Charles H. Bennett, David P. DiVincenzo, John A. Smolin
备注：
4 pages, RevTeX, epsf macros to include 2 eps figures. See also http://vesta.physics.ucla.edu/~smolin/ Version 2, submitted to Phys. Rev. Lett., includes several small changes and corrections
The quantum analog of the classical erasure channel provides a simple example of a channel whose asymptotic capacity for faithful transmission of intact quantum states, with and without the assistance of a two-way classical side channel, can be computed exactly. We derive the quantum and classical capacities for the quantum erasure channel and related channels, and compare them to the depolarizing channel, for which only upper and lower bounds on the capacities are known.
We present a communication protocol for the erasure channel assisted by backward classical communication, which achieves a significantly better rate than the best prior result. In addition, we prove an upper bound for the capacity of the channel. The upper bound is smaller than the capacity of the erasure channel when it is assisted by two-way classical communication. Thus, we prove the separation between quantum capacities assisted by backward classical communication and two-way classical communication.
We consider various forms of a process, which we call {\em gluing}, for combining two or more complementary quantum channel pairs $(\mathcal{B},\mathcal{C})$ to form a composite. One type of gluing combines a perfect channel with a second channel to produce a \emph{generalized erasure channel} pair $(\mathcal{B}_g,\mathcal{C}_g)$. We consider two cases in which the second channel is (i) an amplitude-damping, or (ii) a phase-damping qubit channel; (ii) is the \emph{dephrasure channel} of Leditzky et al. For both (i) and (ii), $(\mathcal{B}_g,\mathcal{C}_g)$ depends on the damping parameter $0\l…
We provide $poly\log$ sparse quantum codes for correcting the erasure channel arbitrarily close to the capacity. Specifically, we provide $[[n, k, d]]$ quantum stabilizer codes that correct for the erasure channel arbitrarily close to the capacity if the erasure probability is at least $0.33$, and with a generating set $\langle S_1, S_2, ... S_{n-k} \rangle$ such that $|S_i|\leq \log^{2+ζ}(n)$ for all $i$ and for any $ζ> 0$ with high probability. In this work we show that the result of Delfosse et al. is tight: one can construct capacity approaching codes with weight almost $O(1)$.
The quantum erasure channel (QEC) is considered. Codes for the QEC have to correct for erasures, i. e., arbitrary errors at known positions. We show that four qubits are necessary and sufficient to encode one qubit and correct one erasure, in contrast to five qubits for unknown positions. Moreover, a family of quantum codes for the QEC, the quantum BCH codes, that can be efficiently decoded is introduced.
Quantum capacity, as the key figure of merit for a given quantum channel, upper bounds the channel's ability in transmitting quantum information. Identifying different type of channels, evaluating the corresponding quantum capacity and finding the capacity-approaching coding scheme are the major tasks in quantum communication theory. Quantum channel in discrete variables has been discussed enormously involving various error models, while error model in the continuous variable channel has been less studied due to the infinite dimensional problem. In this paper, we investigate a general continuo…
Losses in quantum communication lines severely affect the rates of reliable information transmission and are usually considered to be state-independent. However, the loss probability does depend on the system state in general, with the polarization dependent losses being a prominent example. Here we analyze biased trace decreasing quantum operations that assign different loss probabilities to states and introduce the concept of a generalized erasure channel. We find lower and upper bounds for the classical and quantum capacities of the generalized erasure channel as well as characterize its de…
By exploiting a generalization of recent results on environment-assisted channel correction, we show that, whenever a quantum system undergoes a channel realized as an interaction with a probe, the more efficiently the information about the input state can be erased from the probe, the higher is the corresponding entanglement fidelity of the corrected channel, and vice-versa. The present analysis applies also to channels for which perfect quantum erasure is impossible, thus extending the original quantum eraser arrangement, and naturally embodies a general information-disturbance tradeoff.
We calculate the information capacities of a time-correlated amplitude-damping channel, provided the sender and receiver share prior entanglement. Our analytical results show that the noisy channel with zero capacity can transmit information if it has finite memory. The capacities increase as the memory increases attaining maximum value for perfect memory channel.
Topological subsystem color codes (TSCCs) are an important class of topological subsystem codes that allow for syndrome measurement with only 2-body measurements. It is expected that such low complexity measurements can help in fault tolerance. While TSCCs have been studied over depolarizing noise model, their performance over the erasure channel has not been studied as much. Recently, we proposed erasure decoders for TSCCs and reported a threshold of 9.7%. In this paper, we continue our study of TSCCS over the erasure channel. We propose two erasure decoders for topological subsystem color co…
We survey what is known about the information transmitting capacities of quantum channels, and give a proposal for how to calculate some of these capacities using linear programming.
We study the communication capacities of bosonic broadband channels in the presence of different sources of noise. In particular we analyze lossy channels in presence of white noise and thermal bath. In this context, we provide a numerical solution for the entanglement assisted capacity and upper and lower bounds for the classical and quantum capacities.
By using the Holevo-Schumacher-Westmoreland (HSW) theorem and through solving eigenvalues of states out from the quantum noisy channels directly, or with the help of the Bloch sphere representation, or Stokes parametrization representation, we investigate the classical information capacities of some well-known quantum noisy channels.
We calculate the entanglement-assisted and unassisted channel capacities of an exactly solvable spin star system, which models the quantum dephasing channel. The capacities for this non-Markovian model exhibit a strong dependence on the coupling strengths of the bath spins with the system, the bath temperature, and the number of bath spins. For equal couplings and bath frequencies, the channel becomes periodically noiseless.
Entanglement shared between the two ends of a quantum communication channel has been shown to be a useful resource in increasing both the quantum and classical capacities for these channels. The entanglement-assisted capacities were derived assuming an unlimited amount of shared entanglement per channel use. In this paper, bounds are derived on the minimum amount of entanglement required per use of a channel, in order to asymptotically achieve the capacity. This is achieved by introducing a class of entanglement-assisted quantum codes. Codes for classes of qubit channels are shown to achieve t…
We propose a method to detect lower bounds to quantum capacities of a noisy quantum communication channel by means of few measurements. The method is easily implementable and does not require any knowledge about the channel. We test its efficiency by studying its performance for most well known single qubit noisy channels and for the generalised Pauli channel in arbitrary finite dimension.
Classical capacity of unital qubit channels is well known, whereas that of nonunital qubit channels is not. We find lower and upper bounds on classical capacity of nonunital qubit channels by using a recently developed decomposition technique relating nonunital and unital positive qubit maps.
The maximum rates for information transmission through noisy quantum channels has primarily been developed for memoryless channels, where the noise on each transmitted state is treated as independent. Many real world communication channels experience noise which is modelled better by errors that are correlated between separate channel uses. In this paper, upper bounds on the classical information capacities of a class of quantum memory channels are derived. The class of channels consists of indecomposable quantum memory channels, a generalization of classical indecomposable finite-state channe…
For quantum states of two subsystems, entanglement measures are related to capacities of communication tasks -- highly entangled states give higher capacity of transmitting classical as well as quantum information. However, we show that this is no more the case in general: quantum capacities of multi-access channels, motivated by communication in quantum networks, do not have any relation with genuine multiparty entanglement measures. Along with revealing the structural richness of multi-access channel capacities, this gives us a tool to classify multiparty quantum states from the perspective …
We prove that a broad array of capacities of a quantum channel are continuous. That is, two channels that are close with respect to the diamond norm have correspondingly similar communication capabilities. We first show that the classical capacity, quantum capacity, and private classical capacity are continuous, with the variation on arguments epsilon apart bounded by a simple function of epsilon and the channel's output dimension. Our main tool is an upper bound of the variation of output entropies of many copies of two nearby channels given the same initial state; the bound is linear in the …
We review in a unified way a recently proposed method to detect properties of unknown quantum channels and lower bounds to quantum capacities, without resorting to full quantum process tomography. The method is based on the preparation of a fixed bipartite entangled state at the channel input or, equivalently, an ensemble of an overcomplete set of single-system states, along with few local measurements at the channel output.
We study classical capacities of quantum multi-access channels in geometric terms revealing breaking of additivity of Holevo-like capacity. This effect is purely quantum since, as one points out, any classical multi-access channels have their regions additive. The observed non-additivity in quantum version presented here seems to be the first effect of this type with no additional resources like side classical or quantum information (or entanglement) involved. The simplicity of quantum channels involved resembles butterfly effect in case of classical channel with two senders and two receivers.
We introduce potential capacities of quantum channels in an operational way and provide upper bounds for these quantities, which quantify the ultimate limit of usefulness of a channel for a given task in the best possible context. Unfortunately, except for a few isolated cases, potential capacities seem to be as hard to compute as their "plain" analogues. We thus study upper bounds on some potential capacities: For the classical capacity, we give an upper bound in terms of the entanglement of formation. To establish a bound for the quantum and private capacity, we first "lift" the channel to a…
We test a general method to detect lower bounds of the quantum channel capacity for two-qubit correlated channels. We consider in particular correlated dephasing, depolarising and amplitude damping channels. We show that the method is easily implementable, it does not require a priori knowledge about the channels, and it is very efficient, since it does not rely on full quantum process tomography.
Quantum entropy inequalities are studied. Some quantum entropy inequalities are obtained by several methods. For entanglement breaking channel, we show that the entanglement-assisted classical capacity is upper bounded by $\log d$. A relationship between entanglement-assisted and one-shot unassisted capacities is obtained. This relationship shows the entanglement-assisted channel capacity is upper bounded by the sum of $\log d$ and the one-shot unassisted classical capacity.
Reciprocal pairs of quantum channels are defined as completely positive transformations which admit a rigid, distance-preserving, yet not completely-positive transformation that allows to reproduce the outcome of one from the corresponding outcome of the other. From a classical perspective these transmission lines should exhibit the same communication efficiency. This is no longer the case in the quantum setting: explicit asymmetric behaviours are reported studying the classical communication capacities of reciprocal pairs of depolarizing and Weyl-covariant channels.
We study information transmission over a fully correlated amplitude damping channel acting on two qubits. We derive the single-shot classical channel capacity and show that entanglement is needed to achieve the channel best performance. We discuss the degradability properties of the channel and evaluate the quantum capacity for any value of the noise parameter. We finally compute the entanglement-assisted classical channel capacity.
We introduce a general model for a lossy bosonic memory channel and calculate the classical and the quantum capacity, proving that coherent state encoding is optimal. The use of a proper set of collective field variables allows to unravel the memory, showing that the n-fold concatenation of the memory channel is unitarily equivalent to the direct product of n single-mode lossy bosonic channels.
We investigate the capacity of bosonic quantum channels for the transmission of quantum information. Achievable rates are determined from measurable moments of the channel by showing that every channel can asymptotically simulate a Gaussian channel which is characterized by second moments of the initial channel. We calculate the quantum capacity for a class of Gaussian channels, including channels describing optical fibers with photon losses, by proving that Gaussian encodings are optimal. Along the way we provide a complete characterization of degradable Gaussian channels and those arising fr…
We extend a recent method to detect lower bounds to the quantum capacity of quantum communication channels by considering realistic scenarios with general input probe states and arbitrary detection procedures at the output. Realistic certification relies on a new bound for the coherent information of a quantum channel that can be applied with arbitrary bipartite mixed input states and generalized output measurements.
Surface codes are among the best candidates to ensure the fault-tolerance of a quantum computer. In order to avoid the accumulation of errors during a computation, it is crucial to have at our disposal a fast decoding algorithm to quickly identify and correct errors as soon as they occur. We propose a linear-time maximum likelihood decoder for surface codes over the quantum erasure channel. This decoding algorithm for dealing with qubit loss is optimal both in terms of performance and speed.
We determine the capacity of the classical compound quantum wiretapper channel with channel state information at the transmitter. Moreover we derive a lower bound on the capacity of this channel without channel state information and determine the capacity of the classical quantum compound wiretap channel with channel state information at the transmitter.
We study quantum channels that vary on time in a deterministic way, that is, they change in an independent but not identical way from one to another use. We derive coding theorems for the classical entanglement assisted and unassisted capacities. We then specialize the theory to lossy bosonic quantum channels and show the existence of contrasting examples where capacities can or cannot be drawn from the limiting behavior of the lossy parameter.
A strong converse theorem for channel capacity establishes that the error probability in any communication scheme for a given channel necessarily tends to one if the rate of communication exceeds the channel's capacity. Establishing such a theorem for the quantum capacity of degradable channels has been an elusive task, with the strongest progress so far being a so-called "pretty strong converse". In this work, Morgan and Winter proved that the quantum error of any quantum communication scheme for a given degradable channel converges to a value larger than $1/\sqrt{2}$ in the limit of many cha…
We consider classical message transmission under entanglement assistance for compound memoryless and arbitrarily varying quantum channels. In both cases, we prove general coding theorems together with corresponding weak converse bounds. In this way, we obtain single-letter characterizations of the entanglement-assisted classical capacities for both channel models. Moreover, we show that the entanglement-assisted classical capacity does exhibit no strong converse property for some compound quantum channels for the average as well as the maximal error criterion. A strong converse to the entangle…
A generalization of the superactivation of quantum channel capacities to the case of n>2 channels is considered. An explicit example of such superactivation for the 1-shot quantum zero-error capacity is constructed for n=3. Some implications of this example and its reformulation on terms of quantum measurements are described.
A quantum communication channel can be put to many uses: it can transmit classical information, private classical information, or quantum information. It can be used alone, with shared entanglement, or together with other channels. For each of these settings there is a capacity that quantifies a channel's potential for communication. In this short review, I summarize what is known about the various capacities of a quantum channel, including a discussion of the relevant additivity questions. I also give some indication of potentially interesting directions for future research.
Quantum amplifier channels are at the core of several physical processes. Not only do they model the optical process of spontaneous parametric down-conversion, but the transformation corresponding to an amplifier channel also describes the physics of the dynamical Casimir effect in superconducting circuits, the Unruh effect, and Hawking radiation. Here we study the communication capabilities of quantum amplifier channels. Invoking a recently established minimum output-entropy theorem for single-mode phase-insensitive Gaussian channels, we determine capacities of quantum-limited amplifier chann…
Using combinatorial arguments, we determine an upper bound on achievable rates of stabilizer codes used over the quantum erasure channel. This allows us to recover the no-cloning bound on the capacity of the quantum erasure channel, R is below 1-2p, for stabilizer codes: we also derive an improved upper bound of the form : R is below 1-2p-D(p) with a function D(p) that stays positive for 0 < p < 1/2 and for any family of stabilizer codes whose generators have weights bounded from above by a constant - low density stabilizer codes. We obtain an application to percolation theory for a family of …
We consider energy-constrained infinite-dimensional quantum channels from a given system (satisfying a certain condition) to any other systems. We show that dealing with basic capacities of these channels we may assume (accepting arbitrarily small error $\varepsilon$) that all channels have the same finite-dimensional input space -- the subspace corresponding to the $m(\varepsilon)$ minimal eigenvalues of the input Hamiltonian. We also show that for the class of energy-limited channels (mapping energy-bounded states to energy-bounded states) the above result is valid with substantially smaller…
We focus on classical and quantum list decoding. The capacity of list decoding was obtained by Nishimura in the case when the number of list does not increase exponentially. However, the capacity of the exponential-list case is open even in the classical case while its converse part was obtained by Nishimura. We derive the channel capacities in the classical and quantum case with an exponentially increasing list. The converse part of the quantum case is obtained by modifying Nagaoka's simple proof for strong converse theorem for channel capacity. The direct part is derived by a quite simple ar…
Two new classes of quantum channels, which we call more capable and less noisy, are introduced. The more capable class consists of channels such that the quantum capacities of the complementary channels to the environments are zero. The less noisy class consists of channels such that the private capacities of the complementary channels to the environment are zero. For the more capable class, it is clarified that the private capacity and quantum capacity coincide. For the less noisy class, it is clarified that the private capacity and quantum capacity can be single letter characterized.
We propose examples of low dimensional quantum channels demonstrating different forms of superactivation of one-shot zero-error capacities, in particular, the extreme superactivation (this complements the recent result of T.S.Cubitt and G.Smith). We also describe classes of quantum channels whose zero-error classical and quantum capacities cannot be superactivated. We consider implications of the superactivation of one-shot zero-error capacities to analysis of reversibility of a tensor-product channel with respect to families of pure states. Our approach based on the notions of complementary c…
We study universal quantum codes for entanglement-assisted quantum communication over compound quantum channels. In this setting, sender and receiver do not know the specific channel that will be used for communication, but only know the set that the channel is selected from. We investigate different variations of the problem: uninformed users, informed receiver, informed sender, and feedback assistance. We derive single-letter formulas for all corresponding channel capacities. Our proofs are based on one-shot decoupling bounds and properties of smooth entropies.
We define the direct and reverse secret-key capacities of a memoryless quantum channel as the optimal rates that entanglement-based quantum key distribution protocols can reach by using a single forward classical communication (direct reconciliation) or a single feedback classical communication (reverse reconciliation). In particular, the reverse secret-key capacity can be positive for antidegradable channels, where no forward strategy is known to be secure. This property is explicitly shown in the continuous variable framework by considering arbitrary one-mode Gaussian channels.
