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Quantum Mrs. Gerber's Lemma

Let X1X_1 and X2X_2 be binary classical random variables with quantum side-information systems B1B_1 and B2B_2, respectively, and let ρX1B1\rho^{X_1B_1} and ρX2B2\rho^{X_2B_2} be classical-quantum states. Define

H1=H(X1∣B1),H2=H(X2∣B2).H_1=H(X_1|B_1),\qquad H_2=H(X_2|B_2).

Here, hh is the binary entropy function, h−1h^{-1} denotes its inverse on [0,1/2][0,1/2], and ∗\ast denotes binary convolution. Quantum Mrs. Gerber's Lemma. The conditional entropy of the binary sum satisfies

H(X1+X2∣B1B2)≥{h(h−1(H1)∗h−1(H2)),H1+H2≤log⁡2,H1+H2−log⁡2+h(h−1(log⁡2−H1)∗h−1(log⁡2−H2)),H1+H2≥log⁡2.H(X_1+X_2|B_1B_2)\geq \begin{cases} h\bigl(h^{-1}(H_1)\ast h^{-1}(H_2)\bigr),&H_1+H_2\leq\log 2,\\ H_1+H_2-\log 2+h\bigl(h^{-1}(\log 2-H_1)\ast h^{-1}(\log 2-H_2)\bigr),&H_1+H_2\geq\log 2. \end{cases}

This conjecture is the quantum-side-information analogue of Mrs. Gerber's Lemma and would give the optimal lower bound for information combining. Its status is not resolved in the supplied source.

Sources & referencesView supporting material

Primary source

Christoph Hirche and David Reeb, “Bounds on Information Combining With Quantum Side Information”, arXiv:1706.09752 (2017).

entropy-inequalitiesinformation-theoryquantum-information

Progress summary

Refreshed
Open

No public source reports a proof or counterexample, so the quantum extension of Mrs. Gerber’s Lemma remains open.

The inequality is stated as Conjecture VII.1 in a 2017 paper on information combining with quantum side information. It proposes the optimal lower bound for the conditional entropy of the binary sum, but the paper explicitly presents it as unproved.

Known results

  • A uniform-probability special case is proved as Theorem VI.1, but it does not establish the conjectured formula for general classical–quantum states.
  • Additional lower bounds, including the bound in equation (62), give an almost-optimal result rather than the conjecture.
  • Numerical tests on several hundred thousand states found no violations, but this is evidence rather than a proof.
  • The proved bounds yield only a sub-exponential blocklength guarantee in the associated polar-coding application, versus the conjecture’s polynomial guarantee.

ar5iv.labs.arxiv.org · fis.uni-hannover.de

Current status (as of August 2026): The conjecture and several special-case lower bounds are established in the literature, but the stated general inequality remains open, with no reported proof, counterexample, or verified resolution.

Sources

Solutions 0

No solutions have been posted yet.