Abstract-In this paper, we show that the conditional min-entropy Hmin(AjB) of a bipartite state AB is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the B-part of AB are allowed. In the special case where A is classical, this overlap corresponds to the probability of guessing A given B. In a similar vein, we connect the conditional max-entropy H max (AjB) to the maximum fidelity of AB with a product state that is completely mixed on A. In the case where A is classical, this corresponds to the security of A when used as a secret key in the presence of an adversary holding B.Because min-and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessing A given B is a lower bound on the number of uniform secret bits that can be extracted from A relative to an adversary holding B.Index Terms-Entropy measures, max-entropy, min-entropy, operational interpretations, quantum information theory, quantum hypothesis testing, singlet fraction, single-shot information theory.

Abstract. Privacy amplification is the art of shrinking a partially secret string Z to a highly secret key S. We show that, even if an adversary holds quantum information about the initial string Z, the key S obtained by two-universal hashing is secure, according to a universally composable security definition. Additionally, we give an asymptotically optimal lower bound on the length of the extractable key S in terms of the adversary's (quantum) knowledge about Z. Our result has applications in quantum cryptography. In particular, it implies that many of the known quantum key distribution protocols are universally composable.

We consider the implementation of two-party cryptographic primitives based on the sole assumption that no largescale reliable quantum storage is available to the cheating party. We construct novel protocols for oblivious transfer and bit commitment, and prove that realistic noise levels provide security even against the most general attack. Such unconditional results were previously only known in the so-called bounded-storage model which is a special case of our setting. Our protocols can be implemented with present-day hardware used for quantum key distribution. In particular, no quantum storage is required for the honest parties.

We propose a general method for studying properties of quantum channels acting on an n-partite system, whose action is invariant under permutations of the subsystems. Our main result is that, in order to prove that a certain property holds for an arbitrary input, it is sufficient to consider the case where the input is a particular de Finetti-type state, i.e., a state which consists of n identical and independent copies of an (unknown) state on a single subsystem. Our technique can be applied to the analysis of information-theoretic problems. For example, in quantum cryptography, we get a simple proof for the fact that security of a discrete-variable quantum key distribution protocol against collective attacks implies security of the protocol against the most general attacks. The resulting security bounds are tighter than previously known bounds obtained with help of the exponential de Finetti theorem.

We prove that constant-depth quantum circuits are more powerful than their classical counterparts. To this end we introduce a non-oracular version of the Bernstein-Vazirani problem which we call the 2D Hidden Linear Function problem. An instance of the problem is specified by a quadratic form q that maps n-bit strings to integers modulo four. The goal is to identify a linear boolean function which describes the action of q on a certain subset of n-bit strings. We prove that any classical probabilistic circuit composed of bounded fan-in gates that solves the 2D Hidden Linear Function problem with high probability must have depth logarithmic in n. In contrast, we show that this problem can be solved with certainty by a constant-depth quantum circuit composed of one-and two-qubit gates acting locally on a two-dimensional grid.building on earlier studies of so-called IQP circuits [6,7,8], gave further evidence that the output distribution of SQCs may be hard to sample classically even if a constant statistical error is allowed; see also Ref. [9]. A more powerful model of computation consisting of logarithmic-depth quantum circuits assisted by polynomial-time classical computation is known to be capable of solving hard problems such as factoring [10].Parallelism and circuit depth are important considerations when designing quantum algorithms that can be executed in the near-future on small quantum computers that may lack error correction capabilities [11,12,13]. While the overhead for encoding and manipulating quantum data fault-tolerantly is asymptotically small [14,15], it remains prohibitive for current technology. A quantum computation without error correction can only compute for a constant amount of time before the qubits decohere and the entropy builds up [16]. In this situation one may wish to parallelize the computation as much as possible to fit within the coherence time.What can we hope to prove concerning the computational power of SQCs ? A rigorous proof that SQCs outperform polynomial-time classical algorithms for some computational task is arguably beyond our reach (as it would imply a separation between the complexity classes BQP and BPP). In this paper we set a less ambitious goal and pose the following question:Can constant-depth quantum circuits solve a computational problem that constant-depth classical circuits cannot?

When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by 2 kd 2 n , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation Uµ+ν ⊂ Uµ ⊗ Uν of the unitary group U(d), for highest weights µ, ν and µ + ν. Let ξµ be the state obtained by tracing out Uν. Then ξµ is close to a convex combination of the coherent states Uµ(g)|vµ , where g ∈ U(d) and |vµ is the highest weight vector in Uµ.For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our "half" theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general.

Abstract. Instantaneous measurements of non-local observables between space-like separated regions can be performed without violating causality. This feat relies on the use of entanglement. Here we propose novel protocols for this task and the related problem of multipartite quantum computation with local operations and a single round of classical communication. Compared to previously known techniques, our protocols reduce the entanglement consumption by an exponential amount. We also prove a linear lower bound on the amount of entanglement required for the implementation of a certain non-local measurement. These results relate to position-based cryptography: an amount of entanglement scaling exponentially with the number of communicated qubits is sufficient to render any such scheme insecure. Furthermore, we show that certain schemes are secure under the assumption that the adversary has less entanglement than a given bound and is restricted to classical communication.

Given a quantum error correcting code, an important task is to find encoded operations that can be implemented efficiently and fault tolerantly. In this Letter we focus on topological stabilizer codes and encoded unitary gates that can be implemented by a constant-depth quantum circuit. Such gates have a certain degree of protection since propagation of errors in a constant-depth circuit is limited by a constant size light cone. For the 2D geometry we show that constant-depth circuits can only implement a finite group of encoded gates known as the Clifford group. This implies that topological protection must be "turned off" for at least some steps in the computation in order to achieve universality. For the 3D geometry we show that an encoded gate U is implementable by a constant-depth circuit only if UPU(†) is in the Clifford group for any Pauli operator P. This class of gates includes some non-Clifford gates such as the π/8 rotation. Our classification applies to any stabilizer code with geometrically local stabilizers and sufficiently large code distance.