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IEEE Conference on Decision and Control (CDC)

Beyond Common Randomness: Quantum Resources in Decentralized Control

Shashank A. Deshpande, Ankur A. Kulkarni

Published Dec 13, 2023• 1 citation
Publisher

Abstract

Ananthram and Borkar [1] showed that there exist strategies that are consistent with the requirements of a decentralized information structure but are unattainable through the use of common randomness. This opens the question of discovering physically realisable mechanisms that provide access to this region of the strategic space. In our previous work we introduced a class of quantum strategies that allow such access in a two-agent setting. In this paper, we consider the problem of optimal allocation of a$k$-partite quantum resource amongst$n$agents,$k < n$. We study the problem of decentralized estimation of a binary source by agents that are informed through independent binary symmetric channels, and face a cost that is homogeneous in their actions. We show a$k$-partite quantum resource produces the maximum advantage over classical strategies when allocated to the agents with the$k$most reliable channels.

DOI
10.1109/cdc49753.2023.10384066

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Ananthram and Borkar [1] showed that there exist strategies that are consistent with the requirements of a decentralized information structure but are unattainable through the use of common randomness. This opens the question of discovering physically realisable mechanisms that provide access to this region of the strategic space. In our previous work we introduced a class of quantum strategies that allow such access in a two-agent setting. In this paper, we consider the problem of optimal allocation of akkk-partite quantum resource amongstnnnagents,k<nk < nk<n. We study the problem of decentralized estimation of a binary source by agents that are informed through independent binary symmetric channels, and face a cost that is homogeneous in their actions. We show akkk-partite quantum resource produces the maximum advantage over classical strategies when allocated to the agents with thekkkmost reliable channels.
It is known in the context of decentralised control that there exist control strategies consistent with the requirements of a given information structure, yet physically unimplementable through any amount of passive common randomness. This imposes a natural set of limitations on what is achievable through common randomness in both cooperative and competitive settings. We show that it is possible to breach these limitations with the use of quantum-physical architectures. In particular, we present a class of stochastic strategies that leverage quantum entanglement to produce strategic distributions which compose a strict superclass of strategies implemented through passive common randomness. We investigate numerically, the `quantum advantage' offered by this new class over a parametric family of cooperative decision problems with static information structure. We demonstrate through variations across the parametric family that fundamental decision theoretic elements such as information and the cost determine the manifestation of quantum advantage in a given control problem. Our work motivates a novel decision and control paradigm with an enlarged space of control policies achievable by means of quantum architectures.
We have shown that entanglement assisted stochastic strategies allow access to strategic measures beyond the classically correlated measures accessible through passive common randomness, and thus attain a quantum advantage in decentralised control. In this two part series of articles, we investigate the decision theoretic origins of the quantum advantage within a broad superstructure of problem classes. Each class in our binary team superstructure corresponds to a parametric family of cost functions with a distinct algebraic structure. In this part, identify the only problem classes that benefit from quantum strategies. We find that these cost structures admit a special decision-theoretic feature -- `the coordination dilemma'. Our analysis hence reveals some intuition towards the utility of non-local quantum correlations in decentralised control.
This work considers the non-interactive source simulation problem (NISS). In the standard NISS scenario, a pair of distributed agents observe a distributed binary memoryless source (Xd,YdX^{d}, Y^{d}Xd,Yd) generated based on the joint distributionPX,YP_{X, Y}PX,Y​. The agents wish to produce a pair of discrete random variables(Ud,Vd)\left(U_{d}, V_{d}\right)(Ud​,Vd​)with joint distributionPUd,VdP_{U_{d}, V_{d}}PUd​,Vd​​, such thatPUd,VdP_{U_{d}, V_{d}}PUd​,Vd​​converges in total variation distance to a target distributionQU,VQ_{U, V}QU,V​. Two variations of the standard NISS scenario are considered. In the first variation, in addition to(Xd,Yd)\left(X^{d}, Y^{d}\right)(Xd,Yd), the agents have access to a shared Bell state. They each measure their respective state, using a measurement of their choice, and use its classical output along with(Xd,Yd)\left(X^{d}, Y^{d}\right)(Xd,Yd)to simulate the target distribution. This scenario is called the entanglement-assisted NISS (EA-NISS). In the second variation, the agents have access to a classical common random bitZZZ, in addition to (Xd,YdX^{d}, Y^{d}Xd,Yd). This scenario is called the common randomness NISS (CR-NISS). It is shown that for binary output NISS scenarios, the set of simulatable distributions for EA-NISS and CR-NISS are equal with each other. Hence, there is no quantum advantage in these EA-NISS scenarios. For non-binary output NISS, it is shown that in a specific class of scenarios, the set of CR-NISS simulatable distributions forms a measure zero subset of EA-NISS simulatable distributions. A numerical example is provided, where the set of EA-NISS simulatable distributions strictly contains the set of CR-NISS simulatable distributions. This shows that there is a quantum advantage in non-binary output EA-NISS.
Consider a decentralized partially-observed Markov decision problem (POMDP) with multiple cooperative agents aiming to maximize a long-term-average reward criterion. We observe that the availability, at a fixed rate, of entangled states of a product quantum system between the agents, where each agent has access to one of the component systems, can result in strictly improved performance even compared to the scenario where common randomness is provided to the agents, i.e. there is a quantum advantage in decentralized control. This observation comes from a simple reinterpretation of the conclusions of the well-known Mermin-Peres square, which underpins the Mermin-Peres game. While quantum advantage has been demonstrated earlier in one-shot team problems of this kind, it is notable that there are examples where there is a quantum advantage for the one-shot criterion but it disappears in the dynamical scenario. The presence of a quantum advantage in dynamical scenarios is thus seen to be a novel finding relative to the current state of knowledge about the achievable performance in decentralized control problems. This paper is dedicated to the memory of Pravin P. Varaiya.
Randomness appears both in classical stochastic physics and in quantum mechanics. A number of communication models and other information protocols entail sharing randomness among distant parties. Here we report a computational scenario of shared randomness processing where quantum sources manifest clear-cut precedence over the corresponding classical counterparts. The advantage is established in a resource theoretic set-up which we formalize in this work. The classical mutual information turns out to be a faithful resource quantifier of shared randomness although it does not sufficiently characterize all possible transitions among the resources. Quantum theory exhibits advantage in shared randomness processing in the sense that the set of resourceful states obtained under free operation from a bipartite quantum system is strictly larger than the set obtained from an analogous classical system. Interestingly, the quantum advantage can be manifested in the optimal payoff of a two players co-operative game -- the `non-monopolize social subsidy' game. We also show that the quantum states resulting to the desired advantage necessitates nonclassicality in the form of quantum discord. We then address the task of distributing shared randomness between two parties. Quite interestingly, it turns out that noisy quantum channel can exhibit advantage even when it has zero quantum capacity and classical capacity much less than the corresponding classical channel. The imperfect channel examples facilitates and welcomes experiment to achieve the reported quantum advantage with presently available quantum devices.
Quantum advantage is notoriously hard to find and even harder to prove. For example the class of functions computable with classical physics exactly coincides with the class computable quantum mechanically. It is strongly believed, but not proven, that quantum computing provides exponential speed-up for a range of problems, such as factoring. Here we address a computational scenario of randomness processing in which quantum theory provably yields, not only resource reduction over classical stochastic physics, but a strictly larger class of problems which can be solved. Beyond new foundational insights into the nature and malleability of randomness, and the distinction between quantum and classical information, these results also offer the potential of developing classically intractable simulations with currently accessible quantum technologies. Although quantum alternatives of various classical tasks are considered advantageous, this is typically extremely difficult to concretely prove. Here, the authors show that a quantum approach to randomness processing provides a reduction in resources and a larger class of solvable problems.