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Transmitting Quantum Information Reliably across Various Quantum Channels

Yingkai Ouyang

Published May 1, 2013• 1 citation
OpenAlex

Abstract

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.

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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.
This dissertation explores covertness in two quantum settings: communication and computing. Covert communication refers to transmitting over a channel such that an adversary cannot detect the presence of the transmission, whereas covert quantum computing refers to running computations on a quantum computer that are undetectable by an adversary who is also using the same device. Although these problems arise from different physical phenomena, they share the common goal of preventing an adversary from detecting their operation. We begin by presenting the preliminaries and mathematical tools from quantum information theory required for covert analysis of communication over quantum channels and covert quantum computing. We then proceed to explore covert communication over the lossy thermal-noise bosonic channel, a quantum-mechanical model of many practical channels, including optical ones. First, we characterize the covert capacity region of the degraded lossy thermal-noise bosonic broadcast channel. We extend prior work on degraded bosonic broadcast channels to incorporate photon loss to an adversarial environment. We derive the classical capacity region and the covert capacity for the two-receiver case, and extend them to MMM receivers. We demonstrate that coherent-state encoding achieves these capacities and establish that time-division is optimal in the covert regime. These results rely on a strong minimum entropy output conjecture, a direct result of the entropy photon-number inequality (EPnI) conjecture. Building on covert communication, we develop a system construction for reliable transmission of covert qubits over an arbitrary quantum channel using single- and dual-rail encodings. We apply this construction to the lossy thermal-noise bosonic channel, thereby deriving achievable rates. We further explore the impact of access to a two-way classical communication link. While a tight upper bound remains an open problem, we provide a converse upper bound by decomposing the channel into a concatenation of a pure-loss channel and a quantum-limited amplifier and discarding the former. These bounds demonstrate a \emph{square root law} (SRL) for covert quantum communication similar to that for classical: M(n)∝nM(n)\propto\sqrt{n}M(n)∝n​ qubits can be transmitted covertly and reliably over nnn uses of the channel. We conjecture that the gap between our lower and upper bounds on the number of reliable covert qubits can be reduced by improving quantum error-correction codes and quantum channel capacity bounds. We then shift our focus to \emph{covert quantum computing}, introducing and formalizing a setting in which an adversary attempts to detect whether computation is occurring on a shared quantum processing unit (QPU) by monitoring a subset of the qubits or qumodes. For superconducting architectures with a nearest-neighbor crosstalk noise model and planar qubit topologies, a circuit remains covert by inserting idle qubits between the qubits running the computation and the remaining qubits on the QPU. Using discrete isoperimetric inequalities, we show that a circuit placement exists such that only O(n)\mathcal{O}(\sqrt{n})O(n​) spectator qubits bordering the circuit provide detection information to an adversary, where nnn is the number of qubits used for computation. Thus, in this setting, the cost of maintaining covertness while also maintaining reliability of the computation is an additional O(n)\mathcal{O}(\sqrt{n})O(n​) idle border qubits. We experimentally explore this model on IQM's Emerald processor, using Ramsey experiments on the adversary's qubits to detect gate operations. While nearest-neighbor crosstalk was detected as expected, long-range coupling effects were also observed beyond the border qubits, identifying a side channel that an adversary can exploit for detection. Finally, we discuss the outlook, future research directions, and open problems.
The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel's ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels, since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds for certain regimes.
This thesis provides bounds on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution. The first two chapters provide a bare-bones introduction to classical and quantum error correcting codes, respectively. The next four chapters present achievable rates for quantum codes in various scenarios. The final chapter is dedicated to an upper bound on the quantum channel capacity. Chapter 3 studies coding for adversarial noise using quantum list codes, showing there exist quantum codes with high rates and short lists. These can be used, together with a very short secret key, to communicate with high fidelity at noise levels for which perfect fidelity is, impossible. Chapter 4 explores the performance of a family of degenerate codes when used to communicate over Pauli channels, showing they can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code and that exceed those of previously known degenerate codes. By studying the scaling of the optimal block length as a function of the channel's parameters, we develop a heuristic for designing even better codes. Chapter 5 describes an equivalence between a family of noisy preprocessing protocols for quantum key distribution and entanglement distillation protocols whose target state belongs to a class of private states called twisted states. In Chapter 6, the codes of Chapter 4 are combined with the protocols of Chapter 5 to provide higher key rates for one-way quantum key distribution than were previously thought possible. Finally, Chapter 7 presents a new upper bound on the quantum channel that is both additive and convex, and which can be interpreted as the of the channel for communication given access to side channels from a class of zero cloning channels. This clone assisted capacity is equal to the unassisted for channels that are degradable, which we use to find new upper bounds on the of a depolarizing channel.
The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel's ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of great interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds.
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.
No quantum system can be perfectly isolated from the environment and, as a result, no physical implementation of quantum information processing tasks can be completely free from noise. The best tool for combating such noise is the use of quantum error correcting codes (QECCs). The general requirements for QECCs have been documented for a while and yet construction of good codes and understanding their effect remains a difficult and active area of study. In this thesis I outline the work I have completed looking into both of these problems. My general technique throughout this work is to first reduce the problem space as much as possible through use of group theory. Then to use numerical methods and bring computational power to bear on the problem. In chapter 2, I investigate the construction of good codes for the amplitude damping channel, using the codeword stabilized quantum code(CWS) framework. Through an exhaustive search method many new codes with better parameters than previously known are found. In chapter 3, I continue constructing good codes for the amplitude damping channel, this time using code concatenation techniques to find results that would be unfeasible to find via an exhaustive search. Finally, chapter 4 broaches the difficult problem of determining if a quantum channel has capacity; the ultimate use of QECCs in a sense. Expanding on the key works done on the problem, I develop the theory surrounding effective noise channels obtained from applying a QECC on multiple uses of a channel in order to determine if capacity exists. Using this framework and making use of computational power available today, these techniques allow us to find many very noisy non-Pauli channels that have positive capacity which previously had not been shown to have capacity.