# The signaling dimension of physical systems

This is a Perspective on "Classical simulations of communication channels" by Péter E. Frenkel, published in Quantum 6, 751 (2022).

By Michele Dall'Arno (Yukawa Institute for Theoretical Physics, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan and Faculty of Education and Integrated Arts and Sciences, Waseda University, Shinjuku-ku, Tokyo, 169-8050, Japan).

The question we start our analysis from is whether a quantum system can outperform a classical system of the same dimension for some communication scenarios. By “communication scenario”, we mean the usual setup in which some classical message 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 overwhelming 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 discrimination problem, and clearly whether or not a quantum system can outperform a classical one depends on the specific case. Or does it?

The anticipated ground-breaking 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 $S$ of linear dimension $\ell \in \mathbb{N}$ can be represented by a triple $(\mathcal{S}, \mathcal{E}, \cdot)$, where $\mathcal{S} \subseteq \mathbb{R}^\ell$ and $\mathcal{E} \subseteq \mathbb{R}^\ell$ are the set of admissible states and effects, respectively, and $\cdot$ denotes an inner product in $\mathbb{R}^\ell$. The probability of measuring the effect $\pi \in \mathcal{E}$ given the state $\rho \in \mathcal{S}$ is given by $\rho \cdot \pi$, and the effect that gives unit probability for any state is called unit effect. For any $d \in \mathbb{N}$, we denote the $d$-dimensional classical and quantum systems by $C_d$ and $Q_d$, respectively. That is, for $C_d$ one has that $\mathcal{S}$ is the $(d – 1)$-dimensional regular simplex, while for $Q_d$ one has that $\mathcal{S}$ is the set of $d \times d$ (vectorized) density matrices; in either case, $\cdot$ is the usual dot product and $\mathcal{E}$ is the set induced
by the requirement that probabilities are non-negative.

Given two finite alphabets $\mathcal{X}$ and $\mathcal{Y}$ with cardinality $m$ and $n$, respectively, an encoding $\boldsymbol{\rho}$ is a map from $\mathcal{X}$ to $\mathcal{S}$, while a decoding $\boldsymbol{\pi}$ is a map from $\mathcal{Y}$ to $\mathcal{E}$, such that $\sum_{y \in \mathcal{Y}} \boldsymbol{\pi} \left( y \right)$ is the unit effect. We consider the set $\mathcal{P}^{m \to n}_S$ of input-output conditional probability distributions $p$ that can be generated by system $S$ with shared randomness. That is, $p$ is an element of $\mathcal{P}^{m \to n}_S$ if and only if there exists a family $\{\boldsymbol{\rho}_\lambda \}_\lambda$ of encodings and a family $\{ \boldsymbol{\pi}_\lambda \}_\lambda$ of decodings such that
\nonumber
p \left(x | y \right) = \sum_{\lambda} q \left( \lambda\right) \boldsymbol{\rho }_{\lambda} \left( x \right)\cdot \boldsymbol{\pi}_{\lambda} \left( y \right),

for some probability distribution $q$.

We are now in a position to introduce the main quantity of interest in our analysis. The \textit{signaling dimension} [4] of a physical system $S$ is 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
\nonumber
\operatorname{sign.dim} \left( S \right) := \min_{d \in
\mathbb{N}} d \qquad \textrm{s.t.} \qquad \mathcal{P}^{m
\to n}_S \subseteq \mathcal{P}^{m \to n}_{C_d}, \;
\forall m, n \in \mathbb{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 dimension 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
\nonumber
\operatorname{sign.dim} \left( Q_d \right) = d, \; \forall
d \in \mathbb{N},

or, in words, no quantum channel can outperform a classical channel of equal dimension in \textit{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 than 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 answer this question a criterion to meaningfully compare systems belonging to different theories had to be derived. This was achieved by observing that, by definition, the signaling dimension of the composition of multiple classical systems is just given by the product of the signaling dimensions of the component systems. Informally, this is to say that using classical systems sequentially allows for the same amount of communication as using them in parallel. As a formula this statement reads
\nonumber
\operatorname{sign.dim} \left( S_0 \otimes S_1 \right) \le
\operatorname{sign.dim} \left( S_0 \right)
\operatorname{sign.dim} \left( S_1 \right),

where $\otimes$ denotes the composition of systems (not necessarily given by the tensor product). This seemingly obvious fact, formally given the name of $\textit{no-hypersignaling principle}$ [4], also holds true for quantum theory, but only as a consequence of the aforementioned result [3] by Frenkel and Weiner, and provides the promised criterion to meaningfully compare systems belonging to different theories.

It must be remarked that the no-hypersignaling principle constrains the time-like correlations that any physical theory can exhibit. This is in contrast with previously known principles, including the no-signaling principle itself, which instead bounds the space-like correlations of any given physical theory. Therefore, it is remarkable to observe that there exist theories [4] that, although equivalent to quantum theory in terms of their space-like correlations, exhibit super-quantum time-like correlations only detectable as violations of the no-hypersignaling principle. Moreover, it was recently shown [5,6] by D’Ariano, Erba, and Perinotti that there exist theories that are locally classical but nevertheless violate the no-hypersignaling principle.

From the resource theoretical point of view, the signaling dimension of a channel — a system can be regarded in this sense as the identity channel — clearly quantifies the cost of classically simulating such a channel. The problem of quantifying the signaling dimension of a given channel has recently drawn considerable attention. While a detailed analysis of the literature is outside the scope of the present perspective, here we just mention a few works: for the case of quantum channels, the works by Hoffmann, Spee, Gühne, and Budroni [7], Dall’Arno, Brandsen, and Buscemi [8], Doolittle and Chitambar [9], and Chitambar, George, Doolittle, and Junge [10]; for entanglement-assisted classical channels, the work by Frenkel and Weiner [11]; and for channels in generalized probabilistic theories, the recent work by Heinosaari, Kerppo, and Leevi Leppäjärvi [12].

Ultimately, the signaling dimension is the result of a comparison between a given generalized probabilistic theory — possibly quantum theory — and classical theory, through their probability ranges. A context in which such a comparison is inherently meaningful is the device-independent approach to quantum information processing, in which classical theory plays the role of the standard reference against which any other theory is compared. This is because classical events — the pressing of a button, the detection of a light bulb lighting up — are assumed as the only events directly accessible by the observer.

Therefore, it comes as no surprise that device-independent tests of quantum channels based on the signaling dimension have been proposed [13,14,15]. Therein, the probability range of a given quantum device is compared with observations in order to rule out, in a device-independent way, that such a device has produced the observations. In this context, Frenkel and Weiner’s result [3] established that, unless one abandons the input-output setup — as done for instance by Gallego, Brunner, Hadley, and Acìn [16] — the identity channels in classical and quantum theories remain indistinguishable. These ideas were further pushed forward [17,18,19] by the same authors in collaboration with Bisio, Tosini, Ho, and Scarani, by developing data-driven techniques for the inference of quantum devices.

Perhaps, one of the most promising research lines originating from the study of the signaling dimension is based on an attempt to unify the aforementioned resource-theoretical and device-independent approaches to quantum information theory. To this end, recently, Dall’Arno, Buscemi, and Scarani laid the foundations [20] of a device-independent approach to resource theory, that is, a scenario in which operationally meaningful majorization relations between resources are established on the basis of observed input-output correlations only.

The new paper [21] by Frenkel, for which this perspective is written, is a timely addition to this rich research landscape. Therein, the author extends the ground-breaking result [3] he obtained almost a decade ago in collaboration with Weiner, thus further pushing the state of the art in the problem of characterizing the signaling dimension. The paper provides bounds on the signaling dimension in terms of another quantifier of information, that is, the information storability [22]. The information storability
quantifies the trade-off between the success probability in the state discrimination and the size $m$ of the alphabet; as a formula:
\nonumber
\operatorname{inf.stor} \left( S \right) :=
\max_{m, \boldsymbol{\rho}, \boldsymbol{\pi}} \sum_{k =
1}^{m} \boldsymbol{\rho} \left( k \right)
\boldsymbol{\pi} \left( k \right),

where the optimization is over any size $m$ and any encoding and decoding. While previous literature on the signaling dimension in general probabilistic theories mostly focused on systems whose state space is a regular polygon, the bounds derived by Frenkel can be used to compute the signaling dimension of regular polyhedrons. Moreover, Frenkel also shows that, when the state space is a ball according to the $k/(k-1)$-norm, the signaling dimension is upper-bounded by $k$, and provides a full characterization of the signaling dimension of several classes of noisy quantum channels.

But perhaps the most significant contribution of Frenkel’s work [21] lies in having pushed forward the mathematical techniques, based on graph theory and mixed discriminants, on which his previous breakthrough [3], in collaboration with Weiner, was based. Recently, these techniques were the topic of a course, held by Frenkel and Weiner, at the Kyoto School on Advanced Topics in Quantum Information and Foundations, whose lectures are freely available online [23]. It is hard not to wonder about the impact these techniques will have in quantum information theory in the years to come.

### ► References

[1] A. S. Holevo. Bounds for the quantity of information transmitted by a quantum communication channel. Problems of Information Transmission, 9, 1973.

[2] N. Elron and Y. C. Eldar. Optimal encoding of classical information in a quantum medium. IEEE Trans. Inf. Theory, 53, 2007. 10.1109/​TIT.2006.894686.
https:/​/​doi.org/​10.1109/​TIT.2006.894686

[3] P. E. Frenkel and M. Weiner. Classical information storage in an n-level quantum system. Comm. Math. Phys., 340, 2015. 10.1007/​s00220-015-2463-0.
https:/​/​doi.org/​10.1007/​s00220-015-2463-0

[4] M. Dall'Arno, S. Brandsen, A. Tosini, F. Buscemi, and V. Vedral. No-hypersignaling principle. Phys. Rev. Lett., 119, 2017a. 10.1103/​PhysRevLett.119.020401.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.020401

[5] G. M. D'Ariano, M. Erba, and P. Perinotti. Classicality without local discriminability: Decoupling entanglement and complementarity. Phys. Rev. A, 102, 2020a. 10.1103/​PhysRevA.102.052216.
https:/​/​doi.org/​10.1103/​PhysRevA.102.052216

[6] G. M. D'Ariano, M. Erba, and P. Perinotti. Classical theories with entanglement. Phys. Rev. A, 101, 2020b. 10.1103/​PhysRevA.101.042118.
https:/​/​doi.org/​10.1103/​PhysRevA.101.042118

[7] J. Hoffmann, C. Spee O. Gühne, and C. Budroni. Structure of temporal correlations of a qubit. New J. Phys., 20, 2018. 10.1088/​1367-2630/​aae87f.
https:/​/​doi.org/​10.1088/​1367-2630/​aae87f

[8] M. Dall'Arno, S. Brandsen, and F. Buscemi. Explicit construction of optimal witnesses for input-output correlations attainable by quantum channels. Open Syst. Inf. Dyn., 27, 2020a. 10.1142/​S1230161220500171.
https:/​/​doi.org/​10.1142/​S1230161220500171

[9] B. Doolittle and E. Chitambar. Certifying the classical simulation cost of a quantum channel. Phys. Rev. Research, 3, 2021. 10.1103/​PhysRevResearch.3.043073.
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.043073

[10] E. Chitambar, I. George, B. Doolittle, and M. Junge. The communication value of a quantum channel. arXiv:2109.11144, 2021. 10.48550/​arXiv.2109.11144.
https:/​/​doi.org/​10.48550/​arXiv.2109.11144
arXiv:2109.11144

[11] P. E. Frenkel and M. Weiner. On entanglement assistance to a noiseless classical channel. Quantum, 6, 2022. 10.22331/​q-2022-03-01-662.
https:/​/​doi.org/​10.22331/​q-2022-03-01-662

[12] T. Heinosaari, O. Kerppo, and L. Leppäjärvi. Communication tasks in operational theories. J. Phys. A: Math. Theor., 53, 2020. 10.1088/​1751-8121/​abb5dc.
https:/​/​doi.org/​10.1088/​1751-8121/​abb5dc

[13] M. Dall'Arno, S. Brandsen, and F. Buscemi. Device-independent tests of quantum channels. Proc. R. Soc. A, 473, 2017b. 10.1098/​rspa.2016.0721.
https:/​/​doi.org/​10.1098/​rspa.2016.0721

[14] M. Dall'Arno, S. Brandsen, F. Buscemi, and V. Vedral. Device-independent tests of quantum measurements. Phys. Rev. Lett., 118, 2017c. 10.1103/​PhysRevLett.118.250501.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.250501

[15] M. Dall'Arno. Device-independent tests of quantum states. Phys. Rev. A, 99, 2019. 10.1103/​PhysRevA.99.052353.
https:/​/​doi.org/​10.1103/​PhysRevA.99.052353

[16] R. Gallego, N. Brunner, C. Hadley, and A. Acìn. Device-independent tests of classical and quantum dimensions. Phys. Rev. Lett., 105, 2010. 10.1103/​PhysRevLett.105.230501.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.230501

[17] F. Buscemi and M. Dall'Arno. Data-driven inference of physical devices: Theory and implementation. New J. Phys., 21, 2019. 10.1088/​1367-2630/​ab5003.
https:/​/​doi.org/​10.1088/​1367-2630/​ab5003

[18] M. Dall'Arno, F. Buscemi, A. Bisio, and A. Tosini. Data-driven inference, reconstruction, and observational completeness of quantum devices. Phys. Rev. A, 102, 2020b. 10.1103/​PhysRevA.102.062407.
https:/​/​doi.org/​10.1103/​PhysRevA.102.062407

[19] M. Dall'Arno, A. Ho, F. Buscemi, and V. Scarani. Data-driven inference and observational completeness of quantum devices. arXiv:1905.04895, 2019.
arXiv:1905.04895

[20] M. Dall'Arno, F. Buscemi, and V. Scarani. Extension of the alberti-ulhmann criterion beyond qubit dichotomies. Quantum, 4, 2020c. 10.22331/​q-2020-02-20-233.
https:/​/​doi.org/​10.22331/​q-2020-02-20-233

[21] P. E. Frenkel. Classical simulations of communication channels. Quantum, 6, 2022. 10.22331/​q-2022-06-29-751.
https:/​/​doi.org/​10.22331/​q-2022-06-29-751

[22] K. Matsumoto and G. Kimura. Information-induced asimmetry of state space in view of general probabilistic theories. arXiv.1802.01162, 2018. 10.48550/​arXiv.1802.01162.
https:/​/​doi.org/​10.48550/​arXiv.1802.01162

[23] P. E. Frenkel and M. Weiner. Kyoto school on advanced topics in quantum information and foundations, 2021. URL https:/​/​www2.yukawa.kyoto-u.ac.jp/​ atqif21/​.
https:/​/​www2.yukawa.kyoto-u.ac.jp/​~atqif21/​

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