Classical Communication Helps Single-Shot Quantum Compression
Abstract
Free classical communication can yield an unbounded single-shot saving in quantum compression, even though it leaves the asymptotic rate of an i.i.d. source unchanged. At every fixed error in (0,1), we exhibit sources whose unassisted compression cost is arbitrarily large, yet a single qubit suffices with classical communication and no preshared entanglement. We establish this separation by determining the exact optimal source fidelity for compression through an m-dimensional noiseless quantum channel, with and without free classical communication, while preserving correlations with an inaccessible reference. Without classical assistance, the optimal fidelity is exactly the Ky Fan norm |ρ|(m), the sum of the m largest eigenvalues of the source state. With classical assistance, we obtain an explicit spectral threshold formula and a protocol attaining it. Classical communication strictly improves fidelity whenever m is smaller than the source rank. The corresponding compression costs are two smoothed versions of the log-rank entropy, also called max-entropy: trace-distance smoothing under local operations and quadratic smoothing under LOCC. For i.i.d. copies of a fixed source with positive entropy variance, however, the two costs have identical first-, second-, and third-order asymptotic terms. For nonlattice spectra, we determine the constant-order advantage: classical communication saves log2(e)−1 bits per block asymptotically, and the ratio of assisted to unassisted minimum communication dimensions converges to 2/e.
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BibTeX
@misc{farber2026classical,
title = {{Classical Communication Helps Single-Shot Quantum Compression}},
author = {Omri Farber and Gilad Gour},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14785},
eprint = {2610.14785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14785},
}
RIS
TY - UNPB
TI - Classical Communication Helps Single-Shot Quantum Compression
AU - Omri Farber
AU - Gilad Gour
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14785
UR - https://arxiv.org/abs/2610.14785
AB - Free classical communication can yield an unbounded single-shot saving in quantum compression, even though it leaves the asymptotic rate of an i.i.d. source unchanged. At every fixed error in (0,1), we exhibit sources whose unassisted compression cost is arbitrarily large, yet a single qubit suffices with classical communication and no preshared entanglement. We establish this separation by determining the exact optimal source fidelity for compression through an m-dimensional noiseless quantum channel, with and without free classical communication, while preserving correlations with an inaccessible reference. Without classical assistance, the optimal fidelity is exactly the Ky Fan norm |ρ|_(m), the sum of the m largest eigenvalues of the source state. With classical assistance, we obtain an explicit spectral threshold formula and a protocol attaining it. Classical communication strictly improves fidelity whenever m is smaller than the source rank. The corresponding compression costs are two smoothed versions of the log-rank entropy, also called max-entropy: trace-distance smoothing under local operations and quadratic smoothing under LOCC. For i.i.d. copies of a fixed source with positive entropy variance, however, the two costs have identical first-, second-, and third-order asymptotic terms. For nonlattice spectra, we determine the constant-order advantage: classical communication saves log_2(e)−1 bits per block asymptotically, and the ratio of assisted to unassisted minimum communication dimensions converges to 2/e.
ER -