We show how entanglement shared between encoder and decoder can simplify the theory of quantum error correction. The entanglement-assisted quantum codes we describe do not require the dual-containing constraint necessary for standard quantum error correcting codes, thus allowing us to "quantize" all of classical linear coding theory. In particular, efficient modern classical codes that attain the Shannon capacity can be made into entanglement-assisted quantum codes attaining the hashing bound (closely related to the quantum capacity). For systems without large amounts of shared entanglement, these codes can also be used as catalytic codes, in which a small amount of initial entanglement enables quantum communication.

Quantum coherence and quantum entanglement represent two fundamental features of non-classical systems that can each be characterized within an operational resource theory. In this paper, we unify the resource theories of entanglement and coherence by studying their combined behavior in the operational setting of local incoherent operations and classical communication (LIOCC). Specifically we analyze the coherence and entanglement trade-offs in the tasks of state formation and resource distillation. For pure states we identify the minimum coherence-entanglement resources needed to generate a given state, and we introduce a new LIOCC monotone that completely characterizes a state's optimal rate of bipartite coherence distillation. This result allows us to precisely quantify the difference in operational powers between global incoherent operations, LIOCC, and local incoherent operations without classical communication. Finally, a bipartite mixed state is shown to have distillable entanglement if and only if entanglement can be distilled by LIOCC, and we strengthen the wellknown Horodecki criterion for distillability.The ability for quantum systems to exist in "superposition states" reveals the wave-like nature of matter and represents a strong departure from classical physics. Systems in such superposition states are often said to possess quantum coherence. There has currently been much interest in constructing a resource theory of quantum coherence [1][2][3][4][5][6][7][8][9][10][11], in part because of recent experimental and numerical findings that suggest quantum coherence alone can enhance or impact physical dynamics in biology [12][13][14][15], transport theory [2,16,17], and thermodynamics [18,19].In a standard resource-theoretic treatment of quantum coherence, the free (or "incoherent") states are those that are diagonal in some fixed reference (or "incoherent") basis Different classes of allowed (or "incoherent") operations have been proposed in the literature [1,3,[9][10][11] (see also [20,21] for comparative studies of these approaches), however an essential requirement is that the incoherent operations act invariantly on the set of diagonal density matrices. Incoherent operations can then be seen as one of the most basic generalizations of classical operations (i.e. stochastic maps) since their action on diagonal states can always be simulated by classical processing. Note also that most experimental setups will have a natural basis to work in, and arbitrary unitary time evolutions might be physically difficult to implement. In these settings, there are practical advantages to identifying "diagonal preserving" operations as being "free" relative to coherentgenerating ones.One does not need to look far to find an important connection between incoherent operations and quantum entanglement, the latter being one of the most important resources in quantum information processing [22]. Consider the task of entanglement generation. This procedure is usually modeled by bringing together two or more quantum syst...

Bender et al. [Phys. Rev. Lett. 80, 5243 (1998)] have developed PT-symmetric quantum theory as an extension of quantum theory to non-Hermitian Hamiltonians. We show that when this model has a local PT symmetry acting on composite systems, it violates the nonsignaling principle of relativity. Since the case of global PT symmetry is known to reduce to standard quantum mechanics A. Mostafazadeh [J. Math. Phys. 43, 205 (2001)], this shows that the PT-symmetric theory is either a trivial extension or likely false as a fundamental theory.

We find a regularized formula for the entanglement-assisted (EA) capacity region for quantum multiple access channels (QMAC). We illustrate the capacity region calculation with the example of the collective phase-flip channel which admits a single-letter characterization. On the way, we provide a first-principles proof of the EA coding theorem based on a packing argument. We observe that the Holevo-Schumacher-Westmoreland theorem may be obtained from a modification of our EA protocol. We remark on the existence of a family hierarchy of protocols for multiparty scenarios with a single receiver, in analogy to the two-party case. In this way, we relate several previous results regarding QMACs.Comment: Published version. 13 pages, 3 figure

We consider the problem of transmitting classical and quantum information reliably over an entanglement-assisted quantum channel. Our main result is a capacity theorem that gives a three-dimensional achievable rate region. Points in the region are rate triples, consisting of the classical communication rate, the quantum communication rate, and the entanglement consumption rate of a particular coding scheme. The crucial protocol in achieving the boundary points of the capacity region is a protocol that we name the classically-enhanced father protocol. The classically-enhanced father protocol is more general than other protocols in the family tree of quantum Shannon theoretic protocols, in the sense that several previously known quantum protocols are now child protocols of it. The classically-enhanced father protocol also shows an improvement over a time-sharing strategy for the case of a qubit dephasing channel-this result justifies the need for simultaneous coding of classical and quantum information over an entanglement-assisted quantum channel. Our capacity theorem is of a multi-letter nature (requiring a limit over many uses of the channel), but it reduces to a singleletter characterization for at least three channels: the completely depolarizing channel, the quantum erasure channel, and the qubit dephasing channel.Index Terms-quantum Shannon theory, entanglement-assisted quantum channel, entanglement-assisted classical-quantum coding, classically-enhanced father protocol

An unexpected breakdown in the existing theory of quantum serial turbo coding is that a quantum convolutional encoder cannot simultaneously be recursive and non-catastrophic. These properties are essential for quantum turbo code families to have a minimum distance growing with blocklength and for their iterative decoding algorithm to converge, respectively. Here, we show that the entanglement-assisted paradigm simplifies the theory of quantum turbo codes, in the sense that an entanglement-assisted quantum (EAQ) convolutional encoder can possess both of the aforementioned desirable properties. We give several examples of EAQ convolutional encoders that are both recursive and non-catastrophic and detail their relevant parameters. We then modify the quantum turbo decoding algorithm of Poulin et al., in order to have the constituent decoders pass along only "extrinsic information" to each other rather than a posteriori probabilities as in the decoder of Poulin et al., and this leads to a significant improvement in the performance of unassisted quantum turbo codes. Other simulation results indicate that entanglement-assisted turbo codes can operate reliably in a noise regime 4.73 dB beyond that of standard quantum turbo codes, when used on a memoryless depolarizing channel. Furthermore, several of our quantum turbo codes are within 1 dB or less of their hashing limits, so that the performance of quantum turbo codes is now on par with that of classical turbo codes. Finally, we prove that entanglement is the resource that enables a convolutional encoder to be both non-catastrophic and recursive because an encoder acting on only information qubits, classical bits, gauge qubits, and ancilla qubits cannot simultaneously satisfy them. Index Terms quantum communication, entanglement-assisted quantum turbo code, entanglement-assisted quantum error correction, recursive, non-catastrophic, entanglement-assisted quantum convolutional code I. INTRODUCTION Classical turbo codes represent one of the great successes of the modern coding era [1], [2], [3],[4]. These near Shannon-limit codes have efficient encodings, they offer astounding performance on memoryless channels, and their iterative decoding algorithm quickly converges to an accurate error estimate. They are "probabilistic codes," meaning that they possess sufficient structure to ensure efficient encoding and decoding, yet they have enough randomness to allow for analysis of their performance with the probabilistic method [2], [4], [5].The theory of quantum turbo codes is much younger than its classical counterpart [6], and we still stand to learn more regarding these codes' performance and structure. Poulin et al. set this theory on a firm foundation [6] in an attempt to construct explicit quantum codes that come close to achieving the quantum capacity of a quantum channel [7], [8], [9], [10]. The structure of a quantum serial turbo code is similar to its classical counterpart-one quantum convolutional encoder [11], [12] followed by a quantum interleaver and another quantu...

No abstract

Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of pre-existing entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard QECCs which can only be constructed from dual-containing codes. Operator quantum error-correcting codes (OQECCs) allow certain errors to be corrected (or prevented) passively, reducing the complexity of the correction procedure. We combine these two extensions of standard quantum error correction into a unified entanglement-assisted quantum error correction formalism. This new scheme, which we call entanglement-assisted operator quantum error correction (EAOQEC), is the most general and powerful quantum error-correcting technique known, retaining the advantages of both entanglementassistance and passive correction. We present the formalism, show the considerable freedom in constructing EAOQECCs from classical codes, and demonstrate the construction with examples.