Squashed entanglement of qubit Bell-diagonal states
- Field
- Topics
Problem
Is there a closed-form formula for the squashed entanglement \(E_{\mathrm{sq}}\) of an arbitrary two-qubit Bell-diagonal state, or a certified approach to determining it numerically? Let \(A\) and \(B\) be qubits, let \(I,X,Y,Z\) be the identity and Pauli matrices, and set \(\lvert\Phi^+\rangle:=(\lvert00\rangle+\lvert11\rangle)/\sqrt2\) and \(\lvert\Phi_P\rangle:=(I\otimes P)\lvert\Phi^+\rangle\) for \(P\in\{I,X,Y,Z\}\). The states under consideration are
With \(S(\tau):=-\operatorname{Tr}(\tau\log_2\tau)\), define the squashed entanglement of the state in Eq. (1) by
The infimum in Eq. (2) ranges over all finite-dimensional quantum systems \(E\), with no fixed bound on their dimension. Values are in ebits; this is the quantum squashed entanglement of [CW04].
A certified numerical approach should, for any specified probabilities \(\mathbf p\) and tolerance \(\varepsilon>0\), produce an estimate of \(E_{\mathrm{sq}}(\rho_{\mathbf p})\) with a rigorously guaranteed absolute error at most \(\varepsilon\). Restricting \(E\) to a classical register, or to a chosen finite dimension, suffices only if the restriction is proved optimal or its approximation error is controlled.
Source
This question specializes the computational obstruction documented by Fawzi and Fawzi, who study two-qubit Werner states in Sec. 3.2 and leave convergence and extension-dimension bounds open in Sec. 4 [FF22]. The present catalog formulation asks for a closed-form formula or certified numerical evaluation for the full Bell-diagonal family; it is not quoted verbatim from that paper.
Progress
Write \(p_\star:=\max_Pp_P\). The separable region is exactly \(p_\star\leq1/2\), as follows from the Bell-diagonal specialization of Wootters’ concurrence formula, Eqs. (8)–(10) [Woo98]. Squashed entanglement is zero there by Theorem 7 of [CW04], and is strictly positive for \(p_\star>1/2\) by faithfulness, Corollary 1 of [BCY11]. At every pure Bell vertex, \(p_\star=1\), it is exactly one ebit by the pure-state evaluation in Example 2 of [CW04].
Define \(H(\mathbf p):=-\sum_Pp_P\log_2p_P\), \(h_2(x):=-x\log_2x-(1-x)\log_2(1-x)\), with \(0\log_2 0:=0\), and \(C_{\mathbf p}:=\max\{0,2p_\star-1\}\). Explicit bounds for every Bell-diagonal state are
\begin{equation} \max\{0,1-H(\mathbf p)\} \leq E_{\mathrm{sq}}(\rho_{\mathbf p}) \leq\min\!\left\{ 1-\frac{H(\mathbf p)}2, h_2\!\left(\frac{1+\sqrt{1-C_{\mathbf p}^2}}2\right) \right\}. \tag{3} \end{equation}The lower bound in Eq. (3) follows from Corollary 12 of [CW04], since both qubit marginals are maximally mixed. The first upper bound uses a trivial extension; the second combines \(E_{\mathrm{sq}}\leq E_F\), Proposition 5 of [CW04], with Wootters’ exact entanglement-of-formation formula [Woo98]. These are bounds on the desired measure, rather than a formula for it.
Fawzi and Fawzi construct semidefinite-programming lower bounds. Their Sec. 3.2 and Fig. 1 show close numerical agreement with heuristic finite-extension upper bounds for two-qubit Werner states, a one-parameter Bell-diagonal subfamily [FF22]. Their Proposition 2.4 controls the entropy-approximation error in an intermediate optimization; it does not bound the gap of the finite semidefinite relaxation. Sec. 4 leaves convergence of that hierarchy open, so these results do not establish the requested certified numerical evaluation.
Ahmed and Smith rule out finite-dimensional attainment on a rank-two Bell-diagonal subfamily (Theorem 2 of a 2026 preprint) [AS26]. Set \(p_I=(1+c)/2\), \(p_Z=(1-c)/2\), and \(p_X=p_Y=0\), where \(0<c<1\), and write \(\rho_c=\rho_{\mathbf p}\). Let \(E_D(\rho_c)\) be the minimum in Eq. (2) with \(\dim E\leq D\). For every positive integer \(D\),
\begin{equation} E_{3D+2}(\rho_c)<E_D(\rho_c). \tag{4} \end{equation}Equation (4) implies that no finite extension dimension attains the infimum, even on this subfamily. It does not determine the value or provide a certified approximation error.
Comment
Squashed entanglement is additive: \(E_{\mathrm{sq}}(\rho_{\mathbf p}^{\otimes n}) =nE_{\mathrm{sq}}(\rho_{\mathbf p})\) for every positive integer \(n\), by Proposition 4 of [CW04]. Thus no regularization is needed; the unknown is the single-state value as a function of the Bell probabilities.
The remaining task is a closed-form formula or certified numerical evaluation of \(E_{\mathrm{sq}}(\rho_{\mathbf p})\) for arbitrary entangled mixed Bell-diagonal states. The known zero region, pure-state values, and numerical bounds do not settle this question.
Finite-dimensional nonattainment does not exclude certified approximation: \(E_{\mathrm{sq}}=\inf_D E_D\) still guarantees a finite-dimensional \(\varepsilon\)-approximation for every \(\varepsilon>0\). The new obstruction provides neither a computable dimension bound nor a convergence rate [AS26].
References
- [CW04]
- M. Christandl and A. Winter, “’Squashed entanglement’: An additive entanglement measure,” Journal of Mathematical Physics 45(3), 829–840 (2004).DOIarXiv
- [Woo98]
- W. K. Wootters, “Entanglement of Formation of an Arbitrary State of Two Qubits,” Physical Review Letters 80, 2245–2248 (1998).DOIarXiv
- [BCY11]
- F. G. S. L. Brandão, M. Christandl, and J. Yard, “Faithful Squashed Entanglement,” Communications in Mathematical Physics 306, 805–830 (2011).DOIarXiv