The signaling dimension of physical systems

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Michele Dall’Arno1,2

1Yukawa Institute for Theoretical Physics, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan

2Faculty of Education and Integrated Arts and Sciences, Waseda University, Shinjuku-ku, Tokyo, 169-

8050, Japan

This is a Perspective on “Classical simulations of communication channels” by

P´eter E. Frenkel, published in Quantum 6, 751 (2022).

The question we start our analysis from is whether a quantum system can out-

perform a classical system of the same dimension for some communication scenarios.

By “communication scenario”, we mean the usual setup in which some classical mes-

sage is encoded into the system and later retrieved, although we do not restrict the

analysis to the discrimination problem. Such a question clearly lies at the heart

of quantum communication theory. Back in 1973, Holevo gave a partial answer [1]

by considering the case of asymptotically many rounds of communication using the

quantum system and proving that the relevant quantiﬁer of information in this case,

that is the Shannon mutual information, indeed cannot outperform that of a classical

system of the same dimension.

During the subsequent four decades, researchers followed in Holevo’s footsteps

by proving analogous results – the impossibility for a quantum system to outperform

a classical system of the same dimension in a communication setup – for diﬀerent

setups, and accordingly diﬀerent quantiﬁers of information. Among the overwhelm-

ing amount of works in this direction, it seems particularly relevant to mention a

result [2] from 2007 by Elron and Eldar, that can be considered a precursor of the

breakthrough that was to come only a few years later. Elron and Eldar showed that

no quantum system can outperform a classical system of equal dimension in any

discrimination scenario (that is to say, technically, for linear games with diagonal

payoﬀ matrix). Still, many communication setups are not instances of a discrim-

ination problem, and clearly whether or not a quantum system can outperform a

classical one depends on the speciﬁc case. Or does it?

The anticipated groundbreaking result is the answer to this question in the most

general case, given by Frenkel and Weiner [3] in 2015. But before proceeding, it is

time to introduce some notation. A physical system Sof linear dimension `∈N

can be represented by a triple (S,E,·), where S ⊆ R`and E ⊆ R`are the set of

admissible states and eﬀects, respectively, and ·denotes an inner product in R`.

The probability of measuring the eﬀect π∈ E given the state ρ∈ S is given by

ρ·π, and the eﬀect that gives unit probability for any state is called unit eﬀect. For

any d∈N, we denote the d-dimensional classical and quantum systems by Cdand

Michele Dall’Arno: dallarno.michele@yukawa.kyoto-u.ac.jp, Report number: YITP-22-129

arXiv:2210.15210v1 [quant-ph] 27 Oct 2022

Qd, respectively. That is, for Cdone has that Sis the (d−1)-dimensional regular

simplex, while for Qdone has that Sis the set of d×d(vectorized) density matrices;

in either case, ·is the usual dot product and Eis the set induced by the requirement

that probabilities are non-negative.

Given two ﬁnite alphabets Xand Ywith cardinality mand n, respectively,

an encoding ρis a map from Xto S, while a decoding πis a map from Yto

E, such that Py∈Y π(y)is the unit eﬀect. We consider the set Pm→n

Sof input-

output conditional probability distributions pthat can be generated by system S

with shared randomness. That is, pis an element of Pm→n

Sif and only if there exists

a family {ρλ}λof encodings and a family {πλ}λof decodings such that

p(x|y) = X

q(λ)ρλ(x)·πλ(y),

for some probability distribution q.

We are now in a position to introduce the main quantity of interest in our anal-

ysis. The signaling dimension [4] of a physical system Sis the minimum dimension

of a classical channel that can reproduce the set of input-output correlations (or

probability range) attainable by system S, that is

sign.dim (S) := min

d∈Nds.t. Pm→n

S⊆ Pm→n

Cd,∀m, n ∈N.

Clearly, whether or not there exists a communication setup in which a quantum

channel can outperform a classical channel of equal dimension can be conveniently

reframed in terms of the signaling dimension; in this sense, the signaling dimen-

sion summarizes the structure of the entire set of input-output correlations that is

consistent with a given system in a single scalar quantity. This is in stark contrast

with previous approaches addressing the same problem that were based on speciﬁc

choices of a witness, that is, that investigated the correlation space along a single

direction only.

Equipped with the deﬁnition of the signaling dimension, we can now announce

the aforementioned breakthrough [3] achieved by Frenkel and Weiner in 2015, that

consisted in proving that

sign.dim (Qd) = d, ∀d∈N,

or, in words, no quantum channel can outperform a classical channel of equal di-

mension in any communication game. Frenkel and Weiner obtained this powerful

result by adopting graph theoretic techniques and mixed discriminants that, to the

best of our knowledge, were a novelty within quantum information theory at that

time. As a consequence of their remarkable ﬁnding, in the same work the authors

also derived a combinatorial Holevo-like bound that turns out to be tighter that the

original in at least some scenarios.

Once it became clear that quantum systems are equivalent to classical systems of

equal dimension in any communication setup, the question immediately arose how

unique quantum theory is in this respect within the set of physical theories or, more

accurately, generalized probabilistic theories. Since for arbitrary systems – that is,

systems other than classical or quantum – there is no concept of “dimension”, to

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摘要：

ThesignalingdimensionofphysicalsystemsMicheleDall'Arno1,21YukawaInstituteforTheoreticalPhysics,KyotoUniversity,Sakyo-ku,Kyoto,606-8502,Japan2FacultyofEducationandIntegratedArtsandSciences,WasedaUniversity,Shinjuku-ku,Tokyo,169-8050,JapanThisisaPerspectiveon\Classicalsimulationsofcommunicationchannel...

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