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 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.
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 work out a theory of approximate quantum error correction that allows us to derive a general lower bound for the entanglement fidelity of a quantum code. The lower bound is given in terms of Kraus operators of the quantum noise. This result is then used to analyze the average error correcting performance of codes that are randomly drawn from unitarily invariant code ensembles. Our results confirm that random codes of sufficiently large block size are highly suitable for quantum error correction. Moreover, employing a lemma of Bennett, Shor, Smolin, and Thapliyal, we prove that random coding attains information rates of the regularized coherent information.
In analogy with its classical counterpart, a noisy quantum channel is characterized by a loss, a quantity that depends on the channel input and the quantum operation performed by the channel. The loss reflects the transmission quality: if the loss is zero, quantum information can be perfectly transmitted at a rate measured by the quantum source entropy. By using block coding based on sequences of n entangled symbols, the average loss (defined as the overall loss of the joint n-symbol channel divided by n, when $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty})$ can be made lower than the loss for a single use of the channel. In this context, we examine several upper bounds on the rate at which quantum information can be transmitted reliably via a noisy channel, that is, with an asymptotically vanishing average loss while the one-symbol loss of the channel is nonzero. These bounds on the channel capacity rely on the entropic Singleton bound on quantum error-correcting codes [Phys. Rev. A 56, 1721 (1997)]. Finally, we analyze the Singleton bounds when the noisy quantum channel is supplemented with a classical auxiliary channel.
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.
The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally “quantum” theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum communication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem (Section 3.5.4) provides the intuition behind the quantum capacity theorem. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information—otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H ( B ) -H ( E )—a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain. We proved achievability of the coherent information for quantum data transmission in Corollary 22.2.1, but the roundabout path that we followed to prove achievability there perhaps does not give much insight into the structure of a quantum code that achieves the coherent information. Our approach in this chapter is different and should shed more light on this structure. Specifically, we show how to make coherent versions of the private classical codes from the previous chapter.
Quantum information decoupling is a pillar of quantum information theory, underlying quantum communication, error correction, and information recovery. However, its existing formulations rely on random unitary operations tied to finite-dimensional assumptions on quantum systems. Here we establish a decoupling framework for arbitrary separable, possibly infinite-dimensional systems. For finite-dimensional inputs and arbitrary separable reference/output systems, we derive error bounds on one-shot decoupling for completely positive maps in terms of sandwiched Rényi conditional entropies. For partial-trace decoupling with finite-dimensional references, the error exponent is optimal up to the known critical rate. To handle infinite-dimensional inputs, we assume finite entropy of the manipulated system. We construct finite-rank projections on independent and identically distributed (IID) states to restrict randomization to a projected finite-dimensional subspace, while ensuring high success probability by including auxiliary components in the discarded subsystem at asymptotically negligible cost. This leads to an infinite-dimensional IID partial-trace decoupling protocol achieving optimal asymptotic dimension rates. As an application, we construct an infinite-dimensional quantum-state-merging protocol, a mother protocol of quantum information theory. Under finite entropy of Alice's marginal, it achieves the same quantum-communication and total-cost rates as in finite dimensions, governed by mutual information and conditional entropy. These results show that the operational interpretation of entropic quantities through these achievable rates constitutes a universal principle beyond finite dimensions. More broadly, our framework provides foundational tools for exploring quantum information regardless of whether we model the physical world using finite- or infinite-dimensional spaces.