Strong converse for general mixed-state quantum compression

Problem

Does general finite-dimensional i.i.d. mixed-state quantum compression obey an unrestricted strong converse? Let \(\rho^{AR}\) be a finite-dimensional source state, where only \(A\) is available to the encoder and \(R\) is an inaccessible reference. Fix a Koashi–Imoto isometry \(U:A\to CNQ\) for which the source has the form

\begin{equation} \omega^{CNQR} :=(U\otimes I_R)\rho^{AR}(U^\dagger\otimes I_R) =\sum_j p_j|j\rangle\!\langle j|^C \otimes\omega_j^N\otimes\rho_j^{QR}, \tag{1} \end{equation}

where \(C\) is classical, \(N\) is redundant relative to \(R\) conditioned on \(C\), and \(Q\) carries the remaining source–reference correlations. At blocklength \(n\), allow arbitrary encoding and decoding channels \(\mathcal E_n:A^{\otimes n}\to M_n\) and \(\mathcal D_n:M_n\to\widehat A^{\otimes n}\), with \(\widehat A\cong A\). Their reference-preserving squared fidelity is

\begin{equation} F_n:=F\!\left( (\rho^{AR})^{\otimes n}, \left[(\mathcal D_n\circ\mathcal E_n) \otimes\operatorname{id}_{R^{\otimes n}}\right] ((\rho^{AR})^{\otimes n}) \right), \qquad F(\tau,\zeta):=\lVert\sqrt\tau\sqrt\zeta\rVert_1^2. \tag{2} \end{equation}

The optimal first-order qubit rate is \(S(CQ)_\omega\). Determine whether every sequence of unrestricted channels defining Eq. (2) satisfies the strong-converse implication

\begin{equation} \limsup_{n\to\infty}\frac1n\log_2|M_n|<S(CQ)_\omega \quad\Longrightarrow\quad \lim_{n\to\infty}F_n=0. \tag{3} \end{equation}

Equation (3) imposes no unitality, isometry, or dimension-expansion condition on the encoder or decoder.

Source

Wilde records the unresolved mixed-state compression problem in Sections 18.4–18.5. Khanian and Winter solve its general finite-dimensional first-order formulation and explicitly leave the strong converse in Eq. (3) open [Wil17], [KW22].

Progress

Koashi and Imoto identify and remove the locally redundant part of a blind mixed-state ensemble. Khanian and Winter use the corresponding Koashi–Imoto decomposition for an arbitrary reference state, as in Eq. (1), and prove the exact first-order rate

\begin{equation} R_{\mathrm{blind}}(\rho^{AR})=S(CQ)_\omega. \tag{4} \end{equation}

Equation (4) supplies both achievability and a weak converse, but it does not force the fidelity to vanish at every rate below the threshold [KI01], [KW22].
A preprint revised in 2025 proves exponential fidelity decay for general visible compression at every rate below

\begin{equation} L_\rho :=\lim_{\alpha\to1^+}E_{\alpha,p}^{\infty}(A{:}R)_\rho, \tag{5} \end{equation}

where \(E_{\alpha,p}^{\infty}\) is the regularized Rényi entanglement of purification used there. Because blind codes form a subclass of visible codes, Eq. (5) also gives an unrestricted CPTP-decoder converse for blind compression below \(L_\rho\). Equality \(L_\rho=E_p^\infty(A{:}R)_\rho\), which would complete the visible strong converse, remains conditional on an unresolved continuity statement; more importantly here, \(L_\rho\) need not reach the blind threshold \(S(CQ)_\omega\) in Eq. (4).

For rates up to the blind threshold, the preprint’s claimed bound assumes that the effective post–Koashi–Imoto decoder \(\widetilde{\mathcal D}_n:M_n\to\widehat C^{\otimes n}\widehat Q^{\otimes n}\) is super-unital in the sense

\begin{equation} I_{\widehat C^{\otimes n}\widehat Q^{\otimes n}} \preceq\widetilde{\mathcal D}_n(I_{M_n}). \tag{6} \end{equation}

If \(\widetilde{\mathcal D}_n\) is trace preserving, taking traces in Eq. (6) forces \(|M_n|\geq|CQ|^n\), so the assumption excludes the relevant dimension-reducing decoders. These results therefore do not prove Eq. (3) for unrestricted codes. The cited version is unrefereed; it also states that its first version contained an error in a lemma and restricts the affected earlier theorem [Kha25].

Comment

The first-order rate in Eq. (4) is settled. The remaining gap is exactly the fidelity conclusion in Eq. (3) for arbitrary CPTP encoders and decoders and an arbitrary finite-dimensional source state \(\rho^{AR}\).

References

[Wil17]: M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press (2017), Secs. 18.4–18.5.DOI arXiv
[KI01]: M. Koashi and N. Imoto, “Compressibility of Quantum Mixed-State Signals,” Physical Review Letters 87, 017902 (2001).DOI arXiv
[KW22]: Z. B. Khanian and A. Winter, “General Mixed State Quantum Data Compression with and without Entanglement Assistance,” IEEE Transactions on Information Theory 68, 3130–3138 (2022).DOI arXiv
[Kha25]: Z. B. Khanian, “Strong Converse Bounds for Compression of Mixed States,” arXiv preprint (2022), version 2 revised in 2025.arXiv

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