Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1108.5065

Entropy of quantum channel in the theory of quantum information

Wojciech Roga
View original source ↗Read source document ↗Export metadata ↓

Abstract

Quantum channels, also called quantum operations, are linear, trace preserving and completely positive transformations in the space of quantum states. Such operations describe discrete time evolution of an open quantum system interacting with an environment. The thesis contains an analysis of properties of quantum channels and different entropies used to quantify the decoherence introduced into the system by a given operation. Part I of the thesis provides a general introduction to the subject. In Part II, the action of a quantum channel is treated as a process of preparation of a quantum ensemble. The Holevo information associated with this ensemble is shown to be bounded by the entropy exchanged during the preparation process between the initial state and the environment. A relation between the Holevo information and the entropy of an auxiliary matrix consisting of square root fidelities between the elements of the ensemble is proved in some special cases. Weaker bounds on the Holevo information are also established. The entropy of a channel, also called the map entropy, is defined as the entropy of the state corresponding to the channel by the Jamiolkowski isomorphism. In Part III of the thesis, the additivity of the entropy of a channel is proved. The minimal output entropy, which is difficult to compute, is estimated by an entropy of a channel which is much easier to obtain. A class of quantum channels is specified, for which additivity of channel capacity is conjectured. The last part of the thesis contains characterization of Davies channels, which correspond to an interaction of a state with a thermal reservoir in the week coupling limit, under the condition of quantum detailed balance and independence of rotational and dissipative evolutions. The Davies channels are characterized for one-qubit and one-qutrit systems.

Explore related subjects

quant-ph

Keep this discovery

BibTeXRIS

Wojciech Roga. 2011-10-03. Entropy of quantum channel in the theory of quantum information. https://arxiv.org/abs/1108.5065

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Probing the Error-Mitigation Threshold with Matrix Product States

Quantum error mitigation relies on accurate noise characterization, but mismatches between the actual and characterized noise can be amplified and drive a sharp threshold between successful and failed mitigation. In random circuits, this threshold maps onto a random-field Ising transition, but previous exact numerics were limited to small one-dimensional and all-to-all systems, leaving explicit two-dimensional architectures unresolved. We develop a fixed-bond-dimension matrix-product-state method for the replicated transfer dynamics that extends threshold calculations beyond exact propagation while retaining the finite-size signatures of the transition. At system sizes beyond previous exact studies, we recover the predicted absence of a threshold for quenched disorder in 1D, obtain a sharper annealed all-to-all critical point, and resolve architecture-dependent finite-depth thresholds in 2D square and heavy-hex circuits. These results establish replicated tensor-network dynamics as a practical tool for probing error-mitigation thresholds in large and higher-dimensional noisy circuits.

quant-ph↗
arXiv2026

Low-cost algorithm-to-execution framework for surface-code quantum computing

The execution of useful quantum algorithms on fault-tolerant processors requires more than a mapping from logical gates to encoded operations: the spatial organization, non-Clifford resource supply, and execution schedule must also be determined while keeping physical overhead within practical limits. Although the theoretical hierarchy from logical circuits to fault-tolerant operations is well established, these implementation choices are often specified and optimized separately. Here we develop a low-cost algorithm-to-execution framework for surface-code quantum computing. From hierarchical algorithm descriptions, it constructs dependency-preserving logical schedules and an executable workload capturing logical interactions, operation parallelism, and time-resolved non-Clifford demand, thereby linking logical computation to surface-code organization, resource-state preparation, and fault-tolerant execution in a traceable workflow. We apply the framework to twenty benchmark circuits across seven algorithm families and a hierarchically composed application-scale elliptic-curve discrete-logarithm workload. Physical costs vary substantially even for circuits with similar logical resource counts. Under our direct-rotation calibration, non-Clifford implementation selection reduces space-time volume by up to 241.5 times versus an all-synthesis baseline for the QAOA amplitude-amplification workload. Circuit-specific surface-code layouts reduce routed-latency estimates for all twenty benchmarks; thirteen also reduce space-time volume because communication savings outweigh added spatial overhead. These results show that low-cost fault-tolerant execution depends on computation scheduling and organization, not aggregate logical resource counts alone.

quant-ph↗
arXiv2026

Sample-optimal learning of stabilizer states

It is well-known that learning a pure $n$-qubit stabilizer state $|\psi\rangle$ both requires, and can be accomplished with, access to a number of copies of $|\psi\rangle$ linear in $n$. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that $L_\delta(n)$, the smallest number of copies from which a quantum procedure can identify any stabilizer state with failure probability at most $0<\delta<1/8$, satisfies $n+\lceil\log_2(1/\delta)\rceil-3\leq L_\delta(n)\leq n+\left\lceil\log_2(1/\delta)\right\rceil+4$. We present a polynomial-time quantum learning algorithm that saturates this bound, achieving a constant factor improvement in sample-complexity over previously known approaches. As an immediate corollary, we obtain via the Choi-Jamiolkowski isomorphism an algorithm for learning an unknown $n$-qubit Clifford unitary from $2n+\left\lceil\log_2(1/\delta)\right\rceil+4$ queries, the $n$-dependence of which we show to be optimal. Our proof technique, which involves Fourier analysis on the abelian group $\mathbb{Z}_4^n \times \mathbb{F}_2^{n(n-1)/2}$, seems to be qualitatively different to previous approaches to stabilizer state learning, and may be of some independent interest; in particular, it admits natural generalisations to further problems in quantum learning theory.

quant-ph↗