Trade-off capacities of the quantum Hadamard channels

Brádler, Kamil; Hayden, Patrick; Touchette, Dave; Wilde, Mark M.

doi:10.1103/physreva.81.062312

Cited by 69 publications

(113 citation statements)

References 67 publications

(169 reference statements)

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“…Concrete examples of degradable channels include erasure channels [14], qubit flip channels, photon number splitting channels [15], cloning channels and Unruh channels [16], [17].…”

Section: B Quantum Channelsmentioning

confidence: 99%

Quantum Broadcast Channels

Yard

Hayden

Devetak

2011

IEEE Trans. Inform. Theory

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104

117

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Abstract-We consider quantum channels with one sender and two receivers, used in several different ways for the simultaneous transmission of independent messages. We begin by extending the technique of superposition coding to quantum channels with a classical input to give a general achievable region. We also give outer bounds to the capacity regions for various special cases from the classical literature and prove that superposition coding is optimal for a class of channels. We then consider extensions of superposition coding for channels with a quantum input, where some of the messages transmitted are quantum instead of classical, in the sense that the parties establish bipartite or tripartite GHZ entanglement. We conclude by using state merging to give achievable rates for establishing bipartite entanglement between different pairs of parties with the assistance of free classical communication.Index Terms-broadcast channels, entanglement, network information theory, shannon theory, quantum information A classical discrete memoryless broadcast channel with one sender and two receivers is modeled by a probability transition matrix p(y, z|x). Broadcast channels were introduced by Cover [1] in 1972, and it is still not known how to compute their capacity regions in full generality. Cover illustrated how to superimpose high-rate information on lowrate information, so that a stronger receiver obtains a refined version of what is available to a weaker receiver. Coding theorems by Bergmans [2] and van der Meulen [3] further developed this idea, leading to the following superposition coding inner bound to the capacity region: Alice, the sender, can transmit a rate R B message to Bob, who sees Y , and a rate R C message to Charlie, who sees Z, while simultaneously sending a rate R common message to both, iffor some p(t, x). Superposition coding works well when Bob receives a stronger signal than Charlie. Making this idea precise led to the characterization of the capacity regions for several special cases.The first case to be solved is when Charlie's output is a degraded attainable by superposition coding simplifies toCover conjectured [1], and Gallager proved [4], that this region is optimal for the degraded broadcast channel. In Section II-A, we prove a coding theorem for quantum broadcast channels with a classical input, establishing the superposition coding inner bound (1) for such channels. In Section II-B, we give an outer bound for the capacity region of degraded broadcast channels with a classical input and give conditions under which superposition coding is optimal. Superposition coding is also optimal for classical channels in several other settings. We recall some of these in Section II-C and illustrate how existing classical results yield outer bounds on the associated capacity regions for channels with quantum outputs. The remainder of the paper considers broadcast channels with a quantum input, as modeled by completely-positive trace-preserving maps. We consider several variants that involve quantum commu...

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“…Concrete examples of degradable channels include erasure channels [14], qubit flip channels, photon number splitting channels [15], cloning channels and Unruh channels [16], [17].…”

Section: B Quantum Channelsmentioning

confidence: 99%

Quantum Broadcast Channels

Yard

Hayden

Devetak

2011

IEEE Trans. Inform. Theory

Self Cite

104

117

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“…Channels of this are also known as Hadamard channels in the literature on quantum Shannon theory, where they represent one of the important classes of channels with tractable capacity regions [104,105]. Energy-preserving channels with nondegenerate Hamiltonian have a number of properties.…”

Section: Discussionmentioning

confidence: 99%

Optimal quantum operations at zero energy cost

Chiribella

Yang

2017

Phys. Rev. A

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Quantum technologies are developing powerful tools to generate and manipulate coherent superpositions of different energy levels. Envisaging a new generation of energy-efficient quantum devices, here we explore how coherence can be manipulated without exchanging energy with the surrounding environment. We start from the task of converting a coherent superposition of energy eigenstates into another. We identify the optimal energy-preserving operations, both in the deterministic and in the probabilistic scenario. We then design a recursive protocol, wherein a branching sequence of energy-preserving filters increases the probability of success while reaching maximum fidelity at each iteration. Building on the recursive protocol, we construct efficient approximations of the optimal fidelity-probability trade-off, by taking coherent superpositions of the different branches generated by probabilistic filtering. The benefits of this construction are illustrated in applications to quantum metrology, quantum cloning, coherent state amplification, and ancilla-driven computation. Finally, we extend our results to transitions where the input state is generally mixed and we apply our findings to the task of purifying quantum coherence.

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“…[32], [31] that the (+, +, −) octant single-letterizes for this channel. The characterization of the dynamic capacity region in Theorem 3 shows that the classically-enhanced father protocol combined with the unit protocols achieves the full region.…”

Section: B Dynamic Casementioning

confidence: 99%

“…This proof builds on earlier work in Refs. [31], [32] to show that single-letterization holds. In a later work [33], we build on the efforts in Ref.…”

Section: Introductionmentioning

confidence: 99%

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Trading classical communication, quantum communication, and entanglement in quantum Shannon theory

Hsieh

Wilde

2010

IEEE Trans. Inform. Theory

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Abstract-We give trade-offs between classical communication, quantum communication, and entanglement for processing information in the Shannon-theoretic setting. We first prove a "unit-resource" capacity theorem that applies to the scenario where only the above three noiseless resources are available for consumption or generation. The optimal strategy mixes the three fundamental protocols of teleportation, super-dense coding, and entanglement distribution. We then provide an achievable rate region and a matching multi-letter converse for the "direct static" capacity theorem. This theorem applies to 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). Our coding strategy involves a protocol that we name the classically-assisted state redistribution protocol and the three fundamental protocols. We finally provide an achievable rate region and a matching mutli-letter converse for the "direct dynamic" capacity theorem. This theorem applies 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. Our coding strategy combines the classically-enhanced father protocol with the three fundamental unit protocols.

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Trade-off capacities of the quantum Hadamard channels

Cited by 69 publications

References 67 publications

Quantum Broadcast Channels

Quantum Broadcast Channels

Optimal quantum operations at zero energy cost

Trading classical communication, quantum communication, and entanglement in quantum Shannon theory

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