Source-linked AI summary
Quantum Communication With Zero-Capacity Channels
Graeme Smith, Jon Yard
TL;DR
Quantum capacity is a fundamental measure of protected quantum information, but its ability to characterize channels remains under question. The paper develops a finite-dimensional-channel argument using a 50%-erasure channel and shows a strictly positive information difference, demonstrating nonzero transmission in the combined setting.
Problem
The practical significance of quantum capacity’s regularized behavior remains uncertain because deviations from single-use capacity were speculated to be small.
Method
The paper uses a key channel relationship and a finite-dimensional 50%-erasure channel to construct the required input state and ensemble.
Results
I(X; B) − I(X; E) > 0.02 for an input and ensemble with two rank-two states.
Takeaways & Limitations
The result shows that quantum capacity can be positive for a convex combination of channels under sufficiently small p.
Takeaways & Limitations
The supporting material assumes basic knowledge of quantum information.
Abstract
from arXiv · showhide
Communication over a noisy quantum channel introduces errors in the transmission that must be corrected. A fundamental bound on quantum error correction is the quantum capacity, which quantifies the amount of quantum data that can be protected. We show theoretically that two quantum channels, each with a transmission capacity of zero, can have a nonzero capacity when used together. This unveils a rich structure in the theory of quantum communications, implying that the quantum capacity does not uniquely specify a channel's ability for transmitting quantum information.
Superactivation with finite dimensional channels
The section replaces an infinite-dimensional auxiliary channel with a finite-dimensional 50%-erasure channel and establishes a state yielding positive coherent information when combined with a suitable channel. A four-dimensional private Horodecki channel provides an explicit example with a mutual-information difference exceeding 0.02.
- Finite-dimensional construction: A finite-dimensional 50%-erasure channel is introduced as a more manageable replacement for the infinite-dimensional auxiliary channel.The erasure channel has finite-dimensional input and output systems.
- Finite-dimensional construction: For a channel N with finite input, the resulting lower bound is achievable using a finite construction.The bound is formulated using a 50%-erasure channel whose input dimension matches that of N's input.
- Explicit example: A four-dimensional private Horodecki channel has a two-state rank-two ensemble satisfying I(X; B) − I(X; E) > 0.02.This yields a corresponding four-dimensional erasure channel and a state for the combined construction.
- Proof strategy: The proof purifies the mixed ensemble, sends the purifying system through a 50%-erasure channel, and evaluates coherent information after sending the original system through N.The entropy inequalities exploit the erasure channel's equal-probability delivery of its input to the output or environment.
A four-dimensional private Horodecki channel
The section explicitly describes a four-dimensional private Horodecki channel using a two-qubit input and six Kraus matrices. An ensemble yields a private-capacity lower bound exceeding 0.02.
- The channel is a four-dimensional private Horodecki channel, denoted N^(4)_H.
- The input is a tensor product of two qubits, A = A1A2, with output system B.
- The channel is specified by six Kraus matrices.
- 0.02 is a lower bound on the private capacity P(N^(4)_H), obtained via an ensemble.
- I(X; B) − I(X; E) ≥ 1 − q log2 q − (1 − q) log2(1 − q) > 0.02.
Nonconvexity of quantum capacity
Quantum capacity is nonconvex: a convex combination of two zero-capacity channels can have positive capacity. Moreover, channels with Q(1)=0 can have arbitrarily large actual capacity.
- Nonconvexity of quantum capacity: Ic(Mp ⊗Mp, ρAC) > 0 whenever 0 < p < Ic(NH ⊗Ae, ρAC) c + Ic(NH ⊗Ae, ρAC).Here c = log2 |E|, arising from the finite environment dimension of NH.
- Nonconvexity of quantum capacity: 0 < p < 0.0041 gives positive capacity for the four-dimensional example, with stronger violations expected in larger examples.For that example, c = log2 6.