generalized Bell-diagonal states

Problem

For every local dimension \(d\geq3\), determine the entanglement of formation of an arbitrary Weyl–Bell-diagonal state. Let \(\omega_d:=\exp(2\pi i/d)\) and define the generalized Pauli operators and their associated Bell basis by

\begin{equation} X\lvert j\rangle:=\lvert j+1\!\!\pmod d\rangle, \qquad Z\lvert j\rangle:=\omega_d^j\lvert j\rangle, \qquad \lvert\Phi_{a,b}\rangle :=(I\otimes X^aZ^b)\lvert\Phi_d\rangle, \qquad \lvert\Phi_d\rangle:=\frac1{\sqrt d}\sum_{j=0}^{d-1}\lvert j,j\rangle, \tag{1} \end{equation}

where \(a,b\in\mathbb Z_d\). In this problem, “Pauli-diagonal” means diagonal in the generalized Bell basis in Eq. (1). Thus the state is

\begin{equation} \rho_{\mathbf p} :=\sum_{a,b\in\mathbb Z_d}p_{a,b} \lvert\Phi_{a,b}\rangle\!\langle\Phi_{a,b}\rvert, \qquad p_{a,b}\geq0, \qquad \sum_{a,b\in\mathbb Z_d}p_{a,b}=1. \tag{2} \end{equation}

For every probability array \(\mathbf p\) in Eq. (2), determine an evaluable exact formula and an optimal pure-state ensemble for

\begin{equation} E_F(\rho_{\mathbf p}) :=\inf_{\rho_{\mathbf p}=\sum_i q_i \lvert\psi_i\rangle\!\langle\psi_i\rvert} \sum_i q_i\, S\!\left(\operatorname{Tr}_B \lvert\psi_i\rangle\!\langle\psi_i\rvert\right), \qquad S(\sigma):=-\operatorname{Tr}(\sigma\log_2\sigma). \tag{3} \end{equation}

The infimum in Eq. (3) is over finite pure-state ensembles with \(q_i\geq0\) and \(\sum_iq_i=1\).

Source

The problem is implicit in Vollbrecht and Werner’s finite-Weyl symmetry construction and general convex-roof reduction [VW01]. Terhal and Vollbrecht’s solution of the isotropic subfamily isolates a one-parameter slice rather than the full generalized Bell simplex [TV00].

Progress

For \(d=2\), Wootters determined \(E_F\) for every two-qubit state and therefore for every ordinary Bell-diagonal state [Woo98]. This formula relies on the two-qubit concurrence and does not extend to the general \(d\geq3\) probability array in Eq. (2).
The finite-Weyl twirl and the symmetry method of Vollbrecht and Werner reduce the full convex roof to a phase optimization followed by a convexification. More precisely, define

\begin{equation} \varepsilon_d(\mathbf p) :=\min_{\boldsymbol\theta\in[0,2\pi)^{d^2}} S\!\left(\operatorname{Tr}_B \lvert\psi_{\mathbf p,\boldsymbol\theta}\rangle \!\langle\psi_{\mathbf p,\boldsymbol\theta}\rvert\right), \qquad \lvert\psi_{\mathbf p,\boldsymbol\theta}\rangle :=\sum_{a,b\in\mathbb Z_d} \sqrt{p_{a,b}}e^{i\theta_{a,b}}\lvert\Phi_{a,b}\rangle. \tag{4} \end{equation}

Their theorem gives

\begin{equation} E_F(\rho_{\mathbf p}) =\operatorname{co}\varepsilon_d(\mathbf p), \tag{5} \end{equation}

where \(\operatorname{co}\) denotes the lower convex envelope on the probability simplex. Equations (4) and (5) are an exact reduction, but the phase minimum and its convex envelope remain unknown for a general \(\mathbf p\) [VW01].
For the isotropic probability array

\begin{equation} p_{0,0}=F, \qquad p_{a,b}=\frac{1-F}{d^2-1}\quad((a,b)\neq(0,0)), \qquad 0\leq F\leq1, \tag{6} \end{equation}

Terhal and Vollbrecht reduced the answer to the convex hull of an explicit function [TV00]; Fei and Li-Jost subsequently proved the conjectured convex-hull shape in every dimension [FLJ06]. Hence the states in Eq. (6) are solved exactly, but they form only a one-dimensional subfamily of the \((d^2-1)\)-dimensional simplex.
A 2023 analysis of the same Weyl–Bell simplex classified Bell-diagonal qutrit and ququart states using analytical and numerical separability criteria, but left \(22.6\%\) of its sampled PPT ququart states unclassified as separable or bound entangled [PH23]. Because \(E_F(\rho)=0\) exactly for separable states, this unresolved zero-versus-positive boundary is already an obstruction to a general exact formula; that work does not evaluate \(E_F\) on the unclassified states.

Comment

The unresolved task is to evaluate the phase minimum in Eq. (4), determine its lower convex envelope in Eq. (5), and construct optimal ensembles for arbitrary \(\mathbf p\) when \(d\geq3\). Problem 7 instead asks for the regularized entanglement cost of qubit Bell-diagonal states.

References

[VW01]: K. G. H. Vollbrecht and R. F. Werner, “Entanglement Measures under Symmetry,” Physical Review A 64, 062307 (2001).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
[Woo98]: W. K. Wootters, “Entanglement of Formation of an Arbitrary State of Two Qubits,” Physical Review Letters 80, 2245–2248 (1998).DOI arXiv
[FLJ06]: S.-M. Fei and X. Li-Jost, “\(R\) Function Related to Entanglement of Formation,” Physical Review A 73, 024302 (2006).DOI arXiv
[PH23]: C. Popp and B. C. Hiesmayr, “Comparing Bound Entanglement of Bell Diagonal Pairs of Qutrits and Ququarts,” Scientific Reports 13, 2037 (2023).DOI arXiv

Entanglement of formation of generalized Bell-diagonal states

Problem

Source

Progress

Comment

References

Cite this problem

BibTeX

Plain text

Problem

Source

Progress

Comment

References

Related problems