LOCC entanglement cost of isotropic states
- Field
- Topics
Problem
What is the entanglement cost of the isotropic state \(\sigma_{d,F}\) for every integer \(d\geq2\) and every fidelity \(1/d<F<1\)? On \(\mathbb C^d\otimes\mathbb C^d\), let \(\lvert\Phi_d\rangle:=d^{-1/2}\sum_{j=0}^{d-1}\lvert jj\rangle\) and \(\Phi_d:=\lvert\Phi_d\rangle\!\langle\Phi_d\rvert\). For \(F\in[0,1]\), the isotropic state with fidelity \(F\) is
The state in Eq. (1) is invariant under \(U\otimes\overline U\) for every unitary \(U\) on \(\mathbb C^d\), is separable for \(F\leq1/d\), and is entangled for \(F>1/d\). 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 \(n\to\infty\). Determine \(E_C(\sigma_{d,F})\) for all integers \(d\geq2\) and all \(1/d<F<1\), including the qubit case \(d=2\).
Source
Wilde notes that the entanglement cost of an isotropic state, the Choi state of a depolarizing channel, is not known (Section IV) [Wil18], and Fang, Fawzi, and Fawzi state that, to the best of their knowledge, the entanglement costs of isotropic and Werner states under LOCC “remain unresolved” (Section 4.2 of the arXiv version) [FFF26].
Progress
For \(F\leq1/d\), the state is separable and \(E_C(\sigma_{d,F})=0\) [TV00]. At \(F=1\), it is maximally entangled, and the cost of a pure state is its entanglement entropy, so \(E_C(\sigma_{d,1})=\log_2d\) [HHT01]. For \(d=2\), the open interval is \(1/2<F<1\).
Rains, in Theorem 7 of the arXiv version 2, evaluates the relative entropy of entanglement with respect to states with positive partial transpose for isotropic states with \(F\geq1/d\), with the separable isotropic state of fidelity \(1/d\) as an optimizer, and proves it additive on tensor powers [Rai99]. Rubboli and Tomamichel prove the same value for the relative entropy of entanglement with respect to separable states, and its additivity whenever one factor is isotropic (Proposition 12 and Section 6.3) [RT24]. Hence the regularized separable relative entropy of entanglement is
\begin{equation} E_R^\infty(\sigma_{d,F}) =\log_2d+F\log_2F+(1-F)\log_2\frac{1-F}{d-1} \qquad\left(\frac1d\leq F\leq1\right), \tag{2} \end{equation}with \(0\log_20:=0\). Donald, Horodecki, and Rudolph show that \(E_R^\infty\leq E_C\) (Proposition 20) [DHR02], so Eq. (2) is a lower bound on the cost for every \(d\geq2\).
Fang, Fawzi, and Fawzi note that the efficiently computable lower bounds of Wang and Duan and of Lami and Regula vanish on full-rank states (Proposition 24 and Section 4.2), which include \(\sigma_{d,F}\) for \(0<F<1\). In their numerical comparison for \(d=3\) (Example 1 and Figure 2), their first-level bound coincides with Eq. (2) and improves the bound of Wang, Jing, and Zhu [FFF26], [WJZ25]. These are lower bounds and do not identify an optimal LOCC preparation rate.
The depolarizing channel \(\mathcal D_{d,p}(X):=(1-p)X+p\operatorname{Tr}(X)I/d\), with \(0\leq p\leq d^2/(d^2-1)\), has normalized Choi state \(\sigma_{d,F}\), where
\begin{equation} F=1-p+\frac{p}{d^2}, \qquad 0<p<\frac{d}{d+1}\iff\frac1d<F<1 . \tag{3} \end{equation}Since \(\mathcal D_{d,p}(UXU^\dagger)=U\,\mathcal D_{d,p}(X)\,U^\dagger\) for every unitary \(U\), the channel is covariant in Wilde’s sense, and his Theorem 1 gives equal sequential and parallel costs, \(E_C(\mathcal D_{d,p})=E_C^{(p)}(\mathcal D_{d,p})=E_C(\sigma_{d,1-p+p/d^2})\) [Wil18]. The correspondence in Eq. (3) was checked directly for this entry.
Comment
Literature checked through 15 September 2026. No exact LOCC entanglement cost of an isotropic state with \(1/d<F<1\) was located for any \(d\geq2\), and no LOCC preparation rate matching the lower bound in Eq. (2) is known. For \(d=2\), \(\sigma_{2,F}\) is the Bell-diagonal state with weight \(F\) on \(\lvert\Phi_2\rangle\) and \((1-F)/3\) on each other Bell state, the isotropic line singled out in the entanglement-cost problem for qubit Bell-diagonal states; up to a local unitary it is the two-qubit Werner state with antisymmetric weight \(F\). The qubit case therefore coincides with the \(d=2\) case of the LOCC entanglement cost of Werner states, while for \(d\geq3\) the \(U\otimes\overline U\)-invariant and \(U\otimes U\)-invariant families differ.
References
- [Wil18]
- M. M. Wilde, “Entanglement Cost and Quantum Channel Simulation,” Physical Review A 98, 042338 (2018).DOIarXiv
- [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).DOIarXiv
- [TV00]
- B. M. Terhal and K. G. H. Vollbrecht, “Entanglement of Formation for Isotropic States,” Physical Review Letters 85, 2625–2628 (2000).DOIarXiv
- [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
- [Rai99]
- E. M. Rains, “Bound on Distillable Entanglement,” Physical Review A 60, 179–184 (1999); Erratum, Physical Review A 63, 019902 (2000).DOIDOIarXiv
- [RT24]
- R. Rubboli and M. Tomamichel, “New Additivity Properties of the Relative Entropy of Entanglement and Its Generalizations,” Communications in Mathematical Physics 405, 162 (2024).DOIarXiv