Communication over a noisy quantum channel introduces errors in the transmission that must be corrected. A fundamental bound on quantum error correction is the quantum capacity, which quantifies the amount of quantum data that can be protected. We show theoretically that two quantum channels, each with a transmission capacity of zero, can have a nonzero capacity when used together.This unveils a rich structure in the theory of quantum communications, implying that the quantum capacity does not uniquely specify a channel's ability for transmitting quantum information. 1 arXiv:0807.4935v2 [quant-ph]

We propose examples of a hybrid quantum-classical simulation where a classical computer assisted by a small quantum processor can efficiently simulate a larger quantum system. First, we consider sparse quantum circuits such that each qubit participates in Oð1Þ two-qubit gates. It is shown that any sparse circuit on n þ k qubits can be simulated by sparse circuits on n qubits and a classical processing that takes time 2 OðkÞ polyðnÞ. Second, we study Pauli-based computation (PBC), where allowed operations are nondestructive eigenvalue measurements of n-qubit Pauli operators. The computation begins by initializing each qubit in the so-called magic state. This model is known to be equivalent to the universal quantum computer. We show that any PBC on n þ k qubits can be simulated by PBCs on n qubits and a classical processing that takes time 2OðkÞ polyðnÞ. Finally, we propose a purely classical algorithm that can simulate a PBC on n qubits in a time 2 αn polyðnÞ, where α ≈ 0.94. This improves upon the brute-force simulation method, which takes time 2 n polyðnÞ. Our algorithm exploits the fact that n-fold tensor products of magic states admit a low-rank decomposition into n-qubit stabilizer states.

Starting from a state of low quantum entanglement, local unitary time evolution increases the entanglement of a quantum many-body system. In contrast, local projective measurements disentangle degrees of freedom and decrease entanglement. We study the interplay of these competing tendencies by considering time evolution combining both unitary and projective dynamics. We begin by constructing a toy model of Bell pair dynamics which demonstrates that measurements can keep a system in a state of low (i.e. area law) entanglement, in contrast with the volume law entanglement produced by generic pure unitary time evolution. While the simplest Bell pair model has area law entanglement for any measurement rate, as seen in certain non-interacting systems, we show that more generic models of entanglement can feature an area-to-volume law transition at a critical value of the measurement rate, in agreement with recent numerical investigations. As a concrete example of these ideas, we analytically investigate Clifford evolution in qubit systems which can exhibit an entanglement transition. We are able to identify stabilizer size distributions characterizing the area law, volume law and critical 'fixed points.' We also discuss Floquet random circuits, where the answers depend on the order of limits -one order of limits yields area law entanglement for any non-zero measurement rate, whereas a different order of limits allows for an area law -volume law transition. Finally, we provide a rigorous argument that a system subjected to projective measurements can only exhibit a volume law entanglement entropy if it also features a subleading correction term, which provides a universal signature of projective dynamics in the high-entanglement phase.Note: The results presented here supersede those of all previous versions of this manuscript, which contained some erroneous claims. CONTENTS

We present a unifying approach to quantum error correcting code design that encompasses additive (stabilizer) codes, as well as all known examples of nonadditive codes with good parameters. We use this framework to generate new codes with superior parameters to any previously known. In particular, we find ((10,18,3)) and ((10,20,3)) codes. We also show how to construct encoding circuits for all codes within our framework.Comment: 5 pages, 1 eps figure, ((11,48,3)) code removed, encoding circuits added, typos corrected in codewords and elsewher

We provide an efficient method for computing the maximum-likelihood mixed quantum state (with density matrix ρ) given a set of measurement outcomes in a complete orthonormal operator basis subject to Gaussian noise. Our method works by first changing basis yielding a candidate density matrix μ which may have nonphysical (negative) eigenvalues, and then finding the nearest physical state under the 2-norm. Our algorithm takes at worst O(d(4)) for the basis change plus O(d(3)) for finding ρ where d is the dimension of the quantum state. In the special case where the measurement basis is strings of Pauli operators, the basis change takes only O(d(3)) as well. The workhorse of the algorithm is a new linear-time method for finding the closest probability distribution (in Euclidean distance) to a set of real numbers summing to one.

An optical network of superconducting quantum bits (qubits) is an appealing platform for quantum communication and distributed quantum computing, but developing a quantum-compatible link between the microwave and optical domains remains an outstanding challenge. Operating at T < 100 mK temperatures, as required for quantum electrical circuits, we demonstrate a mechanically-mediated microwave-optical converter with 47% conversion efficiency, and use a classical feedforward protocol to reduce added noise to 38 photons. The feedforward protocol harnesses our discovery that noise emitted from the two converter output ports is strongly correlated because both outputs record thermal motion of the same mechanical mode. We also discuss a quantum feedforward protocol that, given high system efficiencies, would allow quantum information to be transferred even when thermal phonons enter the mechanical element faster than the electro-optic conversion rate. arXiv:1712.06535v2 [quant-ph]

Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable and present a complete description of anti-degradable unital qubit channels with a new proof.For higher output dimensions we explore the relationship between the output and environment dimensions (d B and d E respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with a environment that is "small" in the sense d E ≤ d B . Such channels include all those with qubit or qutrit output, those that map some pure state to an output with full rank, and all those which can be represented using simultaneously diagonal Kraus operators, even in a non-orthogonal basis. Perhaps surprisingly, we also present examples of degradable channels with "large" environments, in the sense that the minimal dimensionThese examples can also be used to give a negative answer to the question of whether additivity of the coherent information is helpful for establishing additivity for the Holevo capacity of a pair of channels.In the case of channels with diagonal Kraus operators, we describe the subclass which are complements of entanglement breaking channels. We also obtain a number of results for channels in the convex hull of conjugations with generalized Pauli matrices. However, a number of open questions remain about these channels and the more general case of random unitary channels.

A striking feature of quantum error correcting codes is that they can sometimes be used to correct more errors than they can uniquely identify. Such degenerate codes have long been known, but have remained poorly understood. We provide a heuristic for designing degenerate quantum codes for high noise rates, which is applied to generate codes that can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code. The gap between nondegenerate and degenerate code performance is quite large, in contrast to the tiny magnitude of the only previous demonstration of this effect. We also identify a channel for which none of our codes outperform the best nondegenerate code and show that it is nevertheless quite unlike any channel for which nondegenerate codes are known to be optimal. DOI: 10.1103/PhysRevLett.98.030501 PACS numbers: 03.67.Hk, 05.40.Ca It was Shannon [1] who discovered, by a random coding argument, the beautiful fact that the capacity of a noisy channel N is equal to the maximal mutual information between an input variable, X, and its image under the action of the channel:It is remarkable that this maximization is over a single input to the channel; it does not require consideration of inputs correlated over many channel uses. One would hope that, as in the classical case, there is some measure of quantum correlations that can be maximized over inputs to a quantum channel to give the capacity. Unfortunately, this appears not to be the case. The natural generalization of Eq. (1) is to replace the random variable X with a quantum state and the mutual information with the coherent information I c givingwhereHere j AB i is a purification of . Its use reflects the fact that unlike in the classical case, there can be no remaining copy of the channel input with which to compare correlations-instead we must consider the quantum state as a whole. The coherent information is defined by I c AB S B ÿ S AB with S ÿTr log. While we can achieve Q 1 using a random code on the typical subspace of the maximizing , it has long been known that this rate is not always optimal [2,3]. They exhibit codes with rates larger than Q 1 for very noisy depolarizing channels which have Q 1 small or even zero.The correct quantum capacity formula is not Q 1 , but instead is given by [4 -6]where taking the limit n ! 1 means that we must consider the behavior of the channel on inputs entangled across many uses. This multiletter formula is an expression of our ignorance about the structure of capacity achieving codes for a quantum channel.The difference between these single-and multiletter formulas is intimately related to the existence of degenerate quantum codes. Strictly speaking, degeneracy is not a property of a quantum code alone, but a property of a code together with a family of errors it is designed to correct. More formally, one usually says that a code C degenerately corrects a set of errors E if in addition to correcting E, there are multiple errors in E that are mapped to the same ...