Reliable Simulation of Quantum Channels: the Error Exponent
Ke Li, Yongsheng Yao
备注：
V2: presentation improved, details on proofs added, references added, minor corrections. V3: minor changes. V4: whole paper rewritten with a new title, more references added, main results and proofs unchanged. V5: close to published version
The Quantum Reverse Shannon Theorem has been a milestone in quantum information theory. It states that asymptotically reliable simulation of a quantum channel, assisted by unlimited shared entanglement, requires a rate of classical communication equal to the channel's entanglement-assisted classical capacity. In this paper, we study the error exponent of quantum channel simulation, which characterizes the optimal speed of exponential convergence of the performance towards the perfect, as the blocklength increases. Based on channel purified distance, we derive lower and upper bounds for the error exponent. Then we show that the two bounds coincide when the classical communication rate is below a critical value, and hence, we have determined the exact formula of the error exponent in the low-rate case. This enables us to obtain an operational interpretation to the channel's sandwiched Rényi information of order from 1 to 2, since our formula is expressed as a transform of this quantity. In the derivation, we have also obtained an achievability bound for quantum channel simulation in the finite-blocklength setting, which is of realistic significance.
There are different inequivalent ways to define the Rényi capacity of a channel for a fixed input distribution $P$. In a 1995 paper Csiszár has shown that for classical discrete memoryless channels there is a distinguished such quantity that has an operational interpretation as a generalized cutoff rate for constant composition channel coding. We show that the analogous notion of Rényi capacity, defined in terms of the sandwiched quantum Rényi divergences, has the same operational interpretation in the strong converse problem of classical-quantum channel coding. Denoting the constant compositi…
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Quantum hypothesis testing of non-mixed-unitarity: A multifaceted hierarchy of quantum channel discrimination
Harish-Chandra Research Institute（哈里什-钱德拉研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；IT:U Interdisciplinary Transformation University（IT:U 跨学科转型大学）
In this paper, lower bounds on error probability in coding for discrete classical and classical-quantum channels are studied. The contribution of the paper goes in two main directions: i) extending classical bounds of Shannon, Gallager and Berlekamp to classical-quantum channels, and ii) proposing a new framework for lower bounding the probability of error of channels with a zero-error capacity in the low rate region. The relation between these two problems is revealed by showing that Lovász' bound on zero-error capacity emerges as a natural consequence of the sphere packing bound once we move…
The most unstable quantum states and elementary particles possess more than a single decay channel. At the same time, it is well known that typically the decay law is not simply exponential. Therefore, it is natural to ask how to spot the non-exponential decay when (at least) two decay channels are opened. In this work, we study the tunneling phenomenon of an initially localized particle in two spatially opposite directions through two different barriers, mimicking two decay channels. In this framework, through a specific quantum mechanical examples which can be accurately solved, we study gen…
The Gottesman-Knill theorem allows for the efficient simulation of stabilizer-based quantum error-correction circuits. Errors in these circuits are commonly modeled as depolarizing channels by using Monte Carlo methods to insert Pauli gates randomly throughout the circuit. Although convenient, these channels are poor approximations of common, realistic channels like amplitude damping. Here we analyze a larger set of efficiently simulable error channels by allowing the random insertion of any one-qubit gate or measurement that can be efficiently simulated within the stabilizer formalism. Our ne…
Quantum Channel Stein's Lemma with an Exponential Strong Converse
Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）
In the simple quantum hypothesis testing problem, upper bound with asymmetric setting is shown by using a quite useful inequality by Audenaert et al, quant-ph/0610027, which was originally invented for symmetric setting. Using this upper bound, we obtain the Hoeffding bound, which are identical with the classical counter part if the hypotheses, composed of two density operators, are mutually commutative. Our upper bound improves the bound by Ogawa-Hayashi, and also provides a simpler proof of the direct part of the quantum Stein's lemma. Further, using this bound, we obtain a better exponentia…
It was shown by Ahn, Wiseman, and Milburn [PRA {\bf 67}, 052310 (2003)] that feedback control could be used as a quantum error correction process for errors induced by weak continuous measurement, given one perfectly measured error channel per qubit. Here we point out that this method can be easily extended to an arbitrary number of error channels per qubit. We show that the feedback protocols generated by our method encode $n-2$ logical qubits in $n$ physical qubits, thus requiring just one more physical qubit than in the previous case.
We define the quantum zero-error capacity, a new kind of classical capacity of a noisy quantum channel. Moreover, the necessary requirement for which a quantum channel has zero-error capacity greater than zero is also given.
The essence of quantum error-correction is to use redundant Hilbert space to identify and correct errors, and the channel fidelity of the quantum channel does not affect which errors can be identified and corrected. Based on this, it is found that quantum error-correction can be used to classify the independent quantum channel into 5 types, and 4 of the 5 types can be purified. It is found in quantum error-correction, the decoherence of quantum state may be related to the degree of identification for the state under quantum noise, and the results of this work confirmed that the degree of purit…
In quantum error correction, it is an important assumption that errors on different qubits are independent. In our previous work [Phys. Rev. A {\bf 92}, 052320 (2015)], the generality of the concatenated five-qubit code has been investgated when noises of the principal system and the auxiliary environment are assumed to be the same. In the error correction with concatenated code, tiny differences (in fidelity or independent errors) between initial quantum channels may introduce different effective channels in the next level, and therefore, it is necessary to study a meaningful question: Does f…
We study the deviations from the exponential decay law, both in quantum field theory (QFT) and quantum mechanics (QM), for an unstable particle which can decay in (at least) two decay channels. After a review of general properties of non-exponential decay in QFT and QM, we evaluate in both cases the decay probability that the unstable particle decays in a given channel in the time interval between $t$ and $t+dt.$ An important quantity is the ratio of the probability of decay into the first and the second channel: this ratio is constant in the Breit-Wigner limit (in which the decay law is expon…
In the shallow sub-threshold regime, fault-tolerant quantum computation requires a tremendous amount of qubits. In this paper, we study the error correction in the deep sub-threshold regime. We estimate the physical error rate for achieving the logical error rates of $10^{-6} - 10^{-15}$ using few-qubit codes, i.e. short repetition codes, small surface codes and the Steane code. Error correction circuits that are efficient for biased error channels are identified. Using the Steane code, when error channels are biased with a ratio of $10^{-3}$, the logical error rate of $10^{-15}$ can be achiev…
It was shown [T.S. Cubitt et al., IEEE Trans. Inform. Theory 57, 8114 (2011)] that there exist quantum channels where a single use cannot transmit classical information perfectly yet two uses can. This phenomenon is called the superactivation of the zero-error classical capacity which does not occur in classical channels. In this paper, it is shown that qubit channels cannot generate the superactivation.
Channel capacities of quantum channels can be nonadditive even if one of two quantum channels has no channel capacity. We call this phenomenon \emph{activation} of the channel capacity. In this paper, we show that when we use a quantum channel on a qubit system, only a noiseless qubit channel can generate the activation of the zero-error classical capacity. In particular, we show that the zero-error classical capacity of two quantum channels on qubit systems cannot be activated. Furthermore, we present a class of examples showing the activation of the zero-error classical capacity in low-dimen…
A quantum channel models the interaction between the system we are interested in and its environment. Such a model can capture the main features of the interaction but because of the complexity of the environment we can not assume that it is fully accurate. We study the robustness of quantum error correction operations against completely unexpected and subsequently undetermined type of channel uncertainties. We find that a channel-adapted optimal error correction operation does not only give the best possible channel fidelity but it is more robust against channel alterations than any other err…
We develop a theory for finding quantum error correction (QEC) procedures which are optimized for given noise channels. Our theory accounts for uncertainties in the noise channel, against which our QEC procedures are robust. We demonstrate via numerical examples that our optimized QEC procedures always achieve a higher channel fidelity than the standard error correction method, which is agnostic about the specifics of the channel. This demonstrates the importance of channel characterization before QEC procedures are applied. Our main novel finding is that in the setting of a known noise channe…
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We consider error correction procedures designed specifically for the amplitude damping channel. We analyze amplitude damping errors in the stabilizer formalism. This analysis allows a generalization of the [4,1] `approximate' amplitude damping code of quant-ph/9704002. We present this generalization as a class of [2(M+1),M] codes and present quantum circuits for encoding and recovery operations. We also present a [7,3] amplitude damping code based on the classical Hamming code. All of these are stabilizer codes whose encoding and recovery operations can be completely described with Clifford g…
A family of high rate quantum error correcting codes adapted to the amplitude damping channel is presented. These codes are nonadditive and exploit self-complementarity structure to correct all first-order errors. Their rates can be higher than 1/2. The recovery operations of these codes can be generated by a simple algorithm and have a projection nature, which makes them potentially easy to implement.
Quantum Channel Stein Theorem beyond Definite Causal Order
QudeLeap Research, Shanghai 200030, China（科跃研究）；Thrust of Artificial Intelligence, Information Hub,The Hong Kong University of Science and Technology (Guangzhou), Guangzhou 511453, China（香港科技大学（广州）人工智能 thrust 信息枢纽）
We introduce a model of quantum insertion-deletion (insdel) channels. Insdel channels are meant to represent, for example, synchronization errors arising in data transmission. In the classical setting, they represent a strict generalization of the better-understood corruption error channels, and until recently, had mostly resisted effort toward a similar understanding as their corruption counterparts. They have received considerable attention in recent years. Very recently, Haeupler and Shahrasbi developed a framework, using what they call synchronisation strings, that allows one to turn insde…
