Advantage of Hardy’s Nonlocal Correlation in Reverse Zero-Error Channel Coding

The pioneering work of J. S. Bell establishes one of the most striking departures of quantum theory from the profound classical worldview of local-realism Bell1966 (see also Mermin1993 ; Brunner2014 ). Violation of a Bell-type inequality, as demonstrated in several milestone experiments Clauser1969 ; Freedman1972 ; Aspect1981 ; Aspect1982 ; Zukowski1993 ; Weihs1998 , endorses nonlocal nature of the quantum world. Apart from its foundational implications, Bell nonlocality has also been identified as a useful resource for several practical tasks, such as device-independent cryptography Ekert1991 ; Barrett2005 ; Vazirani2014 , reduction of communication complexity Buhrman2010 , and device-independent randomness certification & amplification Pironio2010 ; Cavalcanti2012 ; Chaturvedi2015 ; Colbeck2012 .

A quite popular technique, known by the name ’nonlocality without inequality’ proof, is often used to establish the nonlocal behaviour of quantum theory. Unlike the Bell-type inequalities, where statistics of many events are collected, these proofs focus on a single event whose occurrence shows the incompatibility of quantum theory with the notion of local-realism. While the first proof of this kind for tripartite quantum systems is due to Greenberger-Horne-Zeilinger Greenberger1989 (see also Greenberger1990 ), for bipartite systems, such proof was first proposed by Hardy Hardy1992 , which is considered to be “simpler and more compelling than the arguments that underlie the derivation of Bell-CH inequality" Mermin1994 ; Clauser1974 . More recently, Hardy-type nonlocality proofs have also been shown to be useful in several practical tasks Das2013 ; Mukherjee2015 ; Li2015 ; Ramanathan2018 ; Rai2021 ; Rai2022 .

In this work, we report a novel application of Hardy’s nonlocal correlation in the simplest communication scenario. We show that Hardy’s nonlocal correlation shared between two distant parties can empower the communication utility of a perfect classical channel. This is quite striking, as such a correlation by itself cannot be used for information transfer, which otherwise will imply violation of the no signalling (NS) principle. We also argue that the advantage reported in this work is different than the advantage of nonlocal correlations known in communication complexity tasks Buhrman2010 .

We consider a guessing game played between two distant players – a sender and a receiver. First, we show that the expected collaborative payoff of this game cannot be positive whenever only $1$-bit classical communication is allowed from the sender to the receiver, who otherwise can share an unlimited amount of classical correlation between them. Interestingly, assistance of Hardy’s nonlocal correlation to the same classical channel can ensure a strictly positive payoff, establishing a nontrivial advantage of Hardy’s correlation in communication tasks. The advantage can be better understood in the framework of correlation-assisted reverse zero-error coding scenario Cubitt2011 , where the aim is to simulate a higher input-output noisy classical channel by a lower input-output identity channel in the presence of NS correlations. In this scenario, the aforesaid game makes Hardy’s nonlocal correlations special as we show that among all the $2$-input-$2$-output NS correlations only those exhibiting Hardy’s nonlocality can ensure a positive payoff. This further motivates us to show that pure entangled states which are not maximally entangled are preferable over the maximally entangled one for simulating certain noisy classical channels. At the end we show that similar sort of results can also be obtained by considering a generalization of Hardy’s nonlocality argument as proposed by Cabello Cabello2002 . The present work, therefore, establishes that the comparison of entanglement in quantum states even for bipartite systems are quite complex when classical communication among distant parties are treated as costly.

The manuscript is organized as follows. In Section II we introduce a two-party guessing game which we call the Distributed Mine-Hunting game. Here we also discuss the reverse zero-error channel coding setup. In Section III we present our main results, and in Section IV we discuss implications of our results along with future outlooks.

The game involves two distant players, Alice and Bob, and a referee named Charlie. In each run of the game, Charlie provides four closed boxes numbered 1 to 4 to Bob, who has to open one of these boxes. Some of the boxes contain a bomb that will explode upon opening it. Among the boxes that do not contain the bomb, some are empty, some contain a dollar bill, and others may even prompt Bob to pay a dollar bill to Charlie. In each run of the game, Charlie uniformly and randomly picks one of four different arrangements of these boxes, as shown in Fig.1. Charlie then informs Alice about the arrangement of the boxes in that particular run, and Alice tries to help Bob in choosing a box. However, only $1$-bit of classical communication is allowed from Alice to Bob, which may be further assisted with NS correlations shared between them a priori. From now on, we will refer to this as the Distributed Mine-Hunting (DMH) game.

Let $\Omega_{n_{c}+SR}(|\mathcal{M}|,|\mathcal{Z}|)$ denote the set of strategy matrices obtained when $n$-bits of classical communication and an unlimited amount of shared randomness are available. The set $\Omega_{n_{c}+SR}(|\mathcal{M}|,|\mathcal{Z}|)$ forms a polytope with extreme points $S^{e}$’s representing strategy matrices obtained through deterministic encoding $\mathbb{E}:\mathcal{M}\to\{0,1\}^{n}$ at Alice’s end and deterministic decoding $\mathbb{D}:\{0,1\}^{n}\to\mathcal{Z}$ at Bob’s end Frenkel2015 (see also Dallarno2017 ). Our first technical result is to limit the optimal success probability of the game in Eq.(6) for classical strategies with limited communication.

We now consider the scenario where the communication line from Alice to Bob is aided with a generic NS correlation $\mathbb{P}\equiv\{p(a,b|x,y)\}$, where $p(a,b|x,y)\geq 0,\leavevmode\nobreak\ \forall\leavevmode\nobreak\ a,b,x,y\leavevmode\nobreak\ \&\leavevmode\nobreak\ \sum_{a,b}p(a,b|x,y)=1,\leavevmode\nobreak\ \forall\leavevmode\nobreak\ x,y$. Here $p(a,b|x,y)$ denotes the probability of obtaining outcome $a\in\mathcal{A}$ at Alice’s end and $b\in\mathcal{B}$ at Bob’s end for their respective inputs $x\in\mathcal{X}$ and $y\in\mathcal{Y}$. Classical correlations that allow a local-realistic description, $p(a,b|x,y)=\int_{\Lambda}\mu(\lambda)p(a|x,\lambda)p(b|y,\lambda)d\lambda$, forms a strict subset (a sub polytope) of the NS polytope; here $\lambda\in\Lambda$ is some classical variable shared between Alice and Bob and $\mu(\lambda)$ is a probability distribution on $\Lambda$ Brunner2014 . Quite surprisingly, entangled quantum states can lead to correlations that are not local-realistic and nonlocality of those correlations can be certified through violation of some Bell-type inequalities Bell1966 . Given such an NS correlation (possibly nonlocal) as an assistance to $1$-bit classical communication from Alice to Bob, the general strategy to play a game $\mathcal{G}$ is described in Fig.2.

For the binary input-output case, Lucien Hardy proposed an elegant argument according to which any NS correlation $\mathbb{H}\equiv\{h(a,b|x,y)\}$ satisfying the constraints

must be nonlocal in nature Hardy1992 (for the sake of completeness we analyze the argument in Appendix-A). A correlation exhibiting Hardy’s nonlocality is termed as Hardy’s nonlocal correlation. Our next result proves a nontrivial advantage of Hardy correlation while playing the DMH game.

This in turn establishes that the $4$-input $4$-output noisy channel $S_{\mathbb{H}}$ can be perfectly simulated by the $2$-dimensional identity channel (i.e. a $1$-bit perfect channel from Alice to Bob) when assisted with Hardy’s nonlocal correlation. However, as follows form Theorem 1, assistance of arbitrary amount of shared randomness fails to achieve the goal. A correlation with Hardy success $h_{0}$ yields Clauser-Horne-Shimony-Holt (CHSH) Clauser1969(1) value $2+4\leavevmode\nobreak\ h_{0}$ Cereceda2000 . Accordingly, the CHSH value corresponding to the optimal quantum Hardy correlation is strictly less than the Cirel’son bound Rabelo2012 ; Cirelson1980 . Therefore, a natural question is whether other nonlocal quantum correlations can lead to better success in the DMH game. Our next result answers this question in negation.

Although it is known that a two-qubit maximally entangled state does not exhibit Hardy’s nonlocality Goldstein1994 ; Jordan1994 , still Theorem 3 is not sufficient to make a claim that such a state shared between Alice and Bob cannot lead to a strategy yielding strictly positive payoff in DMH game. Increasing the cardinality of the input-output sets $\mathcal{X},\leavevmode\nobreak\ \mathcal{Y},\leavevmode\nobreak\ \mathcal{A},\leavevmode\nobreak\ \mathcal{B}$ Alice and Bob can generate a more general NS correlation and then try to utilize this correlation to assist the $1$-cbit channel to obtain a nonzero payoff in DMH game. However, our next result proves a no-go to this aim.

This theorem has an interesting implication. It shows that there exists a communication task wherein a non-maximally pure entangled state can be preferable over the maximally entangled one even when the entanglement of the former is vanishingly zero. More formally we can deduce the following corollary.

Here $\ket{\phi^{+}}:=(\ket{00}+\ket{11})/\sqrt{2}$ and $\Omega_{1_{c}+SR+\ket{\chi}}(4,4)$ denotes the convex set of strategy matrices simulable with $1$-cbit communication from Alice to Bob when the communication channel is further assisted with preshared quantum state $\ket{\chi}$ and unlimited shared randomness. The Corollary 1 follows when results of Theorems 2 & 4 are combined with the fact that all non maximally pure entangled state exhibits Hardy’s nonlocalityGoldstein1994 . More precisely, for every non maximally entangled state $\ket{\psi}\in\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ there exists a noisy channel of the form $S_{\mathbb{H}}$, that can be perfectly simulated with $1$-bit perfect classical channel when assisted with the state $\ket{\psi}$, but not with $\ket{\phi^{+}}$. A more detailed discussion on the implications of theses results is presented in Appendix-B.

The seminal quantum superdense coding protocol is worth mentioning as it shows that quantum entanglement, pre-shared between a sender and a receiver, can increase the classical communication capacity of a quantum system Bennett1992 (see also Bennett1999 ). While in quantum superdense coding protocol a quantum channel is considered, here we show that quantum entanglement can even empower the communication utility of a perfect classical channel. As already mentioned, such an advantage can be better understood in zero-error and reverse zero-error communication set-up. While such an advantage of quantum entanglement is already known (see Proposition $21$ in Cubitt2011 and see also Cubitt2010 ; Frenkel2022 ; Patra2022 ), the full picture of entanglement assistance is not well understood. Our Theorem 4 proves a nontrivial result to this direction. It shows that in the correlation-assisted reverse zero-error coding scenario, there exists noisy channel simulation tasks wherein non-maximally entangled states, with arbitrarily less amount of entanglement, are preferable over maximally entangled state. In the resource theory of quantum entanglement Plenio2007 , where local operations and classical communication (LOCC) are considered to be free, a maximally entangled state is more useful than a non-maximally entangled one, and a deterministic LOCC transformation is always possible from the former to the latter Nielsen1999 . Our work, however, establishes that within the operational paradigm of local operations and limited classical communication the structure of entangled resources are quite complex to characterize.

A nonlocal advantage is also known in a variant of the communication scenario known as the communication complexity problem Buhrman2010 . In such a scenario, Bob’s goal is not to determine Alice’s data $\mathcal{M}$ but to determine some information that is a function of $\mathcal{M}$ in a way that may depend on the other data $\mathcal{N}$ that resides with Bob while $\mathcal{N}$ is unknown to Alice. In that sense, our scenario is closer to the standard framework of Shannon Shannon1948 (and considered by Holevo in quantum set up Holevo1973 ), where at Bob’s end, no further data set $\mathcal{N}$ is considered, albeit in single-shot setup.

In conclusion, the present work establishes exotic uses of quantum entanglement in zero-error information theory Shannon1956 (see also Korner1998 ) whose motivation arises from the fact that in many real-world critical applications, no errors can be tolerated, and in practice, the communication channel can only be available for a finite number of times. In particular, we show that quantum correlations exhibiting Hardy’s nonlocality can empower the communication utility of a perfect classical communication channel. In Appendix C we show that similar results can be obtained by considering the generalization of Hardy’s nonlocality argument as proposed by Cabello Cabello2002 . Our work also motivates many questions for future study. For instance, it would be interesting to see whether any nonlocal correlation can be made useful as a communication resource in the sense discussed here. It will also be interesting to see whether maximally entangled states of higher dimensions provide some advantage in the DMH game. More generally, characterizing the set $\Omega_{n_{c}+SR+\chi}(|\mathcal{M}|,|\mathcal{Z}|)$ for an arbitrary quantum state $\ket{\chi}$ would be very interesting.