On the von Neumann capacity of noisy quantum channels

We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann capacity of quantum channels, which is a quantum mechanical extension of the Shannon capacity and reverts to it in the classical limit. As such, the von Neumann capacity assumes the role of a classical or quantum capacity depending on the usage of the channel. In analogy to the classical construction, this capacity is defined as the maximum von Neumann mutual entropy processed by the channel, a measure which reduces to the capacity for classical information transmission through quantum channels (the "Kholevo capacity") when known quantum states are sent. The quantum mutual entropy fulfills all basic requirements for a measure of information, and observes quantum data-processing inequalities. We also derive a quantum Fano inequality relating the quantum loss of the channel to the fidelity of the quantum code. The quantities introduced are calculated explicitly for the quantum "depolarizing" channel. The von Neumann capacity is interpreted within the context of superdense coding, and an "extended" Hamming bound is derived that is consistent with that capacity.Comment: 15 pages RevTeX with psfig, 13 figures. Revised interpretation of capacity, added section, changed titl

Entropic bounds on coding for noisy quantum channels

In analogy with its classical counterpart, a noisy quantum channel is characterized by a loss, a quantity that depends on the channel input and the quantum operation performed by the channel. The loss reflects the transmission quality: if the loss is zero, quantum information can be perfectly transmitted at a rate measured by the quantum source entropy. By using block coding based on sequences of n entangled symbols, the average loss (defined as the overall loss of the joint n-symbol channel divided by n, when n tends to infinity) can be made lower than the loss for a single use of the channel. In this context, we examine several upper bounds on the rate at which quantum information can be transmitted reliably via a noisy channel, that is, with an asymptotically vanishing average loss while the one-symbol loss of the channel is non-zero. These bounds on the channel capacity rely on the entropic Singleton bound on quantum error-correcting codes [Phys. Rev. A 56, 1721 (1997)]. Finally, we analyze the Singleton bounds when the noisy quantum channel is supplemented with a classical auxiliary channel.Comment: 20 pages RevTeX, 10 Postscript figures. Expanded Section II, added 1 figure, changed title. To appear in Phys. Rev. A (May 98

Information-theoretic interpretation of quantum error-correcting codes

Quantum error-correcting codes are analyzed from an information-theoretic perspective centered on quantum conditional and mutual entropies. This approach parallels the description of classical error correction in Shannon theory, while clarifying the differences between classical and quantum codes. More specifically, it is shown how quantum information theory accounts for the fact that "redundant" information can be distributed over quantum bits even though this does not violate the quantum "no-cloning" theorem. Such a remarkable feature, which has no counterpart for classical codes, is related to the property that the ternary mutual entropy vanishes for a tripartite system in a pure state. This information-theoretic description of quantum coding is used to derive the quantum analogue of the Singleton bound on the number of logical bits that can be preserved by a code of fixed length which can recover a given number of errors.Comment: 14 pages RevTeX, 8 Postscript figures. Added appendix. To appear in Phys. Rev.

Floquetifying the Colour Code

Floquet codes are a recently discovered type of quantum error correction code. They can be thought of as generalising stabilizer codes and subsystem codes, by allowing the logical Pauli operators of the code to vary dynamically over time. In this work, we use the ZX-calculus to create new Floquet codes that are in a definable sense equivalent to known stabilizer codes. In particular, we find a Floquet code that is equivalent to the colour code, but has the advantage that all measurements required to implement it are of weight one or two. Notably, the qubits can even be laid out on a square lattice. This circumvents current difficulties with implementing the colour code fault-tolerantly, while preserving its advantages over other well-studied codes, and could furthermore allow one to benefit from extra features exclusive to Floquet codes. On a higher level, as in arXiv:2303.08829, this work shines a light on the relationship between 'static' stabilizer and subsystem codes and 'dynamic' Floquet codes; at first glance the latter seems a significant generalisation of the former, but in the case of the codes that we find here, the difference is essentially just a few basic ZX-diagram deformations.Comment: 15 + 24 pages, 18 figures. Comments encouraged - email address in paper

Code deformation and lattice surgery are gauge fixing

International audienceThe large-scale execution of quantum algorithms requires basic quantum operations to be implemented fault-tolerantly. The most popular technique for accomplishing this, using the devices that can be realized in the near term, uses stabilizer codes which can be embedded in a planar layout. The set of fault-tolerant operations which can be executed in these systems using unitary gates is typically very limited. This has driven the development of measurement-based schemes for performing logical operations in these codes, known as lattice surgery and code deformation. In parallel, gauge fixing has emerged as a measurement-based method for performing universal gate sets in subsystem stabilizer codes. In this work, we show that lattice surgery and code deformation can be expressed as special cases of gauge fixing, permitting a simple and rigorous test for fault-tolerance together with simple guiding principles for the implementation of these operations. We demonstrate the accuracy of this method numerically with examples based on the surface code, some of which are novel

Boolean Differential Operators

We consider four combinatorial interpretations for the algebra of Boolean differential operators. We show that each interpretation yields an explicit matrix representation for Boolean differential operators

Molecular study of drought response in the Mediterranean conifer Pinus Pinaster Ait. : differential transcriptomic profiling reveals constitutive water deficit‐independent drought tolerance mechanisms

Adaptation of long‐living forest trees to respond to environmental changes is essential to secure their performance under adverse conditions. Water deficit is one of the most significant stress factors determining tree growth and survival. Maritime pine (Pinus pinaster Ait.), the main source of softwood in southwestern Europe, is subjected to recurrent drought periods which, according to climate change predictions for the years to come, will progressively increase in the Mediterranean region. The mechanisms regulating pine adaptive responses to environment are still largely unknown. The aim of this work was to go a step further in understanding the molecular mechanisms underlying maritime pine response to water stress and drought tolerance at the whole plant level. A global transcriptomic profiling of roots, stems, and needles was conducted to analyze the performance of siblings showing contrasted responses to water deficit from an ad hoc designed full‐sib family. Although P. pinaster is considered a recalcitrant species for vegetative propagation in adult phase, the analysis was conducted using vegetatively propagated trees exposed to two treatments: well‐watered and moderate water stress. The comparative analyses led us to identify organ‐specific genes, constitutively expressed as well as differentially expressed when comparing control versus water stress conditions, in drought‐sensitive and drought‐tolerant genotypes. Different response strategies can point out, with tolerant individuals being pre‐adapted for coping with drought by constitutively expressing stress‐related genes that are detected only in latter stages on sensitive individuals subjected to drought