Characterizing secret communication over noisy quantum channels is an interesting problem from both a practical and theoretical perspective. Suppose Alice and Bob wish to communicate secret information so that an eavesdropper Eve will not suspect any type of encoded communication between the two. Classical or quantum cryptography will not suffice since it is always clear secret communication is taking place. Therefore Alice and Bob must execute what is known as a quantum steganographic protocol. Assuming Eve only has partial knowledge of the channel connecting Alice and Bob, we show that for the bit-flip and depolarizing channels Alice can use Eve's lack of knowledge of the channel parameter to encode quantum information steganographically. We give an explicit encoding procedure and calculate the rate at which Alice and Bob can communicate secretly. We also show that our encoding is optimal for nondegenerate quantum codes. We calculate the rate at which a secret key must be consumed. Finally, we discuss the possibility of steganographic communication over more general quantum channels and conjecture a general formula for the steganographic rate.
Characterizing secret communication over noisy quantum channels is an interesting problem from both a practical and theoretical perspective. Suppose Alice and Bob wish to communicate secret information so that an eavesdropper Eve will not suspect any type of encoded communication between the two. Classical or quantum cryptography will not suffice since it is always clear secret communication is taking place. Therefore Alice and Bob must execute what is known as a quantum steganographic protocol. Assuming Eve only has partial knowledge of the channel connecting Alice and Bob, we show that for the bit-flip and depolarizing channels Alice can use Eve's lack of knowledge of the channel parameter to encode quantum information steganographically. We give an explicit encoding procedure and calculate the rate at which Alice and Bob can communicate secretly. We also show that our encoding is optimal for nondegenerate quantum codes. We calculate the rate at which a secret key must be consumed. Finally, we discuss the possibility of steganographic communication over more general quantum channels and conjecture a general formula for the steganographic rate.
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.
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.
Steganography is the art of concealing secret information by embedding it in an apparently innocent-looking message. Quantum steganography applies the principles of quantum mechanics to traditional steganography and, compared to the latter, offers significant advantages, including heightened security, improved concealment, and increased data-hiding capacity. Traditionally, quantum steganography disguises the covert communication as channel noise, which is corrected using preshared classical randomness. This method requires the steganalytic eavesdropper Eve to overestimate the level of channel noise, so that the bounds on the stego channel capacity depend on this assumed gap in Eve's knowledge of the channel. In this work, we point out that by means of preshared quantum entanglement the secret message can be encoded into nonlocal correlations, obviating the need for such an assumption of Eve's ignorance. Consequently, the capacity bounds on the stego channel can then come from the channel capacity of the quantum communication channel. We introduce three such entanglement-based quantum steganographic protocols that make use of catalytic quantum error-correcting codes (QECCs), degenerate entanglement-assisted QECCs, or the phase bit of preshared entanglement. Here catalytic QECCs enable recycling entanglement, while entanglement assistance allows both sender and receiver to contribute to the protocol's secrecy. We derive upper and lower bounds on the secrecy capacity of each protocol, and demonstrate their practical robustness.
Quantum communication is at the heart of the quantum information theory. We study two types of quantum communication protocols over noisy transmission channels, i.e. super dense coding and cryptography protocols. In the first part of this thesis, for various scenarios, it is discussed how the super dense coding capacity is influenced by noisy quantum channels. The case of memoryless channels as well as those channels with memory which are modelled by uncorrelated and correlated noise, respectively, are considered. Explicitly Pauli channels over arbitrary dimensions are treated and the super dense coding capacity for some resource states is derived. For the qubit depolarizing channel, when noise is uncorrelated, the super dense coding capacity with respect to the input state is also optimized. This illustrates a threshold value of the noise parameter below which the super dense coding capacity is optimized by a maximally entangled initial state, while above the threshold value the super dense coding capacity is optimized by a product state. For Pauli channels, with correlated noise, the case of non unitary encoding is studied and the super dense coding capacity is derived. The first part of this thesis is concluded with an example for multipartite super dense coding.
In the second part of the thesis, the problem of optimal eavesdropping on noisy states in quantum key distribution is investigated. The case of the six state protocol, when the signal states are mixed with white noise, is considered. This situation may arise either when Alice deliberately adds noise to the signal states before they leave her lab, or, in a realistic scenario when the eavesdropper (referred to as Eve) is not assumed to replace the noisy quantum channel by a noiseless one. For individual attacks, we find Eve's optimal mutual information with Alice as a function of the quantum bit error rate. Finally, the results illustrate that adding noise on the quantum level can make quantum key distribution more robust against eavesdropping.
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.