Communication tasks in operational theories

Heinosaari, Teiko; Kerppo, Oskari; Leppäjärvi, Leevi

doi:10.1088/1751-8121/abb5dc

Cited by 12 publications

(14 citation statements)

References 24 publications

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“…The function λ max is an ultraweak monotone on the set of communication matrices. Ultraweak monotones were extensively studied in [15]. However, for our present investigation a more important result is the following.…”

Section: Bounds On Communication Successmentioning

confidence: 86%

“…Thus perfect communication of partial ignorance requires Alice and Bob to minimize the worst case error probability with respect to all possible inputs. The strategy that Alice and Bob must implement leads to the concept of communication matrices, which were extensively studied in [15] along with their operational hierarchy.…”

Section: Partial Ignorance Communication Tasksmentioning

confidence: 99%

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Partial ignorance communication tasks in quantum theory

Kerppo

2022

Preprint

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We introduce a generalization of communication of partial ignorance where both parties of a prepare-and-measure setup receive inputs from a third party before a success metric is maximized over the measurements and preparations. Various methods are used to obtain bounds on the success metrics, including SDPs, ultraweak monotones for communication matrices and frame theory for quantum states. Simplest scenarios in the new generalized prepare-and-measure setting, simply called partial ignorance communication tasks, are analysed exhaustively for bits and qudits. Finally, the new generalized setting allows the introduction of operational equivalences to the preparations and measurements, allowing us to analyse and observe a contextual advantage for quantum theory in one of the communication tasks.

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Section: Bounds On Communication Successmentioning

confidence: 86%

Section: Partial Ignorance Communication Tasksmentioning

confidence: 99%

Partial ignorance communication tasks in quantum theory

Kerppo

2022

Preprint

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“…Having introduced signaling polytopes, we can now define the signaling dimension of a channel. This terminology is adopted from recent work by Dall'Arno et al [22] who defined the signaling dimension of a system in generalized probability theories; an analogous quantity without shared randomness has also been studied by Heinosaari et al [34]. In what follows, we assume that N : S(A) → S(B) is a completely positive tracepreserving (CPTP) map, with S(A) denoting the set of density operators (i.e.…”

Section: Signaling Polytopesmentioning

confidence: 99%

“…( 1) as P N (y|x) = Tr |y y|N |x x| . The channel N can be used to generate another channel N with input and output alphabets X and Y by performing a pre-processing map T : X → X and post-processing map R : Y → Y, thereby yielding the channel P N = RP N T. When this relationship holds, P N is said to be ultraweakly majorized by P N [31,34], and the signaling dimension of P N is no greater than that of P N [15].…”

Section: Signaling Polytopesmentioning

confidence: 99%

Certifying the Classical Simulation Cost of a Quantum Channel

Doolittle,

Chitambar

2021

Preprint

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A fundamental objective in quantum information science is to determine the cost in classical resources of simulating a particular quantum system. The classical simulation cost is quantified by the signaling dimension which specifies the minimum amount of classical communication needed to perfectly simulate a channel's input-output correlations when unlimited shared randomness is held between encoder and decoder. This paper provides a collection of device-independent tests that place lower and upper bounds on the signaling dimension of a channel. Among them, a single family of tests is shown to determine when a noisy classical channel can be simulated using an amount of communication strictly less than either its input or its output alphabet size. In addition, a family of eight Bell inequalities is presented that completely characterize when any four-outcome measurement channel, such as a Bell measurement, can be simulated using one communication bit and shared randomness. Finally, we bound the signaling dimension for all partial replacer channels in d dimensions. The bounds are found to be tight for the special case of the erasure channel.

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“…The channel cv also emerges in the problem of zeroerror channel simulation [1]- [4]. In the general task of channel simulation, Alice and Bob attempt to generate one channel P using another channel Q combined with pre-and post-processing [5]- [9]. Interesting variations to this problem arise when different types of resources are used to coordinate the pre-and post-processing of Q.…”

Section: Introductionmentioning

confidence: 99%

The Communication Value of a Quantum Channel

Chitambar¹,

George²,

Doolittle³

et al. 2021

Preprint

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There are various ways to quantify the communication capabilities of a quantum channel. In this work we study the communication value (cv) of channel, which describes the optimal success probability of transmitting a randomly selected classical message over the channel. The cv also offers a dual interpretation as the classical communication cost for zero-error channel simulation using non-signaling resources. We first provide an entropic characterization of the cv as a generalized conditional minentropy over the cone of separable operators. Additionally, the logarithm of a channel's cv is shown to be equivalent to its max-Holevo information, which can further be related to channel capacity. We evaluate the cv exactly for all qubit channels and the Werner-Holevo family of channels. While all classical channels are multiplicative under tensor product, this is no longer true for quantum channels in general. We provide a family of qutrit channels for which the cv is non-multiplicative. On the other hand, we prove that any pair of qubit channels have multiplicative cv when used in parallel. Even stronger, all entanglement-breaking channels and the partially depolarizing channel are shown to have multiplicative cv when used in parallel with any channel. We then turn to the entanglement-assisted cv and prove that it is equivalent to the conditional min-entropy of the Choi matrix of the channel. Combining with previous work on zero-error channel simulation, this implies that the entanglement-assisted cv is the classical communication cost for perfectly simulating a channel using quantum nonsignaling resources. A final component of this work investigates relaxations of the channel cv to other cones such as the set of operators having a positive partial transpose (PPT). The PPT cv is analytically and numerically investigated for well-known channels such as the Werner-Holevo family and the dephrasure family of channels.

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Communication tasks in operational theories

Cited by 12 publications

References 24 publications

Partial ignorance communication tasks in quantum theory

Partial ignorance communication tasks in quantum theory

Certifying the Classical Simulation Cost of a Quantum Channel

The Communication Value of a Quantum Channel

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