In the current thesis we first talk about the six-qubit quantum error-correcting code and show its connections to entanglement-assisted error-correcting coding theory and then to subsystem codes. This code bridges the gap between the five-qubit (perfect) and Steane codes. We discuss two methods to encode one qubit into six physical qubits. Each of the two examples corrects an arbitrary single-qubit error. The first example is a degenerate six-qubit quantum error-correcting code. We 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 (Alice) and the receiver (Bob) and corrects an arbitrary single-qubit error. In the second half of this thesis we explore the yet uncharted and relatively undiscovered area of quantum steganography. Steganography is the process of hiding secret information by embedding it in an innocent message. We present protocols for hiding quantum information in a codeword of a quantum error-correcting code passing through a channel. Using either a shared classical secret key or shared entanglement Alice disguises her information as errors in the channel. Bob can retrieve the hidden information, but an eavesdropper (Eve) with the power to monitor the channel, but without the secret key, cannot distinguish the message from channel noise. We analyze how difficult it is for Eve to detect the presence of secret messages, and estimate rates of steganographic communication and secret key consumption for certain protocols.
In the current thesis we first talk about the six-qubit quantum error-correcting code and show its connections to entanglement-assisted error-correcting coding theory and then to subsystem codes. This code bridges the gap between the five-qubit (perfect) and Steane codes. We discuss two methods to encode one qubit into six physical qubits. Each of the two examples corrects an arbitrary single-qubit error. The first example is a degenerate six-qubit quantum error-correcting code. We 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 (Alice) and the receiver (Bob) and corrects an arbitrary single-qubit error. In the second half of this thesis we explore the yet uncharted and relatively undiscovered area of quantum steganography. Steganography is the process of hiding secret information by embedding it in an innocent message. We present protocols for hiding quantum information in a codeword of a quantum error-correcting code passing through a channel. Using either a shared classical secret key or shared entanglement Alice disguises her information as errors in the channel. Bob can retrieve the hidden information, but an eavesdropper (Eve) with the power to monitor the channel, but without the secret key, cannot distinguish the message from channel noise. We analyze how difficult it is for Eve to detect the presence of secret messages, and estimate rates of steganographic communication and secret key consumption for certain protocols.
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 generators, encoding circuit, codewords, logical Pauli operators, and logical controlled NOT operator for this code. We also show how to convert this code into a nontrivial 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 nondegenerate 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 receiver's 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 sender's side. We discuss the advantages and disadvantages for each of the two codes.
Steganography is the technique of hiding secret information by embedding it in a seemingly ``innocent'' message. We present protocols for hiding quantum information by disguising it as noise in a codeword of a quantum error-correcting code. The sender (Alice) swaps quantum information into the codeword and applies a random choice of unitary operation, drawing on a secret random key she shares with the receiver (Bob). Using the key, Bob can retrieve the information, but an eavesdropper (Eve) with the power to monitor the channel, but without the secret key, cannot distinguish the message from channel noise. We consider two types of protocols: one in which the hidden quantum information is stored locally in the codeword, and another in which it is embedded in the space of error syndromes. We analyze how difficult it is for Eve to detect the presence of secret messages, and estimate rates of steganographic communication and secret key consumption for specific protocols and examples of error channels. We consider both the case where there is no actual noise in the channel (so that all errors in the codeword result from the deliberate actions of Alice), and the case where the channel is noisy and not controlled by Alice and Bob.
We present and analyze a protocol for quantum steganography where the sender (Alice) encodes her steganographic information into the error syndromes of the perfect (five-qubit) quantum error-correcting code, and sends it to the receiver (Bob) over a depolarizing channel. Alice and Bob share a classical secret key, and hide quantum information in such a way that to an eavesdropper (Eve) without access to the secret key, the quantum message looks like an innocent codeword with a typical sequence of quantum errors. We calculate the average rate of key consumption, and show how the protocol improves in performance as information is spread over multiple codeword blocks. Alice and Bob utilize different encodings to optimize the average number of steganographic bits that they can send to each other while matching the error statistics of the depolarizing channel.
Quantum steganography is the study of hiding secret quantum information by encoding it into what an eavesdropper would perceive as an innocent-looking message. Here we study an explicit steganographic encoding for Alice to hide her secret message in the syndromes of an error-correcting code, so that the encoding simulates a given noisy quantum channel. We calculate achievable rates of steganographic communication over noiseless quantum channels using this encoding. We give definitions of secrecy and reliability for the communication process, and with these assumptions derive upper bounds on the amount of steganographic communication possible and show that these bounds match the communication rates achieved with our encoding. This gives a steganographic capacity for a noiseless channel emulating a given noisy channel.
We show how to protect a stream of quantum information from decoherence induced by a noisy quantum communication channel. We exploit preshared entanglement and a convolutional coding structure to develop a theory of entanglement-assisted quantum convolutional coding. Our construction produces a Calderbank-Shor-Steane (CSS) entanglement-assisted quantum convolutional code from two arbitrary classical binary convolutional codes. The rate and error-correcting properties of the classical convolutional codes directly determine the corresponding properties of the resulting entanglement-assisted quantum convolutional code. We explain how to encode our CSS entanglement-assisted quantum convolutional codes starting from a stream of information qubits, ancilla qubits, and shared entangled bits.
This chapter deals with the subject of quantum error correction and the related codes (QECC), which can be applied to noisy quantum channels and quantum memories with the purpose of preserving or protecting the information integrity. I first describe the basics of quantum repetition codes , as applicable to bit-flip and phase-flip quantum channels. Then I consider the 9-qubit Shor code , which has the capability of diagnosing and correcting any combination of bit-flip and phase-flip errors, up to one error of each type. Furthermore, it is shown that the Shor code is, in fact, capable of fully restoring qubit integrity under a continuum of bit or phase errors, a property that has no counterpart in the classical world of error-correction codes. But the exploration of QECC does not stop here! We shall discover the elegant Calderbank–Shor–Steane (CSS) codes, which have the capability of correcting any number of errors t , both bit-flip and phase-flip. As an application of the CSS code, I then describe the 7-qubit Hadamard–Steane code , which can correct up to one error on single qubits. A corresponding quantum circuit, based on an original generator-matrix example, is presented. Quantum repetition code In Chapter 11, we saw that the simplest form of error-correction code (ECC) is the repetition code , based on the principle of majority logic . The background assumption is that in a given message sequence, or bit string, the probability of a bit error is sufficiently small for the majority of bits to be correctly transmitted through the channel.