Min-entropy sampling gives a bound on the min-entropy of a randomly chosen subset of a string, given a bound on the min-entropy of the whole string. König and Renner showed a min-entropy sampling theorem that holds relative to quantum knowledge. Their result achieves the optimal rate, but it can only be applied if the bits are sampled in blocks, and only gives weak bounds for the non-smooth min-entropy. We give two new quantum min-entropy sampling theorems that do not have the above weaknesses. The first theorem shows that the result by König and Renner also applies to bitwise sampling, and the second theorem gives a strong bound for the non-smooth min-entropy. Our results imply a new lower bound for k-out-of-n random access codes: while previous results by Ben-Aroya, Regev, and de Wolf showed that the decoding probability is exponentially small in k if the storage rate is smaller than 0.7, our results imply that this holds for any storage rate strictly smaller than 1, which is optimal.
Min-entropy sampling gives a bound on the min-entropy of a randomly chosen subset of a string, given a bound on the min-entropy of the whole string. König and Renner showed a min-entropy sampling theorem that holds relative to quantum knowledge. Their result achieves the optimal rate, but it can only be applied if the bits are sampled in blocks, and only gives weak bounds for the non-smooth min-entropy. We give two new quantum min-entropy sampling theorems that do not have the above weaknesses. The first theorem shows that the result by König and Renner also applies to bitwise sampling, and the second theorem gives a strong bound for the non-smooth min-entropy. Our results imply a new lower bound for k-out-of-n random access codes: while previous results by Ben-Aroya, Regev, and de Wolf showed that the decoding probability is exponentially small in k if the storage rate is smaller than 0.7, our results imply that this holds for any storage rate strictly smaller than 1, which is optimal.
LetX1,…,Xnbe a sequence ofnclassical random variables and consider a sampleXs1,…,Xsrofr≤npositions selected at random. Then, except with (exponentially inr) small probability, the min-entropyHmin(Xs1⋯Xsr)of the sample is not smaller than, roughly, a fractionnrof the overall entropyHmin(X1⋯Xn), which is optimal. Here, we show that this statement, originally proved in [S. Vadhan, LNCS 2729, Springer, 2003] for the purely classical case, is still true if the min-entropyHminis measured relative to a quantum system. Because min-entropy quantifies the amount of randomness that can be extracted from a given random variable, our result can be used to prove the soundness of locally computable extractors in a context where side information might be quantum-mechanical. In particular, it implies that key agreement in the bounded-storage model—using a standard sample-and-hash protocol—is fully secure against quantum adversaries, thus solving a long-standing open problem.
A natural measure for the amount of quantum information that a physical system E holds about another system A = A 1 ,..., A n is given by the min-entropy H min ( A | E ). Specifically, the min-entropy measures the amount of entanglement between E and A , and is the relevant measure when analyzing a wide variety of problems ranging from randomness extraction in quantum cryptography, decoupling used in channel coding, to physical processes such as thermalization or the thermodynamic work cost (or gain) of erasing a quantum system. As such, it is a central question to determine the behaviour of the min-entropy after some process M is applied to the system A . Here we introduce a new generic tool relating the resulting min-entropy to the original one, and apply it to several settings of interest, including sampling of subsystems and measuring in a randomly chosen basis. The results on random measurements yield new high-order entropic uncertainty relations with which we prove the optimality of cryptographic schemes in the bounded quantum storage model. This is an abridged version of the paper; the full version containing all proofs and further applications can be found in [13]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A natural measure for the amount of quantum information that a physical system E holds about another system A = A1, .. . , Anis given by the min-entropy Hmin(A|E). In particular, the min-entropy measures the amount of entanglement between E and A, and is the relevant measure when analyzing a wide variety of problems ranging from randomness extraction in quantum cryptography, decoupling used in channel coding, to physical processes such as thermalization or the thermodynamic work cost (or gain) of erasing a quantum system. As such, it is a central question to determine the behavior of the minentropy after some process M. is applied to the system A. Here, we introduce a new generic tool relating the resulting min-entropy to the original one, and apply it to several settings of interest. A simple example of such a process is the one of sampling, where a subset S of the systems A1, ... , An is selected at random. Our tool allows us to quantify the entanglement that E has with the selected systems AS, i.e., Hmin(AS|ES) as a function of the original Hmin(A|E). We give two applications of this result. First, it directly provides the first local quantum-to-classical randomness extractors for use in quantum cryptography, as well as decoupling operations acting on only a small fraction AS of the input A. Moreover, it gives lower bounds on the dimension of k-out-of-n fully quantum random access encodings. Another natural example of such a process is a measurement in, e.g., BB84 bases commonly used in quantum cryptography. We establish the first entropic uncertainty relations with quantum side information that are nontrivial whenever E is not maximally entangled with A. As a consequence, we are able to prove optimality of quantum cryptographic schemes in the noisy-storage model. This model allows for the secure implementation of two-party cryptographic primitives under the assumption that the adversary cannot store quantum information perfectly. A special case is the bounded-quantum-storage model (BQSM), which assumes that the adversary's quantum memory device is noise free but limited in size. Ever since the inception of the BQSM, it has been a vexing open question to determine whether the security is possible as long as the adversary can only.
The entropy accumulation theorem states that the smooth min-entropy of an n-partite system A = (A_1 ,. .. , A_n) is lower-bounded by the sum of the von Neumann entropies of suitably chosen conditional states up to corrections that are sublinear in n. This theorem is particularly suited to proving the security of quantum cryptographic protocols, and in particular so-called device-independent protocols for randomness expansion and key distribution, where the devices can be built and preprogrammed by a malicious supplier. However, while the bounds provided by this theorem are optimal in the first order, the second-order term is bounded more crudely, in such a way that the bounds deteriorate significantly when the theorem is applied directly to protocols where parameter estimation is done by sampling a small fraction of the positions, as is done in most QKD protocols. The objective of this paper is to improve this second-order sublinear term and remedy this problem. On the way, we prove various bounds on the divergence variance, which might be of independent interest.
The entropy accumulation theorem states that the smooth min-entropy of an n-partite system A = (A1, ..., An) is lower-bounded by the sum of the von Neumann entropies of suitably chosen conditional states up to corrections that are sublinear in n. This theorem is particularly suited to proving the security of quantum cryptographic protocols, and in particular so-called device-independent protocols for randomness expansion and key distribution, where the devices can be built and preprogrammed by a malicious supplier. However, while the bounds provided by this theorem are optimal in the first order, the second-order term is bounded more crudely, in such a way that the bounds deteriorate significantly when the theorem is applied directly to protocols where parameter estimation is done by sampling a small fraction of the positions, as is done in most QKD protocols. The objective of this paper is to improve this second-order sublinear term and remedy this problem. On the way, we prove various bounds on the divergence variance, which might be of independent interest.
We calculate an achievable secret key rate for quantum key distribution with a finite number of signals by evaluating the quantum conditional min-entropy explicitly. The min-entropy for a classical random variable is the negative logarithm of the maximal value in its probability distribution. The quantum conditional min-entropy can be expressed in terms of the guessing probability, which we calculate for d-dimensional systems. We compare these key rates to previous approaches using the von Neumann entropy and find nonzero key rates for a smaller number of signals. Furthermore, we improve the secret key rates by modifying the parameter estimation step. Both improvements taken together lead to nonzero key rates for only 104--105 signals. An interesting conclusion can also be drawn from the additivity of the min-entropy and its relation to the guessing probability: for a set of symmetric tensor product states, the optimal minimum-error discrimination (MED) measurement is the optimal MED measurement on each subsystem.