Finally, here is a modern, self-contained text on quantum information theory suitable for graduate-level courses. Developing the subject 'from the ground up' it covers classical results as well as ...major advances of the past decade. Beginning with an extensive overview of classical information theory suitable for the non-expert, the author then turns his attention to quantum mechanics for quantum information theory, and the important protocols of teleportation, super-dense coding and entanglement distribution. He develops all of the tools necessary for understanding important results in quantum information theory, including capacity theorems for classical, entanglement-assisted, private and quantum communication. The book also covers important recent developments such as superadditivity of private, coherent and Holevo information, and the superactivation of quantum capacity. This book will be warmly welcomed by the upcoming generation of quantum information theorists and the already established community of classical information theorists.
This paper establishes several converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state, which is ...a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here, we use this approach along with a "privacy test" to establish a general meta-converse bound for private communication, which has a number of applications. The meta-converse allows for proving that any quantum channel's relative entropy of entanglement is a strong converse rate for private communication. For covariant channels, the meta-converse also leads to second-order expansions of relative entropy of entanglement bounds for private communication rates. For such channels, the bounds also apply to the private communication setting in which the sender and the receiver are assisted by unlimited public classical communication, and as such, they are relevant for establishing various converse bounds for quantum key distribution protocols conducted over these channels. We find precise characterizations for several channels of interest and apply the methods to establish converse bounds on the private transmission capabilities of all phase-insensitive bosonic channels.
To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we ...develop a resource theory for magic quantum channels to characterize and quantify the quantum 'magic' or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.
Magic-state distillation (or nonstabilizer state manipulation) is a crucial component in the leading approaches to realizing scalable, fault-tolerant, and universal quantum computation. Related to ...nonstabilizer state manipulation is the resource theory of nonstabilizer states, for which one of the goals is to characterize and quantify nonstabilizerness of a quantum state. In this Letter, we introduce the family of thauma measures to quantify the amount of nonstabilizerness in a quantum state, and we exploit this family of measures to address several open questions in the resource theory of nonstabilizer states. As a first application, we establish the hypothesis testing thauma as an efficiently computable benchmark for the one-shot distillable nonstabilizerness, which in turn leads to a variety of bounds on the rate at which nonstabilizerness can be distilled, as well as on the overhead of magic-state distillation. We then prove that the max-thauma can be used as an efficiently computable tool in benchmarking the efficiency of magic-state distillation, and that it can outperform previous approaches based on mana. Finally, we use the min-thauma to bound a quantity known in the literature as the "regularized relative entropy of magic." As a consequence of this bound, we find that two classes of states with maximal mana, a previously established nonstabilizerness measure, cannot be interconverted in the asymptotic regime at a rate equal to one. This result resolves a basic question in the resource theory of nonstabilizer states and reveals a difference between the resource theory of nonstabilizer states and other resource theories such as entanglement and coherence.
Quantum entanglement is a key physical resource in quantum information processing that allows for performing basic quantum tasks such as teleportation and quantum key distribution, which are ...impossible in the classical world. Ever since the rise of quantum information theory, it has been an open problem to quantify entanglement in an information-theoretically meaningful way. In particular, every previously defined entanglement measure bearing a precise information-theoretic meaning is not known to be efficiently computable, or if it is efficiently computable, then it is not known to have a precise information-theoretic meaning. In this Letter, we meet this challenge by introducing an entanglement measure that has a precise information-theoretic meaning as the exact cost required to prepare an entangled state when two distant parties are allowed to perform quantum operations that completely preserve the positivity of the partial transpose. Additionally, this entanglement measure is efficiently computable by means of a semidefinite program, and it bears a number of useful properties such as additivity and faithfulness. Our results bring key insights into the fundamental entanglement structure of arbitrary quantum states, and they can be used directly to assess and quantify the entanglement produced in quantum-physical experiments.
Recoverability in quantum information theory Wilde, Mark M.
Proceedings - Royal Society. Mathematical, physical and engineering sciences,
10/2015, Letnik:
471, Številka:
2182
Journal Article
Recenzirano
Odprti dostop
The fact that the quantum relative entropy is non-increasing with respect to quantum physical evolutions lies at the core of many optimality theorems in quantum information theory and has ...applications in other areas of physics. In this work, we establish improvements of this entropy inequality in the form of physically meaningful remainder terms. One of the main results can be summarized informally as follows: if the decrease in quantum relative entropy between two quantum states after a quantum physical evolution is relatively small, then it is possible to perform a recovery operation, such that one can perfectly recover one state while approximately recovering the other. This can be interpreted as quantifying how well one can reverse a quantum physical evolution. Our proof method is elementary, relying on the method of complex interpolation, basic linear algebra and the recently introduced Rényi generalization of a relative entropy difference. The theorem has a number of applications in quantum information theory, which have to do with providing physically meaningful improvements to many known entropy inequalities.
The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, ...conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz-Rényi relative entropies. In this paper, I introduce the optimized quantum f-divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz's original approach. Interestingly, the sandwiched Rényi relative entropies are particular examples of the optimized f-divergence. Thus, one benefit of this paper is that there is now a single, unified approach for establishing the data processing inequality for both the Petz-Rényi and sandwiched Rényi relative entropies, for the full range of parameters for which it is known to hold. This paper discusses other aspects of the optimized f-divergence, such as the classical case, the classical-quantum case, and how to construct optimized f-information measures.
Strong Converse Rates for Quantum Communication Tomamichel, Marco; Wilde, Mark M.; Winter, Andreas
IEEE transactions on information theory,
01/2017, Letnik:
63, Številka:
1
Journal Article
Recenzirano
Odprti dostop
We revisit a fundamental open problem in quantum information theory, namely, whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a ...non-vanishing probability of decoding error. Here, we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication. For any sequence of codes with rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical postprocessing on the transmitted quantum data. As an application of this result, for generalized dephasing channels, we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. Thus, we conclusively settle the strong converse question for a class of quantum channels that have a non-trivial quantum capacity.
The capacities of noisy quantum channels capture the ultimate rates of information transmission across quantum communication lines, and the quantum capacity plays a key role in determining the ...overhead of fault-tolerant quantum computation platforms. Closed formulae for these capacities in bosonic systems were lacking for a key class of non-Gaussian channels, bosonic dephasing channels, which are used to model noise affecting superconducting circuits and fibre-optic communication channels. Here we provide an exact calculation of the quantum, private, two-way assisted quantum and secret-key-agreement capacities of all bosonic dephasing channels. We prove that they are equal to the relative entropy of the distribution underlying the channel with respect to the uniform distribution, solving a problem that was originally posed over a decade ago.An exact solution for the quantum and private capacities of bosonic dephasing channels is provided. The authors prove that these capacities are equal to the relative entropy of the distribution underlying the channel with respect to the uniform distribution.
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