Random access coding is an information task that has been extensively studied and found many applications in quantum information. In this scenario, Alice receives an n-bit string x, and wishes to encode x into a quantum state r x , such that Bob, when receiving the state r x , can choose any bit Î i n [ ] and recover the input bit x i with high probability. Here we study two variants: parity-oblivious random access codes (RACs), where we impose the cryptographic property that Bob cannot infer any information about the parity of any subset of bits of the input apart from the single bits x i ; and evenparity-oblivious RACs, where Bob cannot infer any information about the parity of any even-size subset of bits of the input. In this paper, we provide the optimal bounds for parity-oblivious quantum RACs and show that they are asymptotically better than the optimal classical ones. Our results provide a large non-contextuality inequality violation and resolve the main open problem in a work of Spekkens et al (2009 Phys. Rev. Lett.102 010401). Second, we provide the optimal bounds for evenparity-oblivious RACs by proving their equivalence to a non-local game and by providing tight bounds for the success probability of the non-local game via semidefinite programming. In the case of evenparity-oblivious RACs, the cryptographic property holds also in the device independent model. IntroductionQuantum information theory studies how information is encoded in quantum mechanical systems and how it can be transmitted through quantum channels. A main question is whether quantum information is more powerful than classical information. A celebrated result by Holevo [Hol73] shows that quantum information cannot be used to compress classical information. In high level, in order to transmit n uniformly random classical bits, one needs to transmit no less than n quantum bits. This might imply that quantum information is no more powerful than classical information. This however is wrong in many situations. In the model of communication complexity, one can show that transmitting quantum information may result in exponential savings on the communication needed to solve specific problems [BCWdW01, BJK04, GKK + 08, Raz99, RK11].One specific information task that has been extensively studied in quantum information is the notion of random access codes (RACs) [ANTV02, ANTV99, Nay99]. In this scenario, Alice receives an n-bit string x, drawn from the uniform distribution, and wishes to encode x into a quantum state r x , such that Bob, when receiving the state r x , can choose any bit Î i n [ ] and recover the input bit x i with high probability by performing some general quantum operation on r x .RACs have been used in various situations in quantum information and computation, including in communication complexity, non-locality, extractors and device-independent cryptography [BARdW08, DV10, INRY07, LPY + , PZ10]. Even though this task seems easier than transmitting the entire input string x, it is known that the length of quantum RACs must be ...

Oblivious transfer is a fundamental primitive in cryptography. While perfect information theoretic security is impossible, quantum oblivious transfer protocols can limit the dishonest player's cheating. Finding the optimal security parameters in such protocols is an important open question. In this paper we show that every 1-out-of-2 oblivious transfer protocol allows a dishonest party to cheat with probability bounded below by a constant strictly larger than $1/2$. Alice's cheating is defined as her probability of guessing Bob's index, and Bob's cheating is defined as his probability of guessing both input bits of Alice. In our proof, we relate these cheating probabilities to the cheating probabilities of a bit commitment protocol and conclude by using lower bounds on quantum bit commitment. Then, we present an oblivious transfer protocol with two messages and cheating probabilities at most $3/4$. Last, we extend Kitaev's semidefinite programming formulation to more general primitives, where the security is against a dishonest player trying to force the outcome of the other player, and prove optimal lower and upper bounds for them.

Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this Letter, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on many well-known correlations and discuss how it can rule out correlations of having a finite-dimensional quantum representation. We show that our bound is multiplicative under product correlations and also that it can witness the non-convexity of certain restricted-dimensional quantum correlations.In what ranks as one of the most important achievements of modern physics, it was shown by John Bell in 1964 that some correlations generated within the framework of quantum mechanics can be nonlocal, in the sense that the statistics generated by quantum mechanics cannot always be reproduced by a local hidden-variable model [1,2]. Over the last 40 years, there have been significant efforts in trying to verify this fact experimentally. The first such experimental data [3] was published in 1972 and this remains an active area of research [4]. Moreover, as a central concept in quantum physics and quantum information theory, fully understanding quantum entanglement and nonlocality still remains a very interesting and important problem with far-reaching applications. Indeed, profound relationships between quantum nonlocality and other fundamental quantum concepts or phenomena such as entanglement measures [5,6], entanglement distillation [7,8], and teleportation [9] have been identified. Meanwhile, for many tasks, e.g. in cryptography [10,11], it has been realized that due to quantum nonlocality, quantum strategies enjoy remarkable advantages over their classical counterparts.However, even though quantum nonlocal effects can lead to interesting and often surprising advantages in some applications, this does not paint the full picture. After all, for practical applications, it is just as important to understand the amount of quantum resources required for these advantages to manifest. For instance, if there is an exponential blowup in the amount of resources required, then whatever advantage gained by employing quantum mechanics may not be useful in practice. Quantifying the amount of quantum resources needed to perform a certain task is the central focus of this Letter.We study quantum nonlocality from the viewpoint of two-party quantum correlations that arise from a Bell experiment. A two-party Bell experiment is performed between two parties, Alice and Bob, whose labs are set up in separate locations. Alice (resp. Bob) has in her possession a measurement apparatus whose possible settings are labelled by the elements of a finite set X (resp. Y ) and the possible measurement outcomes are labelled by a finite set A (resp. B). After repeat...

In this work we consider the ground space connectivity problem for commuting local Hamiltonians. The ground space connectivity problem asks whether it is possible to go from one (efficiently preparable) state to another by applying a polynomial length sequence of 2-qubit unitaries while remaining at all times in a state with low energy for a given Hamiltonian H. It was shown in [GS15] that this problem is QCMA-complete for general local Hamiltonians, where QCMA is defined as QMA with a classical witness and BQP verifier. Here we show that the commuting version of the problem is also QCMA-complete. This provides one of the first examples where commuting local Hamiltonians exhibit complexity theoretic hardness equivalent to general local Hamiltonians.

Abstract. An n×n matrix X is called completely positive semidefinite (cpsd) if there exist d×d Hermitian positive semidefinite matrices {P i } n i=1 (for some d ≥ 1) such that X ij = Tr(P i P j ), for all i, j ∈ {1, . . . , n}. The cpsd-rank of a cpsd matrix is the smallest d ≥ 1 for which such a representation is possible. In this work we initiate the study of the cpsd-rank which we motivate twofold. First, the cpsd-rank is a natural non-commutative analogue of the completely positive rank of a completely positive matrix. Second, we show that the cpsd-rank is physically motivated as it can be used to upper and lower bound the size of a quantum system needed to generate a quantum behavior.In this work we present several properties of the cpsd-rank. Unlike the completely positive rank which is at most quadratic in the size of the matrix, no general upper bound is known on the cpsdrank of a cpsd matrix. In fact, we show that the cpsd-rank can be exponential in terms of the size. Specifically, for any n ≥ 1, we construct a cpsd matrix of size 2n whose cpsd-rank is 2 Ω( √ n) . Our construction is based on Gram matrices of Lorentz cone vectors, which we show are cpsd. The proof relies crucially on the connection between the cpsd-rank and quantum behaviors. In particular, we use a known lower bound on the size of matrix representations of extremal quantum correlations which we apply to high-rank extreme points of the n-dimensional elliptope.Lastly, we study cpsd-graphs, i.e., graphs G with the property that every doubly nonnegative matrix whose support is given by G is cpsd. We show that a graph is cpsd if and only if it has no odd cycle of length at least 5 as a subgraph. This coincides with the characterization of cp-graphs.

In this work we study the sets of two-party correlations generated from a Bell scenario involving two spatially separated systems with respect to various physical models. We show that the sets of classical, quantum, no-signaling and unrestricted correlations can be expressed as projections of affine sections of appropriate convex cones. As a by-product, we identify a spectrahedral outer approximation to the set of quantum correlations which is contained in the first level of the Navascués, Pironio and Acín (NPA) hierarchy and also a sufficient condition for the set of quantum correlations to be closed. Furthermore, by our conic formulations, the value of a nonlocal game over the sets of classical, quantum, no-signaling and unrestricted correlations can be cast as a linear conic program. This allows us to show that a semidefinite programming upper bound to the classical value of a nonlocal game introduced by Feige and Lovász is in fact an upper bound to the quantum value of the game and moreover, it is at least as strong as optimizing over the first level of the NPA hierarchy. Lastly, we show that deciding the existence of a perfect quantum (resp. classical) strategy is equivalent to deciding the feasibility of a linear conic program over the cone of completely positive semidefinite matrices (resp. completely positive matrices). By specializing B Antonios Varvitsiotis 123 432 J. Sikora, A. Varvitsiotis the results to synchronous nonlocal games, we recover the conic formulations for various quantum and classical graph parameters that were recently derived in the literature.

We introduce a definition of the fidelity function for multi-round quantum strategies, which we call the strategy fidelity, that is a generalization of the fidelity function for quantum states. We provide many properties of the strategy fidelity including a Fuchs-van de Graaf relationship with the strategy norm. We also provide a general monotonicity result for both the strategy fidelity and strategy norm under the actions of strategy-to-strategy linear maps. We illustrate an operational interpretation of the strategy fidelity in the spirit of Uhlmann's Theorem and discuss its application to the security analysis of quantum protocols for interactive cryptographic tasks such as bit-commitment and oblivious string transfer. Our analysis is general in the sense that the actions of the protocol need not be fully specified, which is in stark contrast to most other security proofs. Lastly, we provide a semidefinite programming formulation of the strategy fidelity.

The study of ground state energies of local Hamiltonians has played a fundamental role in quantum complexity theory. In this article, we take a new direction by introducing the physically motivated notion of “ground state connectivity” of local Hamiltonians, which captures problems in areas ranging from quantum stabilizer codes to quantum memories. Roughly, “ground state connectivity” corresponds to the natural question: Given two ground states |Ψ〉 and |ϕ〉 of a local Hamiltonian H , is there an “energy barrier” (with respect to H ) along any sequence of local operations mapping |Ψ〉 to |ϕ〉? We show that the complexity of this question can range from QCMA-complete to PSPACE-complete, as well as NEXP-complete for an appropriately defined “succinct” version of the problem. As a result, we obtain a natural QCMA-complete problem, a goal which has generally proven difficult since the conception of QCMA over a decade ago. Our proofs rely on a new technical tool, the Traversal Lemma, which analyzes the Hilbert space a local unitary evolution must traverse under certain conditions. We show that this lemma is essentially tight with respect to the length of the unitary evolution in question.