Mark Wilde | Louisiana State University (original) (raw)
Papers by Mark Wilde
Physical Review A, Jan 1, 2010
Arxiv preprint arXiv:1004.0458, Jan 1, 2010
Physical review letters, Jan 1, 2009
IEEE Transactions on Information Theory, Jan 1, 2010
One of the unexpected breakdowns in the existing theory of quantum serial turbo coding is that a ... more One of the unexpected breakdowns in the existing theory of quantum serial turbo coding is that a quantum convolutional encoder cannot simultaneously be recursive and non-catastrophic. These properties are essential for a quantum turbo code to have an unbounded minimum distance and for its iterative decoding algorithm to converge, respectively. Here, we show that the entanglement-assisted paradigm gives a theoretical and practical "turbo boost" to these codes, in the sense that an entanglement-assisted quantum (EAQ) convolutional encoder can possess both of the aforementioned desirable properties, and simulation results indicate that entanglement-assisted turbo codes can operate reliably in a noise regime 5.5 dB beyond that of standard quantum turbo codes. Entanglement is the resource that enables a convolutional encoder to satisfy both properties because an encoder acting on only information qubits, classical bits, gauge qubits, and ancilla qubits cannot simultaneously satisfy them. We give several examples of EAQ convolutional encoders that are both recursive and non-catastrophic and detail their relevant parameters. Finally, simulation results demonstrate that interleaved serial concatenation of EAQ convolutional encoders leads to a powerful code construction with excellent performance on a memoryless depolarizing channel.
The aim of this book is to develop "from the ground up" many of the major, exciting, pre- and pos... more The aim of this book is to develop "from the ground up" many of the major, exciting, pre- and post-millenium developments in the general area of study known as quantum Shannon theory. As such, we spend a significant amount of time on quantum mechanics for quantum information theory (Part II), we give a careful study of the important unit protocols of teleportation, super-dense coding, and entanglement distribution (Part III), and we develop many of the tools necessary for understanding information transmission or compression (Part IV). Parts V and VI are the culmination of this book, where all of the tools developed come into play for understanding many of the important results in quantum Shannon theory.
Arxiv preprint arXiv:1008.0433, Jan 1, 2010
Arxiv preprint arXiv: …, Jan 1, 2011
Channel polarization is a phenomenon in which a particular recursive encoding induces a set of sy... more Channel polarization is a phenomenon in which a particular recursive encoding induces a set of synthesized channels from many instances of a memoryless channel, such that a fraction of the synthesized channels become near perfect for data transmission and the other fraction become near useless for this task. Mahdavifar and Vardy have recently exploited this phenomenon to construct codes that achieve the symmetric private capacity for private data transmission over a degraded wiretap channel. In the current paper, we build on their work and demonstrate how to construct quantum wiretap polar codes that achieve the symmetric private capacity of a degraded quantum wiretap channel with a classical eavesdropper. Due to the Schumacher-Westmoreland correspondence between quantum privacy and quantum coherence, we can construct quantum polar codes by operating these quantum wiretap polar codes in superposition, much like Devetak's technique for demonstrating the achievability of the coherent information rate for quantum data transmission. Our scheme achieves the symmetric coherent information rate for quantum channels that are degradable with a classical environment. This condition on the environment may seem restrictive, but we show that many quantum channels satisfy this criterion, including amplitude damping channels, photon-detected jump channels, dephasing channels, erasure channels, and cloning channels. Our quantum polar coding scheme has the desirable properties of being channel-adapted and symmetric capacity-achieving along with having an efficient encoder, but we have not demonstrated that the decoding is efficient. Also, the scheme may require entanglement assistance, but we show that the rate of entanglement consumption vanishes in the limit of large blocklength if the channel is degradable with classical environment.
Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are ... more Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are capacity-achieving for the task of sending classical data over a channel with classical inputs and quantum outputs. Although they demonstrated the existence of such codes, their proof does not provide an explicit construction of codes for this task. The aim of the present paper is to fill this gap by constructing near-explicit "polar" codes that are capacity achieving. The codes exploit the channel polarization phenomenon observed by Arikan for the case of classical channels. Channel polarization is an effect in which one can synthesize a set of channels, by "channel combining" and "channel splitting," in which a fraction of the synthesized channels are perfect for data transmission while the other fraction are completely useless for data transmission, with the good fraction equal to the capacity of the channel. The channel polarization effect then leads to a simple scheme for data transmission: send the information bits through the perfect channels and "frozen" bits through the useless ones. The main technical contribution of the present paper is to leverage several known results from the quantum information literature to demonstrate that the channel polarization effect occurs for channels with classical inputs and quantum outputs. We then construct linear polar codes based on this effect, and we also demonstrate that a quantum successive cancellation decoder works well, by exploiting Sen's recent "non-commutative union bound" that holds for a sequence of projectors applied to a quantum state. We also speculate on how to construct polar codes for the task of sending quantum data through channels with quantum inputs and quantum outputs.
Physical Review Letters, 2009
The causal self-consistency condition for closed timelike curves can give rise to nonlinear inter... more The causal self-consistency condition for closed timelike curves can give rise to nonlinear interactions on chronology-respecting qubits. We demonstrate that particular unitary interactions between closed timelike curve qubits and chronology-respecting qubits allow perfect ...
The discrete memoryless interference channel is modelled as a conditional probability distributio... more The discrete memoryless interference channel is modelled as a conditional probability distribution with two outputs depending on two inputs and has widespread applications in practical communication scenarios. In this paper, we introduce and study the quantum interference channel, a generalization of a two-input, two-output memoryless channel to the setting of quantum Shannon theory. We discuss three different coding strategies and obtain corresponding achievable rate regions for quantum interference channels. We calculate the capacity regions in the special cases of "very strong" and "strong" interference. The achievability proof in the case of "strong" interference exploits a novel quantum simultaneous decoder for two-sender quantum multiple access channels. We formulate a conjecture regarding the existence of a quantum simultaneous decoder in the three-sender case and use it to state the rates achievable by a quantum Han-Kobayashi strategy.
We provide a solution for the most general setting of information processing in the quantum Shann... more We provide a solution for the most general setting of information processing in the quantum Shannon-theoretic sense by giving optimal trade-offs between classical communication, quantum communication, and entanglement. We begin by showing that a combination of teleportation, superdense coding, and entanglement distribution is the optimal strategy for transmission of information when only the three noiseless resources of classical communication, quantum communication, and entanglement are available. Next, we provide a solution for the scenario where a large number of copies of a noisy bipartite state are available (in addition to consumption or generation of the above three noiseless resources). The coding strategy is an extension of previous techniques in the quantum Shannon-theoretic literature. We finally provide a solution to the scenario where a large number of uses of a noisy quantum channel are available in addition to the consumption or generation of the three noiseless resources. The coding strategy here is the classically-enhanced father protocol, a protocol which we discussed in a previous paper. Our results are of a “ multi-letter” nature, meaning that there might be room for improvement in the coding strategies presented here.
Computing Research Repository, 2010
The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting ... more The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from exchanging the original code's information qubits with its ebits. To introduce this notion, we show how entanglement-assisted (EA) repetition codes and accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes.We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We also establish the Gilbert-Varshamov bound and the Plotkin bound for EAQEC codes, and we use these to examine the existence of some EAQEC codes. Finally, we provide upper bounds on the block error probability when transmitting maximal-entanglement EAQEC codes over the depolarizing channel, and we derive variations of the hashing bound for EAQEC codes, which is a lower bound on the maximum rate at which reliable communication over Pauli channels is possible with the use of pre-shared entanglement.
Physics, 2011
Figure 1 The game board for “Guess That Button.” The game show host gives Alice the game board, a... more Figure 1 The game board for “Guess That Button.” The game show host gives Alice the game board, and Alice pushes one of the four buttons. Observe that there are exactly three edges (green, blue, or red) touching the button that Alice pushes. The game show host ...
Physical Review A, Jan 1, 2010
Arxiv preprint arXiv:1004.0458, Jan 1, 2010
Physical review letters, Jan 1, 2009
IEEE Transactions on Information Theory, Jan 1, 2010
One of the unexpected breakdowns in the existing theory of quantum serial turbo coding is that a ... more One of the unexpected breakdowns in the existing theory of quantum serial turbo coding is that a quantum convolutional encoder cannot simultaneously be recursive and non-catastrophic. These properties are essential for a quantum turbo code to have an unbounded minimum distance and for its iterative decoding algorithm to converge, respectively. Here, we show that the entanglement-assisted paradigm gives a theoretical and practical "turbo boost" to these codes, in the sense that an entanglement-assisted quantum (EAQ) convolutional encoder can possess both of the aforementioned desirable properties, and simulation results indicate that entanglement-assisted turbo codes can operate reliably in a noise regime 5.5 dB beyond that of standard quantum turbo codes. Entanglement is the resource that enables a convolutional encoder to satisfy both properties because an encoder acting on only information qubits, classical bits, gauge qubits, and ancilla qubits cannot simultaneously satisfy them. We give several examples of EAQ convolutional encoders that are both recursive and non-catastrophic and detail their relevant parameters. Finally, simulation results demonstrate that interleaved serial concatenation of EAQ convolutional encoders leads to a powerful code construction with excellent performance on a memoryless depolarizing channel.
The aim of this book is to develop "from the ground up" many of the major, exciting, pre- and pos... more The aim of this book is to develop "from the ground up" many of the major, exciting, pre- and post-millenium developments in the general area of study known as quantum Shannon theory. As such, we spend a significant amount of time on quantum mechanics for quantum information theory (Part II), we give a careful study of the important unit protocols of teleportation, super-dense coding, and entanglement distribution (Part III), and we develop many of the tools necessary for understanding information transmission or compression (Part IV). Parts V and VI are the culmination of this book, where all of the tools developed come into play for understanding many of the important results in quantum Shannon theory.
Arxiv preprint arXiv:1008.0433, Jan 1, 2010
Arxiv preprint arXiv: …, Jan 1, 2011
Channel polarization is a phenomenon in which a particular recursive encoding induces a set of sy... more Channel polarization is a phenomenon in which a particular recursive encoding induces a set of synthesized channels from many instances of a memoryless channel, such that a fraction of the synthesized channels become near perfect for data transmission and the other fraction become near useless for this task. Mahdavifar and Vardy have recently exploited this phenomenon to construct codes that achieve the symmetric private capacity for private data transmission over a degraded wiretap channel. In the current paper, we build on their work and demonstrate how to construct quantum wiretap polar codes that achieve the symmetric private capacity of a degraded quantum wiretap channel with a classical eavesdropper. Due to the Schumacher-Westmoreland correspondence between quantum privacy and quantum coherence, we can construct quantum polar codes by operating these quantum wiretap polar codes in superposition, much like Devetak's technique for demonstrating the achievability of the coherent information rate for quantum data transmission. Our scheme achieves the symmetric coherent information rate for quantum channels that are degradable with a classical environment. This condition on the environment may seem restrictive, but we show that many quantum channels satisfy this criterion, including amplitude damping channels, photon-detected jump channels, dephasing channels, erasure channels, and cloning channels. Our quantum polar coding scheme has the desirable properties of being channel-adapted and symmetric capacity-achieving along with having an efficient encoder, but we have not demonstrated that the decoding is efficient. Also, the scheme may require entanglement assistance, but we show that the rate of entanglement consumption vanishes in the limit of large blocklength if the channel is degradable with classical environment.
Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are ... more Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are capacity-achieving for the task of sending classical data over a channel with classical inputs and quantum outputs. Although they demonstrated the existence of such codes, their proof does not provide an explicit construction of codes for this task. The aim of the present paper is to fill this gap by constructing near-explicit "polar" codes that are capacity achieving. The codes exploit the channel polarization phenomenon observed by Arikan for the case of classical channels. Channel polarization is an effect in which one can synthesize a set of channels, by "channel combining" and "channel splitting," in which a fraction of the synthesized channels are perfect for data transmission while the other fraction are completely useless for data transmission, with the good fraction equal to the capacity of the channel. The channel polarization effect then leads to a simple scheme for data transmission: send the information bits through the perfect channels and "frozen" bits through the useless ones. The main technical contribution of the present paper is to leverage several known results from the quantum information literature to demonstrate that the channel polarization effect occurs for channels with classical inputs and quantum outputs. We then construct linear polar codes based on this effect, and we also demonstrate that a quantum successive cancellation decoder works well, by exploiting Sen's recent "non-commutative union bound" that holds for a sequence of projectors applied to a quantum state. We also speculate on how to construct polar codes for the task of sending quantum data through channels with quantum inputs and quantum outputs.
Physical Review Letters, 2009
The causal self-consistency condition for closed timelike curves can give rise to nonlinear inter... more The causal self-consistency condition for closed timelike curves can give rise to nonlinear interactions on chronology-respecting qubits. We demonstrate that particular unitary interactions between closed timelike curve qubits and chronology-respecting qubits allow perfect ...
The discrete memoryless interference channel is modelled as a conditional probability distributio... more The discrete memoryless interference channel is modelled as a conditional probability distribution with two outputs depending on two inputs and has widespread applications in practical communication scenarios. In this paper, we introduce and study the quantum interference channel, a generalization of a two-input, two-output memoryless channel to the setting of quantum Shannon theory. We discuss three different coding strategies and obtain corresponding achievable rate regions for quantum interference channels. We calculate the capacity regions in the special cases of "very strong" and "strong" interference. The achievability proof in the case of "strong" interference exploits a novel quantum simultaneous decoder for two-sender quantum multiple access channels. We formulate a conjecture regarding the existence of a quantum simultaneous decoder in the three-sender case and use it to state the rates achievable by a quantum Han-Kobayashi strategy.
We provide a solution for the most general setting of information processing in the quantum Shann... more We provide a solution for the most general setting of information processing in the quantum Shannon-theoretic sense by giving optimal trade-offs between classical communication, quantum communication, and entanglement. We begin by showing that a combination of teleportation, superdense coding, and entanglement distribution is the optimal strategy for transmission of information when only the three noiseless resources of classical communication, quantum communication, and entanglement are available. Next, we provide a solution for the scenario where a large number of copies of a noisy bipartite state are available (in addition to consumption or generation of the above three noiseless resources). The coding strategy is an extension of previous techniques in the quantum Shannon-theoretic literature. We finally provide a solution to the scenario where a large number of uses of a noisy quantum channel are available in addition to the consumption or generation of the three noiseless resources. The coding strategy here is the classically-enhanced father protocol, a protocol which we discussed in a previous paper. Our results are of a “ multi-letter” nature, meaning that there might be room for improvement in the coding strategies presented here.
Computing Research Repository, 2010
The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting ... more The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from exchanging the original code's information qubits with its ebits. To introduce this notion, we show how entanglement-assisted (EA) repetition codes and accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes.We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We also establish the Gilbert-Varshamov bound and the Plotkin bound for EAQEC codes, and we use these to examine the existence of some EAQEC codes. Finally, we provide upper bounds on the block error probability when transmitting maximal-entanglement EAQEC codes over the depolarizing channel, and we derive variations of the hashing bound for EAQEC codes, which is a lower bound on the maximum rate at which reliable communication over Pauli channels is possible with the use of pre-shared entanglement.
Physics, 2011
Figure 1 The game board for “Guess That Button.” The game show host gives Alice the game board, a... more Figure 1 The game board for “Guess That Button.” The game show host gives Alice the game board, and Alice pushes one of the four buttons. Observe that there are exactly three edges (green, blue, or red) touching the button that Alice pushes. The game show host ...