Squashed entanglement of the qubit depolarizing channel

Problem

What is the squashed entanglement \(E_{\mathrm{sq}}(\Lambda_p)\) of the qubit depolarizing channel, defined for a mixing parameter \(p\in[0,1]\) by

\begin{equation} \Lambda_p(\rho):=(1-p)\,\rho+p\,\mathrm{Tr}[\rho]\,\frac{I}{2} =\bigl(1-\tfrac{3p}{4}\bigr)\rho +\tfrac{p}{4}\bigl(X\rho X+Y\rho Y+Z\rho Z\bigr), \tag{1} \end{equation}

where \(\rho\) ranges over qubit states, \(I\) is the qubit identity, and \(X,Y,Z\) are the Pauli matrices, so that the total Pauli error probability, read off from the second expression in Eq. (1), is \(\tfrac{3p}{4}\)? Let \(U:A'\to BE\) be an isometric extension of \(\Lambda_p\), let \(\varphi_{RA'}\) range over pure bipartite input states, and let \(S:E\to E'\) range over completely positive trace-preserving maps of arbitrary finite output dimension. For \(\omega_{RBE'}:=(\mathrm{id}_{RB}\otimes S)\bigl[(\mathrm{id}_R\otimes U)\,\varphi_{RA'}\,(\mathrm{id}_R\otimes U^\dagger)\bigr]\) the squashed entanglement of the channel is

\begin{equation} E_{\mathrm{sq}}(\Lambda_p) :=\max_{\varphi_{RA'}}\ \frac{1}{2}\inf_{S}I(R;B|E')_\omega, \tag{2} \end{equation}

where \(I(R;B|E')_\omega\) is the conditional mutual information and all logarithms are base 2 [TGW14]. The quantity in Eq. (2) is additive over tensor products of channels and upper-bounds the two-way-assisted quantum and private capacities, \(Q_2(\Lambda_p)\le E_{\mathrm{sq}}(\Lambda_p)\) and \(P_2(\Lambda_p)\le E_{\mathrm{sq}}(\Lambda_p)\) [TGW14]; this is its operational role, since an exact value would be a single-letter upper bound on secret and quantum transmission over depolarizing noise. Determine \(E_{\mathrm{sq}}(\Lambda_p)\) as an explicit function of \(p\) on \([0,1]\).

Source

The question is implicit in the defining paper of Takeoka, Guha, and Wilde, which upper-bounds the depolarizing channel within a restricted squashing family, describes that bound as not particularly tight, and leaves the exact value uncomputed [TGW14]. Goodenough, Elkouss, and Wehner later determine the exact value for the erasure channel, which their account calls the only class calculated exactly “up until now”, and record that finding exact channel squashed entanglement remains hard in general [GEW16].

Progress

Takeoka, Guha, and Wilde define the quantity in Eq. (2), prove its additivity over tensor products and the bounds \(Q_2\le E_{\mathrm{sq}}\) and \(P_2\le E_{\mathrm{sq}}\), and evaluate for Pauli channels a restricted family of squashing maps built from Bell-pair measurements. For the channel of Eq. (1) the optimizing parameter in that family is indicated only numerically, the resulting bound is described as not particularly tight, and \(E_{\mathrm{sq}}(\Lambda_p)\) itself is left uncomputed [TGW14].
The state measure underlying Eq. (2) is the squashed entanglement of Christandl and Winter, an additive entanglement monotone that vanishes on separable states and never exceeds the entanglement of formation [CW04], and it is faithful, that is, strictly positive on every entangled state, as shown by Brandão, Christandl, and Yard [BCY11]. Consequently the boundary values are known: \(E_{\mathrm{sq}}(\Lambda_0)=1\) for the noiseless channel, while the channel of Eq. (1) is entanglement-breaking precisely for \(p\ge\tfrac{2}{3}\), where every output state is separable and \(E_{\mathrm{sq}}(\Lambda_p)=0\). The open interval is \(0<p<\tfrac{2}{3}\).
Faithfulness gives a positive lower bound on the open interval: the channel of Eq. (1) maps the maximally entangled input to the Bell-diagonal state with Bell fidelity \(1-\tfrac{3p}{4}\), entangled precisely for \(p<\tfrac{2}{3}\), and the maximum over pure inputs in Eq. (2) makes \(E_{\mathrm{sq}}(\Lambda_p)\) at least the squashed entanglement of that output, hence positive throughout \(0<p<\tfrac{2}{3}\) [BCY11]. At \(p=\tfrac{1}{2}\) the clipped symmetric coherent information \(\max\{0,1-H(\mathbf p)\}\) of the Pauli vector \(\mathbf p=\bigl(1-\tfrac{3p}{4},\tfrac{p}{4},\tfrac{p}{4},\tfrac{p}{4}\bigr)\) equals \(0\), so \(E_{\mathrm{sq}}\) already strictly exceeds it there.
Goodenough, Elkouss, and Wehner compute \(E_{\mathrm{sq}}\) and the two-way-assisted capacities of the \(d\)-dimensional erasure channel in closed form, and for the depolarizing channel they improve the upper bound on \(E_{\mathrm{sq}}(\Lambda_p)\) using convexity of \(E_{\mathrm{sq}}\) in the channel and its link to the entanglement-assisted classical capacity [GEW16]. Their aside that, “up until now, this class of channels is the only class whose squashed entanglement has been calculated exactly”, is a census of the 2016 literature rather than a presently verified uniqueness claim.
Flagged extensions bound the quantum capacity of the depolarizing channel directly and surpass squashed-entanglement bounds in some regimes, but they do not evaluate \(E_{\mathrm{sq}}(\Lambda_p)\) [KFG22].
Riemannian optimization of channel extensions over the Stiefel manifold yields the strongest numerical upper bounds on the quantum capacity of the channel in Eq. (1) at large \(p\); the bounds do not meet the achievable rates, and the same work proves that amortization does not enhance the channel coherent information of any channel [ZMFW25].
The most recent dedicated work refines the flagged-extension technique into single-letter upper bounds on the quantum and private capacities of the depolarizing channel, with numerical coherent-information evaluation, and describes the channel as one whose capacity requires multi-letter superadditivity; it likewise leaves \(E_{\mathrm{sq}}(\Lambda_p)\) uncomputed [Nour26].

Comment

The value of \(E_{\mathrm{sq}}(\Lambda_p)\) is known only at the endpoints \(p=0\) and \(p\ge\tfrac{2}{3}\) and is open on \(0<p<\tfrac{2}{3}\). Contributory sub-questions: is the input maximum in Eq. (2) attained at the maximally entangled state for every squashing map, or only within the restricted family used for the known upper bound? Are those squashing constructions optimal, or improvable, on any subinterval of \(p\)? The clipped symmetric coherent information \(\max\{0,1-H(\mathbf p)\}\) of the Pauli vector \(\mathbf p=\bigl(1-\tfrac{3p}{4},\tfrac{p}{4},\tfrac{p}{4},\tfrac{p}{4}\bigr)\) is no candidate at all: wherever the clipping is active with \(p<\tfrac{2}{3}\), at \(p=\tfrac{1}{2}\) for instance, it equals \(0\) while \(E_{\mathrm{sq}}(\Lambda_p)>0\) by faithfulness (see Progress). A closed form would give an additive single-letter upper bound on the two-way-assisted quantum and private capacities of the depolarizing channel and on the two-way secret-key rate of key distribution over depolarizing noise, though not automatically the sharpest computable upper bound among all methods: flagged-extension bounds surpass squashed-entanglement bounds in some regimes [KFG22]. The sibling record 01M20868D5Z2B0W2KKC7QGRGRR asks for the private capacity \(P(\Lambda_p)\) of the same channel, a different quantity: \(E_{\mathrm{sq}}(\Lambda_p)\) upper-bounds the assisted capacity \(P_2(\Lambda_p)\ge P(\Lambda_p)\), and determining either value neither settles nor is settled by the other. The record 01M1HME7809M71BG24CSMYKA8A asks for the unassisted quantum capacity \(\mathcal Q(\Lambda_p)\) of the same channel family, which obeys \(\mathcal Q\le Q_2\le E_{\mathrm{sq}}\), and the record 01M1HME780NTMHKFB95TSXKKFW asks for the distillable entanglement of Bell-diagonal states, which subsumes the two-way quantum capacity of this channel through teleportation covariance, another rate that \(E_{\mathrm{sq}}\) upper-bounds.

References

[TGW14]: M. Takeoka, S. Guha, and M. M. Wilde, “The squashed entanglement of a quantum channel,” IEEE Transactions on Information Theory 60(8), 4987–4998 (2014).DOI arXiv
[GEW16]: K. Goodenough, D. Elkouss, and S. Wehner, “Assessing the performance of quantum repeaters for all phase-insensitive Gaussian bosonic channels,” New Journal of Physics 18, 063005 (2016).DOI arXiv
[KFG22]: F. Kianvash, M. Fanizza, and V. Giovannetti, “Bounding the quantum capacity with flagged extensions,” Quantum 6, 647 (2022).DOI arXiv
[ZMFW25]: C. Zhu, H. Mao, K. Fang, and X. Wang, “Geometric optimization for quantum communication,” arXiv preprint (2025).arXiv
[Nour26]: V. Nourozi, “Flagged Extensions and Numerical Simulations for Quantum Channel Capacity: Bridging Theory and Computation,” arXiv preprint (2026).arXiv
[CW04]: M. Christandl and A. Winter, “’Squashed entanglement’: An additive entanglement measure,” Journal of Mathematical Physics 45(3), 829–840 (2004).DOI arXiv
[BCY11]: F. G. S. L. Brandão, M. Christandl, and J. Yard, “Faithful Squashed Entanglement,” Communications in Mathematical Physics 306, 805–830 (2011).DOI arXiv

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