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arXiv:2401.16312

Symmetry-Enforced Quadratic Approximate-Degradability Bounds for Noisy Landau-Streater Channels

Yun-Feng Lo, Yen-Chi Lee, Min-Hsiu Hsieh

Published Jan 29, 2024• 0 citations
arXivHTMLPDF

Abstract

Understanding the limits of quantum communication is a central problem in quantum information theory. Structural properties of quantum channels provide tools for analyzing these limits. In particular, approximate degradability links quantum-capacity bounds to the accuracy with which a channel's environment can be reconstructed from its output. We study this reconstruction problem for noisy Landau-Streater channels across finite-spin representations. For spin $j$ and noise probability $p$, we construct an explicit degrading map with diamond-norm error at most $[2+8/(j(j+1))]p^2\le(38/3)p^2$ on the common interval $0\le p\le1/8$. A lower bound establishes quadratic order for this construction at each fixed spin. Floating-point semidefinite optimization places its error within $2.48\%$ of the numerical optimum on the sampled grid for four spins. Covariance and conditional-entropy concavity yield explicit two-sided quantum-capacity bounds for this family with a spin-uniform remainder. Building on the low-noise Pauli cancellation strategy, we formulate a sufficient condition based on a common scalar adjoint action on fixed, possibly nonunitary noise Kraus operators. The resulting reconstruction-error bound is uniform in system dimension and Kraus count. The criterion applies to irreducible $SU(N)$ generator channels and noisy Werner-Holevo channels, with the $SU(3)$ adjoint representation providing an example outside the mixed-unitary class.

DOI
10.48550/arxiv.2401.16312

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Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact that the complementary channel can be obtained from the channel by applying a degrading map. In this work we introduce the concept of approximate degradable channels, which satisfy this condition up to some finite ε ≥ 0. That is, there exists a degrading map which upon composition with the channel is ε-close in the diamond norm to the complementary channel. We show that for any fixed channel the smallest such ε can be efficiently determined via a semidefinite program. Moreover, these approximate degradable channels also approximately inherit all other properties of degradable channels. As an application, we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication.
Transmitting quantum information across quantum channels is an important task. However quantum information is delicate, and is easily corrupted. We address the task of protecting quantum information from an information theoretic perspective -- we encode some message qudits into a quantum code, send the encoded quantum information across the noisy quantum channel, then recover the message qudits by decoding. In this dissertation, we discuss the coding problem from several perspectives.} \n \nThe noisy quantum channel is one of the central aspects of the quantum coding problem, and hence quantifying the noisy quantum channel from the physical model is an important problem. \nWe work with an explicit physical model -- a pair of initially decoupled quantum harmonic oscillators interacting with a spring-like coupling, where the bath oscillator is initially in a thermal-like state. In particular, we treat the completely positive and trace preserving map on the system as a quantum channel, and study the truncation of the channel by truncating its Kraus set. We thereby derive the matrix elements of the Choi-Jamiolkowski operator of the corresponding truncated channel, which are truncated transition amplitudes. Finally, we give a computable approximation for these truncated transition amplitudes with explicit error bounds, and perform a case study of the oscillators in the off-resonant and weakly-coupled regime numerically. \n \nIn the context of truncated noisy channels, we revisit the notion of approximate error correction of finite dimension codes. We derive a computationally simple lower bound on the worst case entanglement fidelity of a quantum code, when the truncated recovery map of Leung et. al. is rescaled. As an application, we apply our bound to construct a family of multi-error correcting amplitude damping codes that are permutation-invariant. This demonstrates an explicit example where the specific structure of the noisy channel allows code design out of the stabilizer formalism via purely algebraic means. \n \nWe study lower bounds on the quantum capacity of adversarial channels, where we restrict the selection of quantum codes to the set of concatenated quantum codes. \nThe adversarial channel is a quantum channel where an adversary corrupts a fixed fraction of qudits sent across a quantum channel in the most malicious way possible. The best known rates of communicating over adversarial channels are given by the quantum Gilbert-Varshamov (GV) bound, that is known to be attainable with random quantum codes. We generalize the classical result of Thommesen to the quantum case, thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum GV bound. The outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random independently chosen stabilizer codes, where the rates of the inner and outer codes lie in a specified feasible region. \n \nWe next study upper bounds on the quantum capacity of some low dimension quantum channels. \nThe quantum capacity of a quantum channel is the maximum rate at which quantum information can be transmitted reliably across it, given arbitrarily many uses of it. While it is known that random quantum codes can be used to attain the quantum capacity, the quantum capacity of many classes of channels is undetermined, even for channels of low input and output dimension. For example, depolarizing channels are \n important quantum channels, but do not have tight numerical bounds. \nWe obtain upper bounds on the quantum capacity of some unital and non-unital channels \n -- two-qubit Pauli channels, two-qubit depolarizing channels, two-qubit locally symmetric channels, \n shifted qubit depolarizing channels, and shifted two-qubit Pauli channels -- \n using the coherent information of some degradable channels. We use the notion \n of twirling quantum channels, and Smith and Smolin's method of constructing \n degradable extensions of quantum channels extensively. The degradable channels we \n introduce, study and use are two-qubit amplitude damping channels. Exploiting the \n notion of covariant quantum channels, we give sufficient conditions for the quantum \n capacity of a degradable channel to be the optimal value of a concave program with \n linear constraints, and show that our two-qubit degradable amplitude damping channels have this property.
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 χ2χ^2χ2 divergence. A new integral representation shows how measured χ2χ^2χ2 on a qubit extension recovers quantum χ2χ^2χ2. It lets us pass from complete measured-χ2χ^2χ2 ordering to complete quantum-χ2χ^2χ2, 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-χ2χ^2χ2 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≤p<11/2\le p<11/2≤p<1 and establish the same range for a new four-level family. Reference-assisted measured-χ2χ^2χ2 witnesses show why these activating constructions lack complete comparison.
We study the Landau–Streater quantum channel Φ:B(Hd)↦B(Hd), whose Kraus operators are proportional to the irreducible unitary representation of SU(2) generators of dimension d. We establish SU(2) covariance for all d and U(3) covariance for d = 3. Using the theory of angular momentum, we explicitly find the spectrum and the minimal output entropy of Φ. Negative eigenvalues in the spectrum of Φ indicate that the channel cannot be obtained as a result of Hermitian Markovian quantum dynamics. Degradability and antidegradability of the Landau–Streater channel is fully analyzed. We calculate classical and entanglement-assisted capacities of Φ. Quantum capacity of Φ vanishes if d = 2, 3 and is strictly positive if d ⩾ 4. We show that the channel Φ ⊗ Φ does not annihilate entanglement and preserves entanglement of some states with Schmidt rank 2 if d ⩾ 3.
The quantum capacity captures the value of a quantum channel for transmitting quantum information, establishing the fundamental limits on quantum communication. In spite of its central role in quantum information theory, the quantum capacity of most channels is unknown, with wide gaps between the best upper and lower bounds. Even deciding whether a channel has nonzero capacity -- finding its capacity threshold -- is difficult. In this paper we report significant increases in the capacity thresholds of two prototypical noise models: the depolarizing channel and Pauli channels. In the case of the depolarizing channel, this is the first improvement in 18 years, giving a bigger increase beyond the hashing bound than all previous improvements combined. Our starting point is the representation theoretic framework recently proposed by Bhalerao and Leditzky (2025) to compute coherent information for special permutation invariant states. We generalize their framework to the full symmetric subspace, which allow us to optimize coherent information over rank two states in that space. A representation theoretic calculation shows that exponentially many Kraus operators of the channel annihilate the symmetric space, corresponding to a massive decrease in environment entropy for states on the symmetric space compared to the maximally mixed state. This explains the enhanced coherent information as a manifestation of degeneracy for the resulting codes.