In space-like separated experiments and other scenarios where multiple parties share a classical common cause but no cause-effect relations, quantum theory allows a variety of nonsignaling resources which are useful for distributed quantum information processing. These include quantum states, nonlocal boxes, steering assemblages, teleportages, channel steering assemblages, and so on. Such resources are often studied using nonlocal games, semiquantum games, entanglement-witnesses, teleportation experiments, and similar tasks. We introduce a unifying framework which subsumes the full range of nonsignaling resources, as well as the games and experiments which probe them, into a common resource theory: that of local operations and shared randomness (LOSR). Crucially, we allow these LOSR operations to locally change the type of a resource, so that players can convert resources of any type into resources of any other type, and in particular into strategies for the specific type of game they are playing. We then prove several theorems relating resources and games of different types. These theorems generalize a number of seminal results from the literature, and can be applied to lessen the assumptions needed to characterize the nonclassicality of resources. As just one example, we prove that semiquantum games are able to perfectly characterize the LOSR nonclassicality of every resource of any type (not just quantum states, as was previously shown). As a consequence, we show that any resource can be characterized in a measurement-device-independent manner.
The statistics of local measurements performed on certain entangled states can be reproduced using a local hidden variable (LHV) model. While all known models make use of an infinite amount of shared randomness---the physical relevance of which is questionable---we show that essentially all entangled states admitting a LHV model can be simulated with finite shared randomness. Our most economical model simulates noisy two-qubit Werner states using only 3.58 bits of shared randomness. We also discuss the case of POVMs, and the simulation of nonlocal states with finite shared randomness and finit…
The study of local models using finite shared randomness originates from the consideration about the cost of classically simulating entanglement in composite quantum systems. We construct explicitly two families of local-hidden-state (LHS) models for T-states, by mapping the problem to the Werner state. The continuous decreasing of shared randomness along with entanglement, as the anisotropy increases, can be observed in the one from the most economical model for the Werner state. The construction of the one for separable states shows that the separable boundary of T-states can be generated fr…
It has been shown that the conditional probability distributions obtained by performing measurements on an uncharacterized physical system can be used to infer its underlying dimension in a device-independent way both in the classical and quantum setting. We analyze several aspects of the structure of the sets of probability distributions corresponding to a certain dimension taking into account whether shared randomness is available as a resource or not. We first consider the so-called prepare-and-measure scenario. We show that quantumness and shared randomness are not comparable resources. Th…
Within the simultaneous message passing model of communication complexity, under a public-coin assumption, we derive the minimum achievable worst-case error probability of a classical fingerprinting protocol with one-sided error. We then present entanglement-assisted quantum fingerprinting protocols attaining worst-case error probabilities that breach this bound.
Sharing correlated random variables is a resource for a number of information theoretic tasks such as privacy amplification, simultaneous message passing, secret sharing and many more. In this article, we show that to establish such a resource called shared randomness, quantum systems provide an advantage over their classical counterpart. Precisely, we show that appropriate albeit fixed measurements on a shared two-qubit state can generate correlations which cannot be obtained from any possible state on two classical bits. In a resource theoretic set-up, this feature of quantum systems can be …
We study shared randomness in the context of multi-party number-in-hand communication protocols in the simultaneous message passing model. We show that with three or more players, shared randomness exhibits new interesting properties that have no direct analogues in the two-party case. First, we demonstrate a hierarchy of modes of shared randomness, with the usual shared randomness where all parties access the same random string as the strongest form in the hierarchy. We show exponential separations between its levels, and some of our bounds may be of independent interest. For example, we show…
Enhancing Entanglement Purification with Shared Randomness
Pritzker School of Molecular Engineering, University of Chicago（芝加哥大学普里茨克分子工程学院）；Department of Electrical and Computer Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校电气与计算机工程系）；IonQ Inc.（IonQ公司）
We consider several models of 1-round classical and quantum communication, some of these models have not been defined before. We "almost separate" the models of simultaneous quantum message passing with shared entanglement and the model of simultaneous quantum message passing with shared randomness. We define a relation which can be efficiently exactly solved in the first model but cannot be solved efficiently, either exactly or in 0-error setup in the second model. In fact, our relation is exactly solvable even in a more restricted model of simultaneous classical message passing with shared e…
Multi-party local quantum operations with shared quantum entanglement or shared classical randomness are studied. The following facts are established: (i) There is a ball of local operations with shared randomness lying within the space spanned by the no-signaling operations and centred at the completely noisy channel. (ii) The existence of the ball of local operations with shared randomness is employed to prove that the weak membership problem for local operations with shared entanglement is strongly NP-hard. (iii) Local operations with shared entanglement are characterized in terms of linear…
In quantum cryptography, device-independent (DI) protocols can be certified secure without requiring assumptions about the inner workings of the devices used to perform the protocol. In order to display nonlocality, which is an essential feature in DI protocols, the device must consist of at least two separate components sharing entanglement. This raises a fundamental question: how much entanglement is needed to run such DI protocols? We present a two-device protocol for DI random number generation (DIRNG) which produces approximately $n$ bits of randomness starting from $n$ pairs of arbitrari…
We consider a communication method, where the sender encodes n classical bits into 1 qubit and sends it to the receiver who performs a certain measurement depending on which of the initial bits must be recovered. This procedure is called (n,1,p) quantum random access code (QRAC) where p > 1/2 is its success probability. It is known that (2,1,0.85) and (3,1,0.79) QRACs (with no classical counterparts) exist and that (4,1,p) QRAC with p > 1/2 is not possible. We extend this model with shared randomness (SR) that is accessible to both parties. Then (n,1,p) QRAC with SR and p > 1/2 exists for any …
Operational Coherent Measurements with Steering and Randomness
National Center for Theoretical Sciences, National Taiwan University（台湾大学理论科学中心）；National Taiwan Normal University（台湾师范大学）；Department of Physics and Center for Theoretical Physics, National Taiwan University（台湾大学物理系理论物理中心）；Center for Quantum Science and Engineering, National Taiwan University（台湾大学量子科学与工程研究中心）
Quantum random number generators (QRNGs) can provide genuine randomness based on the inherent unpredictable nature of quantum physics. The extracted randomness relies not only on the physical parts of the QRNG, such as the entropy source and the measurement device, but also on appropriate postprocessing method. The m-least significant bits (m-LSBs) operation is one of the simplest randomness extraction method, which has the advantage of easy implementations. Nonetheless, a detailed analysis of the m-LSBs operation in QRNGs is still missing. In this work we give a physical explanation of the m-…
We consider the problem of quantum measurement compression with side information in the one-shot setting with shared randomness. In this problem, Alice shares a pure state with Reference and Bob and she performs a measurement on her registers. She wishes to communicate the outcome of this measurement to Bob using shared randomness and classical communication, in such a way that the outcome that Bob receives is correctly correlated with Reference and Bob's own registers. Our goal is to simultaneously minimize the classical communication and randomness cost. We provide a protocol based on convex…
This work uses a simple quantum computer model to discuss the randomness of bit strings originated from integer sequences. The considered quantum computer model has three elements: a processing unit responsible for a mathematical operation, an initial equally weighted superposition and a quantum state used as resource. The randomness depends on the last.
Simulating quantum nonlocality and steering requires augmenting pre-shared randomness with non-vanishing communication cost. This prompts the question of how one may provide such an operational characterization for the quantumness of correlations due even to unentangled states. Here we show that for a certain class of states, such quantumness can be pointed out by superlocality, the requirement for a larger dimension of the pre-shared randomness to simulate the correlations than that of the quantum state that generates them. This provides an approach to define the nonclassicality of local mult…
It is known from Bell's theorem that quantum predictions for some entangled states cannot be mimicked using local hidden variable (LHV) models. From a computer science perspective, LHV models may be interpreted as classical computers operating on a potentially infinite number of correlated bits originating from a common source. As such, Bell inequality violations achieved through entangled states are able to characterise the quantum advantage of certain tasks, so long as the task itself imposes no restriction on the availability of correlated bits. However, if the number of shared bits is limi…
We develop the resource theory of private randomness extraction in the distributed and device-dependent scenario. We begin by introducing the notion of independent random bits, which are bipartite states containing ideal private randomness for each party, and motivate the natural set of free operations. As a conceptual tool, we introduce Virtual Quantum State Merging, which is essentially the flip side of Quantum State Merging, without communication. We focus on the bipartite case and find the rate regions achievable in different settings. Surprisingly, it turns out that local noise can boost …
We study the Hamiltonian associated with the quantum adiabatic algorithm with a random cost function. Because the cost function lacks structure we can prove results about the ground state. We find the ground state energy as the number of bits goes to infinity, show that the minimum gap goes to zero exponentially quickly, and we see a localization transition. We prove that there are no levels approaching the ground state near the end of the evolution. We do not know which features of this model are shared by a quantum adiabatic algorithm applied to random instances of satisfiability since despi…
Randomness can help one to implement quantum maps that cannot be realized in a deterministic fashion. Recently, it was discovered that explicitly treating a randomness source as a quantum system could double the efficiency as a catalyst for some tasks. In this work, we first show that every quantum channel that can be implemented with a randomness source without leaking information to it must be a catalysis. For that purpose, we prove a new no-go theorem that generalizes the no-hiding theorem, the no-secret theorem that states no quantum information can be shared with other system as a secret …
Quantum key distribution is a cryptographic primitive for the distribution of symmetric encryption keys between two parties that possess a pre-shared secret. Since the pre-shared secret is a requirement, quantum key distribution may be viewed as a key growing protocol. We note that the use of pre-shared secrets coupled with access to randomness beacons may enable key growing which, though not secure from an information-theoretic standpoint, remains quantum safe.
Private information retrieval systems (PIRs) allow a user to extract an item from a database that is replicated over k>=1 servers, while satisfying various privacy constraints. We exhibit quantum k-server symmetrically-private information retrieval systems (QSPIRs) that use sublinear communication, do not use shared randomness among the servers, and preserve privacy against honest users and dishonest servers. Classically, SPIRs without shared randomness do not exist at all.
We show that some tripartite quantum correlations are inexplicable by any causal theory involving bipartite nonclassical common causes and unlimited shared randomness. This constitutes a device-independent proof that Nature's nonlocality is fundamentally at least tripartite in every conceivable physical theory - no matter how exotic. To formalize this claim we are compelled to substitute Svetlichny's historical definition of genuine tripartite nonlocality with a novel theory-agnostic definition tied to the framework of Local Operations and Shared Randomness (LOSR). A companion article [PRA. 10…
Two parties sharing entangled quantum systems can generate correlations that cannot be produced using only shared classical resources. These nonlocal correlations are a fundamental feature of quantum theory but also have practical applications. For instance, they can be used for device-independent (DI) random number generation, whose security is certified independently of the operations performed inside the devices. The amount of certifiable randomness that can be generated from some given non-local correlations is a key quantity of interest. Here we derive tight analytic bounds on the maximum…
We consider the problem of bounded-error quantum state identification: given either state α_0 or state α_1, we are required to output `0', `1' or `?' ("don't know"), such that conditioned on outputting `0' or `1', our guess is correct with high probability. The goal is to maximize the probability of not outputting `?'. We prove a direct product theorem: if we're given two such problems, with optimal probabilities a and b, respectively, and the states in the first problem are pure, then the optimal probability for the joint bounded-error state identification problem is O(ab). Our proof is based…
We introduce the class of Genuinely Local Operation and Shared Randomness (LOSR) Multipartite Nonlocal correlations, that is, correlations between N parties that cannot be obtained from unlimited shared randomness supplemented by any composition of (N-1)-shared causal Generalized-Probabilistic-Theory (GPT) resources. We then show that noisy N-partite GHZ quantum states as well as the 3-partite W quantum state can produce such correlations. This proves, if the operational predictions of quantum theory are correct, that Nature's nonlocality must be boundlessly multipartite in any causal GPT. We …
This work addresses two problems in the context of two-party communication complexity of functions. First, it concludes the line of research, which can be viewed as demonstrating qualitative advantage of quantum communication in the three most common communication "layouts": two-way interactive communication; one-way communication; simultaneous message passing (SMP). We demonstrate a functional problem, whose communication complexity is $O((\log n)^2)$ in the quantum version of SMP and $\tildeΩ(\sqrt n)$ in the classical (randomised) version of SMP. Second, this work contributes to understandi…
Semi-device-independent randomness certification on discretized continuous-variable platforms
Federal University of Rio Grande do Norte（北里奥格兰德联邦大学）；International Institute of Physics, Federal University of Rio Grande do Norte（北里奥格兰德联邦大学国际物理研究所）；QuIIN - Quantum Industrial Innovation, EMBRAPII CIMATEC Competence Center in Quantum Technologies, SENAI CIMATEC（QuIIN量子工业创新，EMBRAPII CIMATEC量子技术能力中心，SENAI CIMATEC）；Center for Macroscopic Quantum States bigQ, Department of Physics, Technical University of Denmark（丹麦技术大学宏观量子态bigQ研究中心）
Cryptographic protocols are often based on the two main resources: private randomness and private key. In this paper, we develop a relationship between these two resources. First, we show that any state containing perfect, directly accessible, private key (a private state) is a particular case of the state containing perfect, directly accessible, private randomness (an independent state). We then demonstrate a fundamental limitation on the possibility of transferring the privacy of random bits in quantum networks with an intermediate repeater station. More precisely, we provide an upper bound …
Quantum theory predicts the existence of genuinely tripartite-entangled states, which cannot be obtained from local operations over any bipartite entangled states and unlimited shared randomness. Some of us recently proved that this feature is a fundamental signature of quantum theory. The state $\left|{GHZ}_3\right\rangle=(\left|000\right\rangle+\left|111\right\rangle)/\sqrt{2}$ gives rise to tripartite quantum correlations which cannot be explained by any causal theory limited to bipartite nonclassical common causes of any kind (generalising entanglement) assisted with unlimited shared rando…
Two qubit operation sharing schemes [J. Phys. B 44 (2011) 165508] are generalized to qutrit ones. Operations to be shared are classified into three different classes in terms of different probabilities (i.e, 1/3, 2/3 and 1). For the latter two classes, ten and three restricted sets of operations are found out, respectively. Moreover, the two generalized schemes are amply compared from four aspects, namely, quantum and classical resource consumption, necessaryoperation complexity, success probability and efficiency. It is found that the second scheme is overall more optimal than the first one a…
We introduce and analyze a task in which a tripartite quantum state is transformed to an approximately recoverable state by a randomizing operation on one of the three subsystems. We consider cases where the initial state is a tensor product of $n$ copies of a tripartite state $ρ^{ABC}$, and is transformed by a random unitary operation on $A^n$ to another state which is approximately recoverable from its reduced state on $A^nB^n$ (Case 1) or $B^nC^n$ (Case 2). We analyze the minimum cost of randomness per copy required for the task in an asymptotic limit of infinite copies and vanishingly smal…
Generating and characterising randomness is fundamentally important in both classical and quantum information science. Here we report the experimental demonstration of ensembles of pseudorandom optical processes comprising what are known as t-designs. We show that in practical scenarios, certain finite ensembles of two-mode transformations---1- and 2-designs---are indistinguishable from truly random operations for 1- and 2-photon quantum interference, but they fail to mimic randomness for 2- and 3-photon cases, respectively. This provides a novel optical test of pseudo randomness. We make use …
The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure theory, and therefore any operational characterization of the notion of probability is still missing in quantum mechanics. In this paper, based on the toolkit of algorithmic randomness, we present a refinement of the Born rule, as an alternative rule to it, for specifying the property of the results of quantum measurements in an operational way. Algorithmic ran…
Based on the theory of quantum mechanics, intrinsic randomness in measurement distinguishes quantum effects from classical ones. From the perspective of states, this quantum feature can be summarized as coherence or superposition in a specific (classical) computational basis. Recently, by regarding coherence as a physical resource, Baumgratz et al.~present a comprehensive framework for coherence measures. Here, we propose a quantum coherence measure essentially using the intrinsic randomness of measurement. The proposed coherence measure provides an answer to the open question in completing th…
We analyze and demonstrate the feasibility and superiority of linear optical single-qubit fingerprinting over its classical counterpart. For one-qubit fingerprinting of two-bit messages, we prepare `tetrahedral' qubit states experimentally and show that they meet the requirements for quantum fingerprinting to exceed the classical capability. We prove that shared entanglement permits 100% reliable quantum fingerprinting, which will outperform classical fingerprinting even with arbitrary amounts of shared randomness.
