LOCC entanglement cost of Werner states

Problem

What is the entanglement cost of the Werner state \(\rho_{d,p}\) for every integer \(d\geq2\) and every antisymmetric weight \(1/2<p\leq1\), and in particular, does it equal the entanglement of formation? On \(\mathbb C^d\otimes\mathbb C^d\), let \(\mathbb F\) be the swap operator, \(\mathbb F\lvert x\rangle\lvert y\rangle=\lvert y\rangle\lvert x\rangle\), and let \(P_{\mathrm s}:=(I+\mathbb F)/2\) and \(P_{\mathrm a}:=(I-\mathbb F)/2\) be the projectors onto the symmetric and antisymmetric subspaces. For \(p\in[0,1]\), the Werner state with antisymmetric weight \(p\) is

\begin{equation} \rho_{d,p}:=p\,\frac{2P_{\mathrm a}}{d(d-1)} +(1-p)\,\frac{2P_{\mathrm s}}{d(d+1)}. \tag{1} \end{equation}

The state in Eq. (1) is invariant under \(U\otimes U\) for every unitary \(U\) on \(\mathbb C^d\), satisfies \(\operatorname{Tr}(\rho_{d,p}\mathbb F)=1-2p\), and is entangled exactly when \(p>1/2\). The entanglement cost \(E_C(\rho)\) of a bipartite state \(\rho\) is the infimum of the rates \(R\) for which local operations and classical communication (LOCC) transform \(\lceil nR\rceil\) ebits into states whose trace distance from \(\rho^{\otimes n}\) tends to zero as \(n\to\infty\). Write \(E_F\) for the entanglement of formation and \(h_2(x):=-x\log_2x-(1-x)\log_2(1-x)\). For \(1/2<p\leq1\), the one-copy value of \(E_F\) gives the upper bound

\begin{equation} E_C(\rho_{d,p})\leq E_F(\rho_{d,p}) =h_2\!\left(\frac12-\sqrt{p(1-p)}\right), \tag{2} \end{equation}

whose right-hand side does not depend on \(d\). Determine \(E_C(\rho_{d,p})\) for all \(d\geq2\) and \(1/2<p\leq1\), and decide in particular whether equality holds in Eq. (2).

Source

Fang, Fawzi, and Fawzi identified the general LOCC entanglement-cost questions for Werner and isotropic states as unresolved (Section 4.2 of the arXiv version) [FFF26]. For the antisymmetric endpoint \(p=1\), Christandl, Schuch, and Winter explicitly leave open whether the cost equals one ebit (Section VI) [CSW12].

Progress

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Vollbrecht and Werner, Section IV C, prove that a \(U\otimes U\)-invariant state with \(f:=\operatorname{Tr}(\rho\mathbb F)\leq0\) has entanglement of formation \(h_2\bigl(\tfrac12(1-\sqrt{1-f^2})\bigr)\) in every dimension, and that the state is separable when \(f\geq0\) [VW01]. Since \(f=1-2p\) for the state in Eq. (1), this gives the value in Eq. (2), and \(E_C(\rho_{d,p})=0\) for \(p\leq1/2\).
Hayden, Horodecki, and Terhal prove that the entanglement cost is the regularized entanglement of formation (Theorem 1),

\begin{equation} E_C(\rho)=\lim_{n\to\infty}\frac1n\,E_F\bigl(\rho^{\otimes n}\bigr) \tag{3} \end{equation}

[HHT01]. Because \(E_F\) is subadditive, the limit in Eq. (3) is an infimum over \(n\); hence equality holds in Eq. (2) exactly when \(E_F(\rho_{d,p}^{\otimes n})=nE_F(\rho_{d,p})\) for every \(n\geq1\). This equivalence is a direct deduction.
Earlier exact results covered two entangled parameter points. At \((d,p)=(2,1)\) the state is a singlet, whose cost is its entanglement entropy, one ebit [HHT01]. Yura proves that every state \(\rho\) supported on the antisymmetric subspace of \(\mathbb C^3\otimes\mathbb C^3\) satisfies \(E_F(\rho^{\otimes n})=n\) for all \(n\) (Theorem 1 and Corollary 3), so \(E_C(\rho_{3,1})=1\) [Yur03]. Matsumoto and Yura generalize this to antisymmetric states of \(d-1\) qudits divided as \(\mathbb C^d\otimes(\mathbb C^d)^{\otimes(d-2)}\); for \(d\geq4\) these are not the bipartite states \(\rho_{d,1}\) [MY04].
For the antisymmetric state \(\rho_{d,1}\), Christandl, Schuch, and Winter prove a dimension-independent lower bound (Theorem 2), so that, with Eq. (2),

\begin{equation} \log_2\frac43\leq E_C(\rho_{d,1})\leq E_F(\rho_{d,1})=1 . \tag{4} \end{equation}

Their Section VI leaves open whether \(E_C(\rho_{d,1})=1\) and notes that \(E_C(\rho_{d,1})<1\) would give the first explicit counterexample to the additivity of the entanglement of formation [CSW12].
Audenaert, Eisert, Jané, Plenio, Virmani, and De Moor compute the regularized relative entropy of entanglement with respect to states with positive partial transpose (PPT) in the same parameterization,

\begin{equation} E_{R,\mathrm{PPT}}^\infty(\rho_{d,p})= \begin{cases} 1-h_2(p), & \frac12<p\leq\frac{d+2}{2d},\\[4pt] \log_2\frac{d+2}{d}+(1-p)\log_2\frac{d-2}{d+2}, & \frac{d+2}{2d}<p\leq1, \end{cases} \tag{5} \end{equation}

where only the first branch occurs for \(d=2\) [AEJ+01]. Since every separable state is PPT, Eq. (5) is at most the separable regularized relative entropy of entanglement, which Donald, Horodecki, and Rudolph show is at most \(E_C\) (Proposition 20) [DHR02]. At \(p=1\) this lower bound equals \(\log_2(1+2/d)\) and vanishes as \(d\to\infty\), whereas the bound \(\log_2(4/3)\) in Eq. (4) is available only at \(p=1\).
Audenaert, Plenio, and Eisert, and Wang and Wilde, show that the exact (zero-error) entanglement cost of Werner states under PPT-preserving operations is the logarithmic negativity,

\begin{equation} E_{\mathrm{PPT}}^{\mathrm{exact}}(\rho_{d,p}) =\log_2\!\left(1+\frac{2(2p-1)}{d}\right) \qquad\left(\tfrac12<p\leq1\right) \tag{6} \end{equation}

[APE03], [WW23]. Since LOCC operations are PPT-preserving, any lower bound on \(E_C\) that also bounds the vanishing-error cost under PPT-preserving operations is at most Eq. (6), which decays like \(1/d\). Fang, Fawzi, and Fawzi also note that the efficiently computable lower bounds of Wang and Duan and of Lami and Regula vanish on full-rank states, which include \(\rho_{d,p}\) for \(0<p<1\) [FFF26]. The comparison of these bounds is a deduction made for this entry.
Define

\begin{equation} \mathcal N_{d,p}(X):=p\,\frac{\operatorname{Tr}(X)I-X^{\mathsf T}}{d-1} +(1-p)\,\frac{\operatorname{Tr}(X)I+X^{\mathsf T}}{d+1}. \tag{7} \end{equation}

The map in Eq. (7) is a channel whose normalized Choi state \((\operatorname{id}\otimes\mathcal N_{d,p})(\Phi_d)\), with \(\Phi_d\) the maximally entangled state, is \(\rho_{d,p}\), and \(\mathcal N_{d,p}(UXU^\dagger)=\overline U\,\mathcal N_{d,p}(X)\,U^{\mathsf T}\) for every unitary \(U\). Restricting \(U\) to the Heisenberg–Weyl group, a unitary one-design, shows that \(\mathcal N_{d,p}\) is covariant in Wilde’s sense, so his Theorem 1 gives equal sequential and parallel entanglement costs, \(E_C(\mathcal N_{d,p})=E_C^{(p)}(\mathcal N_{d,p})=E_C(\rho_{d,p})\) [Wil18]. Wilde’s Section IV C treats only the Werner–Holevo channel \(\mathcal N_{d,1}\), recording Eq. (4) and the exact value one for \(d=2\) and \(d=3\). The Choi-state and covariance identities were checked directly for this entry.
At fixed \(p\), the cost is nonincreasing in the dimension: \(E_C(\rho_{d',p})\leq E_C(\rho_{d,p})\) for \(d'\geq d\). An isometric embedding of \(\mathbb C^d\) into \(\mathbb C^{d'}\) on both sides, followed by the \(U\otimes U\) twirl, is an LOCC operation mapping \(\rho_{d,p}\) to \(\rho_{d',p}\), because both steps preserve \(\operatorname{Tr}(\rho\mathbb F)=1-2p\) and a \(U\otimes U\)-invariant state is determined by this value [VW01]. With Eq. (2), this gives \(E_C(\rho_{d,p})\leq E_C(\rho_{2,p})=E_F(\rho_{2,p})=E_F(\rho_{d,p})\), where the qubit equality follows from [Fan26]. Hence equality in Eq. (2) at \((d,p)\) implies equality at \((d',p)\) for all \(2\leq d'\leq d\), and a strict inequality at \((d,p)\) implies a strict inequality for all \(d'\geq d\). This monotonicity is a deduction made for this entry.
For \(d=2\), the Bell weights of \(\rho_{2,p}\) are \(p\) and three copies of \((1-p)/3\). Fang’s preprint, Theorem 7, Eq. (40), therefore gives

\begin{equation} E_C(\rho_{2,p})=E_F(\rho_{2,p}) =h_2\!\left(\frac12-\sqrt{p(1-p)}\right), \qquad \frac12<p\leq1. \tag{8} \end{equation}

Equation (8) settles the entire entangled qubit range [Fan26].

Comment

The remaining question concerns \(d\geq3\) and \(1/2<p\leq1\), apart from the known endpoint \((d,p)=(3,1)\): determine the regularization in Eq. (3) and whether it equals Eq. (2). The qubit case is settled by Eq. (8), a specialization of the resolved qubit Bell-diagonal entanglement-cost problem. The LOCC entanglement cost of isotropic states coincides with this problem up to a local unitary at \(d=2\), with fidelity \(p\), but concerns a different invariant family for \(d\geq3\). The higher-dimensional Werner question therefore retains Unsolved status.

References

[FFF26]: K. Fang, H. Fawzi, and O. Fawzi, “Efficient Approximation of Regularized Relative Entropies and Applications,” IEEE Transactions on Information Theory 72, 2330–2342 (2026).DOI arXiv
[CSW12]: M. Christandl, N. Schuch, and A. Winter, “Entanglement of the Antisymmetric State,” Communications in Mathematical Physics 311, 397–422 (2012).DOI arXiv
[VW01]: K. G. H. Vollbrecht and R. F. Werner, “Entanglement Measures under Symmetry,” Physical Review A 64, 062307 (2001).DOI arXiv
[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
[Yur03]: F. Yura, “Entanglement Cost of Three-Level Antisymmetric States,” Journal of Physics A: Mathematical and General 36, L237–L242 (2003).DOI arXiv
[MY04]: K. Matsumoto and F. Yura, “Entanglement Cost of Antisymmetric States and Additivity of Capacity of Some Quantum Channels,” Journal of Physics A: Mathematical and General 37, L167–L171 (2004).DOI arXiv
[AEJ+01]: K. Audenaert, J. Eisert, E. Jané, M. B. Plenio, S. Virmani, and B. De Moor, “Asymptotic Relative Entropy of Entanglement,” Physical Review Letters 87, 217902 (2001).DOI arXiv
[DHR02]: M. J. Donald, M. Horodecki, and O. Rudolph, “The Uniqueness Theorem for Entanglement Measures,” Journal of Mathematical Physics 43, 4252–4272 (2002).DOI arXiv
[APE03]: K. Audenaert, M. B. Plenio, and J. Eisert, “Entanglement Cost under Positive-Partial-Transpose-Preserving Operations,” Physical Review Letters 90, 027901 (2003).DOI arXiv
[WW23]: X. Wang and M. M. Wilde, “Exact Entanglement Cost of Quantum States and Channels under Positive-Partial-Transpose-Preserving Operations,” Physical Review A 107, 012429 (2023).DOI arXiv
[Wil18]: M. M. Wilde, “Entanglement Cost and Quantum Channel Simulation,” Physical Review A 98, 042338 (2018).DOI arXiv
[Fan26]: K. Fang, “Entanglement cost of quantum depolarization,” preprint (2026).arXiv

LOCC entanglement cost of Werner states

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