Using random Gaussian vectors and an information-uncertainty relation, we give a proof that the coherent information is an achievable rate for entanglement transmission through a noisy quantum channel. The codes are random subspaces selected according to the Haar measure, but distorted as a function of the sender's input density operator. Using large deviations techniques, we show that classical data transmitted in either of two Fourier-conjugate bases for the coding subspace can be decoded with low probability of error. A recently discovered information-uncertainty relation then implies that the quantum mutual information for entanglement encoded into the subspace and transmitted through the channel will be high. The monogamy of quantum correlations finally implies that the environment of the channel cannot be significantly coupled to the entanglement which, concluding, ensures the existence of a decoding by the receiver.
Using random Gaussian vectors and an information-uncertainty relation, we give a proof that the coherent information is an achievable rate for entanglement transmission through a noisy quantum channel. The codes are random subspaces selected according to the Haar measure, but distorted as a function of the sender's input density operator. Using large deviations techniques, we show that classical data transmitted in either of two Fourier-conjugate bases for the coding subspace can be decoded with low probability of error. A recently discovered information-uncertainty relation then implies that the quantum mutual information for entanglement encoded into the subspace and transmitted through the channel will be high. The monogamy of quantum correlations finally implies that the environment of the channel cannot be significantly coupled to the entanglement which, concluding, ensures the existence of a decoding by the receiver.
We give a short proof that the coherent information is an achievable rate for the transmission of quantum information through a noisy quantum channel. Our method is to produce random codes by performing a unitarily covariant projective measurement on a typical subspace of a tensor power state. We show that, provided the rank of each measurement operator is sufficiently small, the transmitted data will, with high probability, be decoupled from the channel environment. We also show that our construction leads to random codes whose average input is close to a product state and outline a modification yielding unitarily invariant ensembles of maximally entangled codes.
We construct explicit codes for transmitting classical or quantum information over the thermal-noise Bosonic Gaussian channel. For classical information transmission, our codes achieve the classical capacity, recently characterized by Giovannetti et al. [Nature Photonics 8, 796 (2014)], while in the quantum case they transmit at a rate given by the Gaussian coherent information. The construction is based on concatenating polar codes with a suitable discretization of the channel input into a finite constellation of coherent states. The codes have efficient encoders, but the complexity of decoding is unknown. Our results are based on a quantum version of Wu and Verd\'u's technique of determining the achievable rates of finite constellations for the classical Gaussian-noise channel [Allerton 2010, pp. 620].
Suppose that m senders want to transmit classical information to n receivers with zero probability of error using a noisy multipartite communication channel. The senders are allowed to exchange classical, but not quantum, messages among themselves, and the same holds for the receivers. If the channel is classical, a single use can transmit information if and only if multiple uses can. In sharp contrast, we exhibit, for each m and n with m>1 or n>1, a quantum channel of which a single use is not able to transmit information yet two uses can. This latter property requires and is enabled by quantum entanglement.
The possibility is explicitly shown that the average probability of error is reduced and the transmission rate of information is enhanced by means of entangled input states to a communication channel. When two senders transmit some information to a receiver, they encode the information in the quadrature amplitudes of both modes of a two-mode squeezed vacuum state. The receiver performs a simultaneous measurement of the two quadrature amplitudes of the received two-mode state. In this case, if the average value of the photon number supplied to the communication system is not so small, the average probability of error becomes smaller than that of the communication system without entanglement. Furthermore, the transmission rate of information becomes larger than the quantum capacity of classical information of the communication system without entanglement.
Transmitting quantum information across quantum channels is an important task. However quantum information is delicate, and is easily corrupted. We address the task of protecting quantum information from an information theoretic perspective -- we encode some message qudits into a quantum code, send the encoded quantum information across the noisy quantum channel, then recover the message qudits by decoding. In this dissertation, we discuss the coding problem from several perspectives.} \n \nThe noisy quantum channel is one of the central aspects of the quantum coding problem, and hence quantifying the noisy quantum channel from the physical model is an important problem. \nWe work with an explicit physical model -- a pair of initially decoupled quantum harmonic oscillators interacting with a spring-like coupling, where the bath oscillator is initially in a thermal-like state. In particular, we treat the completely positive and trace preserving map on the system as a quantum channel, and study the truncation of the channel by truncating its Kraus set. We thereby derive the matrix elements of the Choi-Jamiolkowski operator of the corresponding truncated channel, which are truncated transition amplitudes. Finally, we give a computable approximation for these truncated transition amplitudes with explicit error bounds, and perform a case study of the oscillators in the off-resonant and weakly-coupled regime numerically. \n \nIn the context of truncated noisy channels, we revisit the notion of approximate error correction of finite dimension codes. We derive a computationally simple lower bound on the worst case entanglement fidelity of a quantum code, when the truncated recovery map of Leung et. al. is rescaled. As an application, we apply our bound to construct a family of multi-error correcting amplitude damping codes that are permutation-invariant. This demonstrates an explicit example where the specific structure of the noisy channel allows code design out of the stabilizer formalism via purely algebraic means. \n \nWe study lower bounds on the quantum capacity of adversarial channels, where we restrict the selection of quantum codes to the set of concatenated quantum codes. \nThe adversarial channel is a quantum channel where an adversary corrupts a fixed fraction of qudits sent across a quantum channel in the most malicious way possible. The best known rates of communicating over adversarial channels are given by the quantum Gilbert-Varshamov (GV) bound, that is known to be attainable with random quantum codes. We generalize the classical result of Thommesen to the quantum case, thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum GV bound. The outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random independently chosen stabilizer codes, where the rates of the inner and outer codes lie in a specified feasible region. \n \nWe next study upper bounds on the quantum capacity of some low dimension quantum channels. \nThe quantum capacity of a quantum channel is the maximum rate at which quantum information can be transmitted reliably across it, given arbitrarily many uses of it. While it is known that random quantum codes can be used to attain the quantum capacity, the quantum capacity of many classes of channels is undetermined, even for channels of low input and output dimension. For example, depolarizing channels are \n important quantum channels, but do not have tight numerical bounds. \nWe obtain upper bounds on the quantum capacity of some unital and non-unital channels \n -- two-qubit Pauli channels, two-qubit depolarizing channels, two-qubit locally symmetric channels, \n shifted qubit depolarizing channels, and shifted two-qubit Pauli channels -- \n using the coherent information of some degradable channels. We use the notion \n of twirling quantum channels, and Smith and Smolin's method of constructing \n degradable extensions of quantum channels extensively. The degradable channels we \n introduce, study and use are two-qubit amplitude damping channels. Exploiting the \n notion of covariant quantum channels, we give sufficient conditions for the quantum \n capacity of a degradable channel to be the optimal value of a concave program with \n linear constraints, and show that our two-qubit degradable amplitude damping channels have this property.
We construct an explicit entanglement distillation scheme which achieves the coherent information when used to send quantum information over a noisy quantum channel. For Pauli and erasure channels we present efficient encoding and decoding algorithms based on polar codes. Unlike previous constructions, this scheme does not require the sender and receiver to share noiseless entanglement before the protocol begins. It is possible, but still unproven, that the scheme even achieves a rate beyond the coherent information, due to degeneracies of certain error correcting codes. Finally we discuss how the scheme can be used for secret key distillation and private channel coding.