Visible quantum compression with free shared randomness at the Holevo rate
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Problem
Does free shared classical randomness make the Holevo information achievable for visible compression of every finite mixed-state ensemble? Let \(p\) be a probability distribution with full support on a finite alphabet \(\mathcal X\), and let \(W_x\) be density operators on a finite-dimensional system \(B\). Alice knows the independent and identically distributed label sequence \(x^n\), with probability \(p^n(x^n)\); the target is \(W_{x^n}:=\bigotimes_{t=1}^nW_{x_t}\). Alice and Bob share a finite random variable \(S_n\), independent of \(x^n\), with arbitrary distribution \(q_n\) and no bound on its size. For each seed \(s\), Alice prepares a density operator \(\tau_{x^n,s}\) on a quantum message system \(M_n\) and sends it through a noiseless quantum channel. Bob applies a completely positive trace-preserving map \(\mathcal D_{n,s}:\mathcal L(M_n)\to\mathcal L(B^{\otimes n})\). No initially shared entanglement or additional communication is available. The reproduced state and block error are
Thus the shared seed is averaged before taking the trace norm in Eq. (1). Define the rate, in qubits per signal, and the ensemble Holevo information by
where \(S\) is the von Neumann entropy in bits. Is \(R_{\mathrm{vis},q,\mathrm{cr}}(p,W)=\chi(p,W)\) in Eq. (2) for every ensemble, or is there a finite ensemble with a strict gap?
Source
This is the free-randomness communication-rate part of Winter’s Conjecture 23, p. 11, with the visible quantum model defined in Eqs. (15)–(16) of [Win02]. The original conjecture additionally specifies a shared-randomness rate of \(\sum_xp_xS(W_x)\); the present question leaves that resource unrestricted and is consequently weaker.
Progress
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The converse \(R_{\mathrm{vis},q,\mathrm{cr}}(p,W)\ge\chi(p,W)\) follows from Holevo information and data processing, consistent with Winter’s quantum lower bounds [Win02], Theorems 20–21. For the shared-randomness model specifically, independence of \(S_n\) and the labels gives \(I(X^n;M_nS_n)=I(X^n;M_n\mid S_n)\le\log_2\dim M_n\); vanishing block error and entropy continuity then yield the claimed asymptotic converse.
For the analogous protocol with a classical message, Nator and Pereg’s Corollary 3 gives
\begin{equation} R_{\mathrm{vis},c,\mathrm{cr}}(p,W)=\min_{P(u\mid x),\theta_u:\,W_x=\sum_uP(u\mid x)\theta_u}I(X;U), \tag{3} \end{equation}where \(\theta_u\) are density operators, \(P_{XU}(x,u)=p_xP(u\mid x)\), and one may take \(|\mathcal U|\le |\mathcal X|^2(\dim B)^2+1\). Their joint-state trace-norm criterion, Eqs. (3)–(4), equals Eq. (1) by the trace norm of block-diagonal operators [NP24]. A classical message can be encoded in orthogonal quantum states, so Eq. (3) is an upper bound on the quantum rate. For commuting \(W_x\), this value is \(\chi(p,W)\), settling that subclass.
Without shared randomness, Hayashi’s main theorem, Eq. (8), identifies the visible qubit rate with \(E_P^\infty(\omega_{XB})\), where \(\omega_{XB}=\sum_xp_x|x\rangle\langle x|\otimes W_x\), \(E_P\) is the minimum entropy across a bipartition of a purification, and \(E_P^\infty(\omega):=\lim_{n\to\infty}E_P(\omega^{\otimes n})/n\). Ignoring shared randomness therefore gives the upper bound \(R_{\mathrm{vis},q,\mathrm{cr}}\le E_P^\infty(\omega_{XB})\). For pure signal states, Eq. (10) gives \(E_P^\infty(\omega_{XB})=S(\sum_xp_xW_x)=\chi(p,W)\), so that subclass is also settled [Hay06].
Comment
The remaining issue is quantum achievability at the Holevo rate for general noncommuting mixed signals. The classical communication formula in Eq. (3) settles a different resource restriction and does not supply this quantum coding theorem. Likewise, entanglement-assisted reverse Shannon protocols use a resource excluded here. This question is distinct from additivity of entanglement of purification: that problem controls the unassisted visible rate, whereas the present task supplies shared randomness [Hay06]. The placement of the average in Eq. (1) is essential: averaging the trace-norm errors of individual seeds would allow a good seed to be fixed separately at each blocklength, removing any benefit from shared randomness. Literature audited on 9 September 2026. The cited primary definitions and theorem passages were checked, and no full resolution was identified in the later sources examined.