We show that quantum-to-classical channels, i.e., quantum measurements, can be asymptotically sim... more We show that quantum-to-classical channels, i.e., quantum measurements, can be asymptotically simulated by an amount of classical communication equal to the quantum mutual information of the measurement, if sufficient shared randomness is available. This result generalizes Winter's measurement compression theorem for fixed independent and identically distributed inputs [Winter, CMP 244 , 2004] to arbitrary inputs, and more importantly, it identifies the quantum mutual information of a measurement as the information gained by performing it, independent of the input state on which it is performed. Our result is a generalization of the classical reverse Shannon theorem to quantum-to-classical channels. In this sense, it can be seen as a quantum reverse Shannon theorem for quantum-to-classical channels, but with the entanglement assistance and quantum communication replaced by shared randomness and classical communication, respectively. The proof is based on a novel one-shot state merging protocol for "classically coherent states" as well as the post-selection technique for quantum channels, and it uses techniques developed for the quantum reverse Shannon theorem [Berta et al., CMP 306 (579), 2011]. * berta@phys.ethz.ch
Uncertainty relations capture the essence of the inevitable randomness associated with the outcom... more Uncertainty relations capture the essence of the inevitable randomness associated with the outcomes of two incompatible quantum measurements. Recently, Berta et al. [Nature Phys. 6, 659 (2010)] have shown that the lower bound on the uncertainties of the measurement outcomes depends on the correlations between the observed system and an observer who possesses a quantum memory. If the system is maximally entangled with its memory, the outcomes of two incompatible measurements made on the system can be predicted precisely. Here, we obtain an uncertainty relation that tightens the lower bound of Berta et al. by incorporating an additional term that depends on the quantum discord and the classical correlations of the joint state of the observed system and the quantum memory. We discuss several examples of states for which our lower bound is tighter than the bound of Berta et al. On the application side, we discuss the relevance of our inequality for the security of quantum key distribution and show that it can be used to provide bounds on the distillable common randomness and the entanglement of formation of bipartite quantum states.
Suppose that a polynomial-time mixed-state quantum circuit, described as a sequence of local unit... more Suppose that a polynomial-time mixed-state quantum circuit, described as a sequence of local unitary interactions followed by a partial trace, generates a quantum state shared between two parties. One might then wonder, does this quantum circuit produce a state that is separable or entangled? Here, we give evidence that it is computationally hard to decide the answer to this question, even if one has access to the power of quantum computation. We begin by exhibiting a two-message quantum interactive proof system that can decide the answer to a promise version of the question. We then prove that the promise problem is hard for the class of promise problems with "quantum statistical zero knowledge" (QSZK) proof systems by demonstrating a polynomial-time Karp reduction from the QSZK-complete promise problem "quantum state distinguishability" to our quantum separability problem. By exploiting Knill's efficient encoding of a matrix description of a state into a description of a circuit to generate the state, we can show that our promise problem is NP-hard with respect to Cook reductions. Thus, the quantum separability problem (as phrased above) constitutes the first nontrivial promise problem decidable by a two-message quantum interactive proof system while being hard for both NP and QSZK. We also consider a variant of the problem, in which a given polynomial-time mixed-state quantum circuit accepts a quantum state as input, and the question is to decide if there is an input to this circuit which makes its output separable across some bipartite cut. We prove that this problem is a complete promise problem for the class QIP of problems decidable by quantum interactive proof systems. Finally, we show that a two-message quantum interactive proof system can also decide a multipartite generalization of the quantum separability problem.
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.
Bennett and Schumacher's postselected quantum teleportation is a model of closed timelike curves ... more Bennett and Schumacher's postselected quantum teleportation is a model of closed timelike curves (CTCs) that leads to results physically different from Deutsch's model. We show that even a single qubit passing through a postselected CTC (P-CTC) is sufficient to do any postselected quantum measurement with certainty, and we discuss an important difference between "Deutschian" CTCs (D-CTCs) and P-CTCs in which the future existence of a P-CTC might affect the present outcome of an experiment. Then, based on a suggestion of Bennett and Smith, we explicitly show how a party assisted by P-CTCs can distinguish a set of linearly independent quantum states, and we prove that it is not possible for such a party to distinguish a set of linearly dependent states. The power of P-CTCs is thus weaker than that of D-CTCs because the Holevo bound still applies to circuits using them, regardless of their ability to conspire in violating the uncertainty principle. We then discuss how different notions of a quantum mixture that are indistinguishable in linear quantum mechanics lead to dramatically differing conclusions in a nonlinear quantum mechanics involving P-CTCs. Finally, we give explicit circuit constructions that can efficiently factor integers, efficiently solve any decision problem in the intersection of NP and coNP, and probabilistically solve any decision problem in NP. These circuits accomplish these tasks with just one qubit traveling back in time, and they exploit the ability of postselected closed timelike curves to create grandfather paradoxes for invalid answers.
Calculating the capacity of interference channels is a notorious open problem in classical inform... more Calculating the capacity of interference channels is a notorious open problem in classical information theory. Such channels have two senders and two receivers, and each sender would like to communicate with a partner receiver. The capacity of such channels is known exactly in the settings of "very strong" and "strong" interference, while the Han-Kobayashi coding strategy gives the best known achievable rate region in the general case.
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.
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 ...
We analyze the practical performance of quantum polar codes, by computing rigorous bounds on bloc... more We analyze the practical performance of quantum polar codes, by computing rigorous bounds on block error probability and by numerically simulating them. We evaluate our bounds for quantum erasure channels with coding block lengths between 2 10 and 2 20 , and we report the results of simulations for quantum erasure channels, quantum depolarizing channels, and "BB84" channels with coding block lengths up to N = 1024. For quantum erasure channels, we observe that high quantum data rates can be achieved for block error rates Pe ≤ 10 −4 and that somewhat lower quantum data rates can be achieved for quantum depolarizing and BB84 channels. Our results here also serve as bounds for and simulations of private classical data transmission over these channels, essentially due to Renes' duality bounds for privacy amplification and classical data transmission of complementary observables. Future work might be able to improve upon our numerical results for quantum depolarizing and BB84 channels by employing a polar coding rule other than the heuristic used here.
Recent work has precisely characterized the achievable trade-offs between three key information p... more Recent work has precisely characterized the achievable trade-offs between three key information processing tasks-classical communication (generation or consumption), quantum communication (generation or consumption), and shared entanglement (distribution or consumption), measured in bits, qubits, and ebits per channel use respectively. Slices and corner points of this three-dimensional region reduce to several well-known quantum communication protocols over noisy channels. A single trade-off coding technique can attain any point in the region and can outperform timesharing between the best known protocols for accomplishing each information processing task alone. Previously, the benefits of trade-off coding that had been found were too small to be of much practical value (for the dephasing and the universal cloning machine channels, for instance). In this article, we demonstrate that the associated performance gains are nonetheless remarkably high for several physically relevant bosonic channels that model free-space / fiber-optic links, thermalizing channels, or amplifiers (or even relativistic communication). We show that significant performance gains from trade-off coding also apply when trading photon-number resources between transmitting public and private classical information simultaneously over secret-key-assisted bosonic channels.
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.
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.
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 ...
We show that quantum-to-classical channels, i.e., quantum measurements, can be asymptotically sim... more We show that quantum-to-classical channels, i.e., quantum measurements, can be asymptotically simulated by an amount of classical communication equal to the quantum mutual information of the measurement, if sufficient shared randomness is available. This result generalizes Winter's measurement compression theorem for fixed independent and identically distributed inputs [Winter, CMP 244 , 2004] to arbitrary inputs, and more importantly, it identifies the quantum mutual information of a measurement as the information gained by performing it, independent of the input state on which it is performed. Our result is a generalization of the classical reverse Shannon theorem to quantum-to-classical channels. In this sense, it can be seen as a quantum reverse Shannon theorem for quantum-to-classical channels, but with the entanglement assistance and quantum communication replaced by shared randomness and classical communication, respectively. The proof is based on a novel one-shot state merging protocol for "classically coherent states" as well as the post-selection technique for quantum channels, and it uses techniques developed for the quantum reverse Shannon theorem [Berta et al., CMP 306 (579), 2011]. * berta@phys.ethz.ch
Uncertainty relations capture the essence of the inevitable randomness associated with the outcom... more Uncertainty relations capture the essence of the inevitable randomness associated with the outcomes of two incompatible quantum measurements. Recently, Berta et al. [Nature Phys. 6, 659 (2010)] have shown that the lower bound on the uncertainties of the measurement outcomes depends on the correlations between the observed system and an observer who possesses a quantum memory. If the system is maximally entangled with its memory, the outcomes of two incompatible measurements made on the system can be predicted precisely. Here, we obtain an uncertainty relation that tightens the lower bound of Berta et al. by incorporating an additional term that depends on the quantum discord and the classical correlations of the joint state of the observed system and the quantum memory. We discuss several examples of states for which our lower bound is tighter than the bound of Berta et al. On the application side, we discuss the relevance of our inequality for the security of quantum key distribution and show that it can be used to provide bounds on the distillable common randomness and the entanglement of formation of bipartite quantum states.
Suppose that a polynomial-time mixed-state quantum circuit, described as a sequence of local unit... more Suppose that a polynomial-time mixed-state quantum circuit, described as a sequence of local unitary interactions followed by a partial trace, generates a quantum state shared between two parties. One might then wonder, does this quantum circuit produce a state that is separable or entangled? Here, we give evidence that it is computationally hard to decide the answer to this question, even if one has access to the power of quantum computation. We begin by exhibiting a two-message quantum interactive proof system that can decide the answer to a promise version of the question. We then prove that the promise problem is hard for the class of promise problems with "quantum statistical zero knowledge" (QSZK) proof systems by demonstrating a polynomial-time Karp reduction from the QSZK-complete promise problem "quantum state distinguishability" to our quantum separability problem. By exploiting Knill's efficient encoding of a matrix description of a state into a description of a circuit to generate the state, we can show that our promise problem is NP-hard with respect to Cook reductions. Thus, the quantum separability problem (as phrased above) constitutes the first nontrivial promise problem decidable by a two-message quantum interactive proof system while being hard for both NP and QSZK. We also consider a variant of the problem, in which a given polynomial-time mixed-state quantum circuit accepts a quantum state as input, and the question is to decide if there is an input to this circuit which makes its output separable across some bipartite cut. We prove that this problem is a complete promise problem for the class QIP of problems decidable by quantum interactive proof systems. Finally, we show that a two-message quantum interactive proof system can also decide a multipartite generalization of the quantum separability problem.
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.
Bennett and Schumacher's postselected quantum teleportation is a model of closed timelike curves ... more Bennett and Schumacher's postselected quantum teleportation is a model of closed timelike curves (CTCs) that leads to results physically different from Deutsch's model. We show that even a single qubit passing through a postselected CTC (P-CTC) is sufficient to do any postselected quantum measurement with certainty, and we discuss an important difference between "Deutschian" CTCs (D-CTCs) and P-CTCs in which the future existence of a P-CTC might affect the present outcome of an experiment. Then, based on a suggestion of Bennett and Smith, we explicitly show how a party assisted by P-CTCs can distinguish a set of linearly independent quantum states, and we prove that it is not possible for such a party to distinguish a set of linearly dependent states. The power of P-CTCs is thus weaker than that of D-CTCs because the Holevo bound still applies to circuits using them, regardless of their ability to conspire in violating the uncertainty principle. We then discuss how different notions of a quantum mixture that are indistinguishable in linear quantum mechanics lead to dramatically differing conclusions in a nonlinear quantum mechanics involving P-CTCs. Finally, we give explicit circuit constructions that can efficiently factor integers, efficiently solve any decision problem in the intersection of NP and coNP, and probabilistically solve any decision problem in NP. These circuits accomplish these tasks with just one qubit traveling back in time, and they exploit the ability of postselected closed timelike curves to create grandfather paradoxes for invalid answers.
Calculating the capacity of interference channels is a notorious open problem in classical inform... more Calculating the capacity of interference channels is a notorious open problem in classical information theory. Such channels have two senders and two receivers, and each sender would like to communicate with a partner receiver. The capacity of such channels is known exactly in the settings of "very strong" and "strong" interference, while the Han-Kobayashi coding strategy gives the best known achievable rate region in the general case.
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.
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 ...
We analyze the practical performance of quantum polar codes, by computing rigorous bounds on bloc... more We analyze the practical performance of quantum polar codes, by computing rigorous bounds on block error probability and by numerically simulating them. We evaluate our bounds for quantum erasure channels with coding block lengths between 2 10 and 2 20 , and we report the results of simulations for quantum erasure channels, quantum depolarizing channels, and "BB84" channels with coding block lengths up to N = 1024. For quantum erasure channels, we observe that high quantum data rates can be achieved for block error rates Pe ≤ 10 −4 and that somewhat lower quantum data rates can be achieved for quantum depolarizing and BB84 channels. Our results here also serve as bounds for and simulations of private classical data transmission over these channels, essentially due to Renes' duality bounds for privacy amplification and classical data transmission of complementary observables. Future work might be able to improve upon our numerical results for quantum depolarizing and BB84 channels by employing a polar coding rule other than the heuristic used here.
Recent work has precisely characterized the achievable trade-offs between three key information p... more Recent work has precisely characterized the achievable trade-offs between three key information processing tasks-classical communication (generation or consumption), quantum communication (generation or consumption), and shared entanglement (distribution or consumption), measured in bits, qubits, and ebits per channel use respectively. Slices and corner points of this three-dimensional region reduce to several well-known quantum communication protocols over noisy channels. A single trade-off coding technique can attain any point in the region and can outperform timesharing between the best known protocols for accomplishing each information processing task alone. Previously, the benefits of trade-off coding that had been found were too small to be of much practical value (for the dephasing and the universal cloning machine channels, for instance). In this article, we demonstrate that the associated performance gains are nonetheless remarkably high for several physically relevant bosonic channels that model free-space / fiber-optic links, thermalizing channels, or amplifiers (or even relativistic communication). We show that significant performance gains from trade-off coding also apply when trading photon-number resources between transmitting public and private classical information simultaneously over secret-key-assisted bosonic channels.
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.
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.
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 ...
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Papers by Mark Wilde