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 state that, to the best of their knowledge, the entanglement costs of Werner and isotropic states under LOCC “remain 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

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.
Two entangled parameter points have known cost. 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 \(E_C(\rho_{d,p})\leq E_C(\rho_{2,p})\leq E_F(\rho_{2,p})=E_F(\rho_{d,p})\). 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 \(d'\geq d\); in particular, a two-qubit gap would give a gap in every dimension. This monotonicity is a deduction made for this entry.

Comment

Literature checked through 15 September 2026. Apart from the separable range and the parameter points \((d,p)=(2,1)\) and \((3,1)\), no exact LOCC entanglement cost of a Werner state was located, and no Werner state with \(E_C<E_F\) was found. Elsewhere on the range, the lower bounds recorded in Eqs. (4) and (5) stay strictly below Eq. (2), so the open task is to evaluate the regularization in Eq. (3) or to prove a strict gap. For \(d=2\), \(\rho_{2,p}\) is the Bell-diagonal state with weight \(p\) on the singlet and \((1-p)/3\) on each other Bell state, so this qubit line lies inside the entanglement-cost problem for qubit Bell-diagonal states; up to a local unitary it is the two-qubit isotropic state with fidelity \(p\). The LOCC entanglement cost of isotropic states asks the analogous question for the \(U\otimes\overline U\)-invariant family; the two questions coincide at \(d=2\) and differ for \(d\geq3\).

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

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