We present a general theory of entanglement-assisted quantum convolutional coding. The codes have a convolutional or memory structure, they assume that the sender and receiver share noiseless entanglement prior to quantum communication, and they are
not restricted to possess the Calderbank-Shor-Steane structure as in previous work. We provide two significant advances for quantum convolutional coding theory. We first show how to expand a given set of quantum convolutional generators. This expansion step acts as a preprocessor for a polynomial symplectic Gram-Schmidt orthogonalization procedure that simplifies the commutation relations of the expanded generators to be the same as those of entangled Bell states (ebits) and ancilla qubits. The above two steps produce a set of generators with equivalent error-correcting properties to those of the original generators. We then demonstrate how to perform online encoding and decoding for a stream of information qubits, halves of ebits, and ancilla qubits. The upshot of our theory is that the quantum code designer can engineer quantum convolutional codes with desirable error-correcting properties without having to worry about the commutation relations of these generators.
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 ra
te 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 single-letter characterization for at least three channels: the completely depolarizing channel, the quantum erasure channel, and the qubit dephasing channel.
A modified quantum teleportation protocol broadens the scope of the classical forbidden-interval theorems for stochastic resonance. The fidelity measures performance of quantum communication. The sender encodes the two classical bits for quantum tele
portation as weak bipolar subthreshold signals and sends them over a noisy classical channel. Two forbidden-interval theorems provide a necessary and sufficient condition for the occurrence of the nonmonotone stochastic resonance effect in the fidelity of quantum teleportation. The condition is that the noise mean must fall outside a forbidden interval related to the detection threshold and signal value. An optimal amount of classical noise benefits quantum communication when the sender transmits weak signals, the receiver detects with a high threshold, and the noise mean lies outside the forbidden interval. Theorems and simulations demonstrate that both finite-variance and infinite-variance noise benefit the fidelity of quantum teleportation.
We provide several formulas that determine the optimal number of entangled bits (ebits) that a general entanglement-assisted quantum code requires. Our first theorem gives a formula that applies to an arbitrary entanglement-assisted block code. Corol
laries of this theorem give formulas that apply to a code imported from two classical binary block codes, to a code imported from a classical quaternary block code, and to a continuous-variable entanglement-assisted quantum block code. Finally, we conjecture two formulas that apply to entanglement-assisted quantum convolutional codes.
We propose a linear-optical implementation of a hyperentanglement-assisted quantum error-correcting code. The code is hyperentanglement-assisted because the shared entanglement resource is a photonic state hyperentangled in polarization and orbital a
ngular momentum. It is possible to encode, decode, and diagnose channel errors using linear-optical techniques. The code corrects for polarization flip errors and is thus suitable only for a proof-of-principle experiment. The encoding and decoding circuits use a Knill-Laflamme-Milburn-like scheme for transforming polarization and orbital angular momentum photonic qubits. A numerical optimization algorithm finds a unit-fidelity encoding circuit that requires only three ancilla modes and has success probability equal to 0.0097.
We show that qubits traveling along closed timelike curves are a resource that a party can exploit to distinguish perfectly any set of quantum states. As a result, an adversary with access to closed timelike curves can break any prepare-and-measure q
uantum key distribution protocol. Our result also implies that a party with access to closed timelike curves can violate the Holevo bound.
We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer ge
nerators, encoding circuit, codewords, logical Pauli operators, and logical CNOT operator for this code. We also show how to convert this code into a non-trivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender and the receiver and corrects an arbitrary single-qubit error. The code we obtain is globally equivalent to the Steane seven-qubit code and thus corrects an arbitrary error on the receivers half of the ebit as well. We prove that this code is the smallest code with a CSS structure that uses only one ebit and corrects an arbitrary single-qubit error on the senders side. We discuss the advantages and disadvantages for each of the two codes.
We incorporate active and passive quantum error-correcting techniques to protect a set of optical information modes of a continuous-variable quantum information system. Our method uses ancilla modes, entangled modes, and gauge modes (modes in a mixed
state) to help correct errors on a set of information modes. A linear-optical encoding circuit consisting of offline squeezers, passive optical devices, feedforward control, conditional modulation, and homodyne measurements performs the encoding. The result is that we extend the entanglement-assisted operator stabilizer formalism for discrete variables to continuous-variable quantum information processing.
Quantum error-correcting codes will be the ultimate enabler of a future quantum computing or quantum communication device. This theory forms the cornerstone of practical quantum information theory. We provide several contributions to the theory of qu
antum error correction--mainly to the theory of entanglement-assisted quantum error correction where the sender and receiver share entanglement in the form of entangled bits (ebits) before quantum communication begins. Our first contribution is an algorithm for encoding and decoding an entanglement-assisted quantum block code. We then give several formulas that determine the optimal number of ebits for an entanglement-assisted code. The major contribution of this thesis is the development of the theory of entanglement-assisted quantum convolutional coding. A convolutional code is one that has memory and acts on an incoming stream of qubits. We explicitly show how to encode and decode a stream of information qubits with the help of ancilla qubits and ebits. Our entanglement-assisted convolutional codes include those with a Calderbank-Shor-Steane structure and those with a more general structure. We then formulate convolutional protocols that correct errors in noisy entanglement. Our final contribution is a unification of the theory of quantum error correction--these unified convolutional codes exploit all of the known resources for quantum redundancy.
