Recent examples have shown that zero private capacity need not imply antidegradability and that two channels with zero private capacity can nevertheless transmit private information together. Existing entropic channel orders compare what a receiver and its environment can learn and thereby bound capacities. We connect these orders to the recent constructions through binary estimation, whose minimum mean-square error is governed by measured $χ^2$ divergence. A new integral representation shows how measured $χ^2$ on a qubit extension recovers quantum $χ^2$ . It lets us pass from complete measured- $χ^2$ ordering to complete quantum- $χ^2$ , relative-entropy, and less-noisy ordering. Zhu and Wang used a signed lift to show that their qutrit channel has zero private capacity. We show that Hermitian-smoothed measured- $χ^2$ ordering characterizes when such lifts exist, even with a quantum reference. Environmental dominance in binary estimation at every blocklength yields a finite-code reliability--secrecy bound. These results explain why complete comparison prevents activation with antidegradable helpers whereas regularized comparison alone does not. Building on that qutrit example, we establish this stability for a range of noisy Werner--Holevo channels, determine sharp private-capacity and antidegradability thresholds, and compute exact complementary capacities in a nondegradable range. By contrast, we extend Pauli half-erasure activation to all $1/2\le p<1$ and establish the same range for a new four-level family. Reference-assisted measured- $χ^2$ witnesses show why these activating constructions lack complete comparison.Recent examples have shown that zero private capacity need not imply antidegradability and that two channels with zero private capacity can nevertheless transmit private information together. Existing entropic channel orders compare what a receiver and its environment can learn and thereby bound capacities. We connect these orders to the recent constructions through binary estimation, whose minimum mean-square error is governed by measured $χ^2$ divergence. A new integral representation shows how measured $χ^2$ on a qubit extension recovers quantum $χ^2$. It lets us pass from complete measured-$χ^2$ ordering to complete quantum-$χ^2$, relative-entropy, and less-noisy ordering. Zhu and Wang used a signed lift to show that their qutrit channel has zero private capacity. We show that Hermitian-smoothed measured-$χ^2$ ordering characterizes when such lifts exist, even with a quantum reference. Environmental dominance in binary estimation at every blocklength yields a finite-code reliability--secrecy bound. These results explain why complete comparison prevents activation with antidegradable helpers whereas regularized comparison alone does not. Building on that qutrit example, we establish this stability for a range of noisy Werner--Holevo channels, determine sharp private-capacity and antidegradability thresholds, and compute exact complementary capacities in a nondegradable range. By contrast, we extend Pauli half-erasure activation to all $1/2\le p<1$ and establish the same range for a new four-level family. Reference-assisted measured-$χ^2$ witnesses show why these activating constructions lack complete comparison.

quant-pharXiv:2610.03706Oct 5, 20260 citations

Robust and efficient quantum private comparison of equality with collective detection over collective-noise channels

Wei Huang, Qiaoyan Wen, Bin Liu +2 more

Science China Physics Mechanics and AstronomyJul 25, 201351 citations

Remarks on Private and Quantum Capacities of More Capable and Less Noisy Quantum Channels

Shun Watanabe

Oct 26, 20112 citations

Quantum Private Comparison without Classical Computation

Yan‐Feng Lang

International Journal of Theoretical PhysicsAug 7, 20209 citations

Private and quantum capacities of more capable and less noisy quantum channels

Shun Watanabe

Two classes of quantum channels, which we call more capable and less noisy, are introduced. The more capable class consists of channels such that the quantum capacities of the complementary channels to the environments are zero. The less noisy class consists of channels such that the private capacities of the complementary channels to the environment are zero. For the more capable class, it is clarified that the private capacity and quantum capacity coincide. For the less noisy class, it is clarified that the private capacity and quantum capacity can be single letter characterized.Two classes of quantum channels, which we call more capable and less noisy, are introduced. The more capable class consists of channels such that the quantum capacities of the complementary channels to the environments are zero. The less noisy class consists of channels such that the private capacities of the complementary channels to the environment are zero. For the more capable class, it is clarified that the private capacity and quantum capacity coincide. For the less noisy class, it is clarified that the private capacity and quantum capacity can be single letter characterized.

arXiv:1110.5746Physical Review AJan 24, 201250 citations

χ^2

divergence. A new integral representation shows how measured

χ^2

on a qubit extension recovers quantum

χ^2

. It lets us pass from complete measured-

χ^2

ordering to complete quantum-

χ^2

, relative-entropy, and less-noisy ordering. Zhu and Wang used a signed lift to show that their qutrit channel has zero private capacity. We show that Hermitian-smoothed measured-

χ^2

ordering characterizes when such lifts exist, even with a quantum reference. Environmental dominance in binary estimation at every blocklength yields a finite-code reliability--secrecy bound. These results explain why complete comparison prevents activation with antidegradable helpers whereas regularized comparison alone does not. Building on that qutrit example, we establish this stability for a range of noisy Werner--Holevo channels, determine sharp private-capacity and antidegradability thresholds, and compute exact complementary capacities in a nondegradable range. By contrast, we extend Pauli half-erasure activation to all

1/2\le p<1

and establish the same range for a new four-level family. Reference-assisted measured-

χ^2

witnesses show why these activating constructions lack complete comparison.