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Register a new userQuantifying the magic of quantum channels
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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.
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The unextendibility or monogamy of entangled states is a key property of quantum entanglement. Unlike conventional ways of expressing entanglement monogamy via entanglement measure inequalities, we develop a state-dependent resource theory to quantify the unextendibility of bipartite entangled states. First, we introduce a family of entanglement measures called unextendible entanglement. Given a bipartite state $rho_{AB}$, the key idea behind these measures is to minimize a divergence between $rho_{AB}$ and any possibly reduced state $rho_{AB}$ of an extension $rho_{ABB}$ of $rho_{AB}$. These measures are intuitively motivated by the fact that the more a bipartite state is entangled, the less that each of its individual systems can be entangled with a third party. Second, we show that the unextendible entanglement is an entanglement monotone under two-extendible operations, which include local operations and one-way classical communication as a special case. Unextendible entanglement has several other desirable properties, including normalization and faithfulness. As applications, we show that the unextendible entanglement provides efficiently computable benchmarks for the rate of perfect secret key distillation or entanglement distillation, as well as for the overhead of probabilistic secret key or entanglement distillation.
Magic-state distillation (or non-stabilizer state manipulation) is a crucial component in the leading approaches to realizing scalable, fault-tolerant, and universal quantum computation. Related to non-stabilizer state manipulation is the resource theory of non-stabilizer states, for which one of the goals is to characterize and quantify non-stabilizerness of a quantum state. In this paper, we introduce the family of thauma measures to quantify the amount of non-stabilizerness in a quantum state, and we exploit this family of measures to address several open questions in the resource theory of non-stabilizer states. As a first application, we establish the hypothesis testing thauma as an efficiently computable benchmark for the one-shot distillable non-stabilizerness, which in turn leads to a variety of bounds on the rate at which non-stabilizerness 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 pervious 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 non-stabilizerness 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 non-stabilizer states and reveals a difference between the resource theory of non-stabilizer states and other resource theories such as entanglement and coherence.
Quantum addition channels have been recently introduced in the context of deriving entropic power inequalities for finite dimensional quantum systems. We prove a reverse entropy power equality which can be used to analytically prove an inequality conjectured recently for arbitrary dimension and arbitrary addition weight. We show that the relative entropic difference between the output of such a quantum additon channel and the corresponding classical mixture quantitatively captures the amount of coherence present in a quantum system. This new coherence measure admits an upper bound in terms of the relative entropy of coherence and is utilized to formulate a state-dependent uncertainty relation for two observables. Our results may provide deep insights to the origin of quantum coherence for mixed states that truly come from the discrepancy between quantum addition and the classical mixture.
We introduce various measures of forward classical communication for bipartite quantum channels. Since a point-to-point channel is a special case of a bipartite channel, the measures reduce to measures of classical communication for point-to-point channels. As it turns out, these reduced measures have been reported in prior work of Wang et al. on bounding the classical capacity of a quantum channel. As applications, we show that the measures are upper bounds on the forward classical capacity of a bipartite channel. The reduced measures are upper bounds on the classical capacity of a point-to-point quantum channel assisted by a classical feedback channel. Some of the various measures can be computed by semi-definite programming.
We show that the new quantum extension of Renyis alpha-relative entropies, introduced recently by Muller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013), and Wilde, Winter, Yang, Commun. Math. Phys. 331, (2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Renyi relative entropies depends on the parameter alpha: for alpha<1, the right choice seems to be the traditional definition, whereas for alpha>1 the right choice is the newly introduced version. As a sideresult, we show that the new Renyi alpha-relative entropies are asymptotically attainable by measurements for alpha>1, and give a new simple proof for their monotonicity under completely positive trace-preserving maps.
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