results for au:Hayashi_M in:quant-ph

- We derive an attainable bound on the precision of quantum state estimation for finite dimensional systems, providing a construction for the asymptotically optimal measurement. Our results hold under an assumption called local asymptotic covariance, which is weaker than unbiasedness or local unbiasedness. The derivation is based on an analysis of the limiting distribution of the estimator's deviation from the true value of the parameter, and takes advantage of quantum local asymptotic normality, a duality between sequences of identically prepared states and Gaussian states of continuous variable systems. We first prove our results for the mean square error of a special class of models, called D-invariant, and then extend the results to arbitrary models, generic cost functions, and global state estimation, where the unknown parameter is not restricted to a local neighbourhood of the true value. The extension includes a treatment of nuisance parameters, namely parameters that are not of interest to the experimenter but nevertheless affect the estimation. As an illustration of the general approach, we provide the optimal estimation strategies for the joint measurement of two qubit observables, for the estimation of qubit states in the presence of amplitude damping noise, and for noisy multiphase estimation.
- Jan 30 2018 quant-ph arXiv:1801.09158v1We focus on a hidden Markovian process whose internal hidden system is given as a quantum system, and we address a sequence of data obtained from this process. Using a quantum version of the Perron-Frobenius theorem, we derive novel upper and lower bounds for the cumulant generating function of the sample mean of the data. Using these bound, we derive the central limit theorem and large and moderate deviations for the tail probability. Then, we give the asymptotic variance is given by using the second derivative of the cumulant generating function. We also derive another expression for the asymptotic variance by considering the quantum version of fundamental matrix. Further, we explain how to extend our results to a general probabilistic system.
- Jan 15 2018 quant-ph arXiv:1801.03988v2We investigate the ergodic and mixing properties for dynamics in the framework of general probabilistic theory as well as the decoupling properties with an asymptotic setting, where the ergodicity is defined as the convergence of the long time average state and the mixing condition is defined as the combination of the ergodicity and aperiodicity. In this paper, we give relations between the mixing and asymptotically decoupling conditions. In addition, we give conditions equivalent to the mixing one. One of them is useful in determining whether a given dynamical map is mixing or not, by calculation because this condition results in a system of linear equations. We apply our result to the studies on the irreducibility and the primitivity.
- Jan 11 2018 quant-ph arXiv:1801.03306v2We consider the secure quantum communication over a network with the presence of the malicious adversary who can eavesdrop and contaminate the states. As the main result, when the maximum number $m_1$ of the attacked edges is less than a half of network transmission rate $m_0$ (i.e., $m_1 < m_0/2$), our protocol achieves secret and correctable quantum communication of rate $m_0-2m_1$ by asymptotic $n$ uses of the network. Our protocol requires no classical communication and no knowledge of network structure, but instead, a node operation is limited to the application of an invertible matrix to bit basis state. Our protocol can be thought as a generalization of verifiable quantum secret sharing.
- In this work we study the problem of secure communication over a fully quantum Gel'fand-Pinsker channel. The best known achievability rate for this channel model in the classical case was proven by Goldfeld, Cuff and Permuter in [Goldfeld, Cuff, Permuter, 2016]. We generalize the result of [Goldfeld, Cuff, Permuter, 2016]. One key feature of the results obtained in this work is that all the bounds obtained are in terms of error exponent. We obtain our achievability result via the technique of simultaneous pinching. This in turn allows us to show the existence of a simultaneous decoder. Further, to obtain our encoding technique and to prove the security feature of our coding scheme we prove a bivariate classical-quantum channel resolvability lemma and a conditional classical-quantum channel resolvability lemma. As a by product of the achievability result obtained in this work, we also obtain an achievable rate for a fully quantum Gel'fand-Pinsker channel in the absence of Eve. The form of this achievable rate matches with its classical counterpart. The Gel'fand-Pinsker channel model had earlier only been studied for the classical-quantum case and in the case where Alice (the sender) and Bob (the receiver) have shared entanglement between them.
- Sep 25 2017 quant-ph arXiv:1709.07701v1Recently, it is well recognized that hypothesis testing has deep relations with other topics in quantum information theory as well as in classical information theory. These relations enable us to derive precise evaluation in the finite-length setting. However, such usefulness of hypothesis testing is not limited to information theoretical topics. For example, it can be used for verification of entangled state and quantum computer as well as guaranteeing the security of keys generated via quantum key distribution. In this talk, we overview these kinds of applications of hypothesis testing.
- Sep 20 2017 quant-ph arXiv:1709.06112v2A quantum measurement is Fisher symmetric if it provides uniform and maximal information on all parameters that characterize the quantum state of interest. Using (complex projective) 2-designs, we construct measurements on a pair of identically prepared quantum states that are Fisher symmetric for all pure states. Such measurements are optimal in achieving the minimal statistical error without adaptive measurements. We then determine all collective measurements on a pair that are Fisher symmetric for the completely mixed state and for all pure states simultaneously. For a qubit, these measurements are Fisher symmetric for all states. The minimal optimal measurements are tied to the elusive symmetric informationally complete measurements, which reflects a deep connection between local symmetry and global symmetry. In the study, we derive a fundamental constraint on the Fisher information matrix of any collective measurement on a pair, which offers a useful tool for characterizing the tomographic efficiency of collective measurements.
- To guarantee the security of uniform random numbers generated by a quantum random number generator, we study secure extraction of uniform random numbers when the environment of a given quantum state is controlled by the third party, the eavesdropper. Here we restrict our operations to incoherent strategies that are composed of the measurement on the computational basis and incoherent operations (or incoherence-preserving operations). We show that the maximum secure extraction rate is equal to the relative entropy of coherence. By contrast, the coherence of formation gives the extraction rate when a certain constraint is imposed on eavesdropper's operations. The condition under which the two extraction rates coincide is then determined. Furthermore, we find that the exponential decreasing rate of the leaked information is characterized by Rényi relative entropies of coherence. These results clarify the power of incoherent strategies in random number generation, and can be applied to guarantee the quality of random numbers generated by a quantum random number generator.
- We study systematically resource measures of coherence and entanglement based on Rényi relative entropies, which include the logarithmic robustness of coherence, geometric coherence, and conventional relative entropy of coherence together with their entanglement analogues. First, we show that each Rényi relative entropy of coherence is equal to the corresponding Rényi relative entropy of entanglement for any maximally correlated state. By virtue of this observation, we establish a simple operational connection between entanglement measures and coherence measures based on Rényi relative entropies. We then prove that all these coherence measures, including the logarithmic robustness of coherence, are additive. Accordingly, all these entanglement measures are additive for maximally correlated states. In addition, we derive analytical formulas for Rényi relative entropies of entanglement of maximally correlated states and bipartite pure states, which reproduce a number of classic results on the relative entropy of entanglement and logarithmic robustness of entanglement in a unified framework. Several nontrivial bounds for Rényi relative entropies of coherence (entanglement) are further derived, which improve over results known previously. Moreover, we determine all states whose relative entropy of coherence is equal to the logarithmic robustness of coherence. As an application, we provide an upper bound for the exact coherence distillation rate, which is saturated for pure states.
- May 04 2017 quant-ph arXiv:1705.01474v1Quantum network coding on the butterfly network has been studied as a typical example of quantum multiple cast network. We propose secure quantum network coding on the butterfly network in the multiple unicast setting based on a secure classical network coding. This protocol certainly transmits quantum states when there is no attack. We also show the secrecy even when the eavesdropper wiretaps one of the channels in the butterfly network.
- Apr 11 2017 quant-ph arXiv:1704.02896v2Quantum coherence plays a central role in various research areas. The $l_1$-norm of coherence is one of the most important coherence measures that are easily computable, but it is not easy to find a simple interpretation. We show that the $l_1$-norm of coherence is uniquely characterized by a few simple axioms, which demonstrates in a precise sense that it is the analog of negativity in entanglement theory and sum negativity in the resource theory of magic-state quantum computation. We also provide an operational interpretation of the $l_1$-norm of coherence as the maximum entanglement, measured by the negativity, produced by incoherent operations acting on the system and an incoherent ancilla. To achieve this goal, we clarify the relation between the $l_1$-norm of coherence and negativity for all bipartite states, which leads to an interesting generalization of maximally correlated states. Surprisingly, all entangled states thus obtained are distillable. Moreover, their entanglement cost and distillable entanglement can be computed explicitly for a qubit-qudit system.
- Mar 20 2017 quant-ph arXiv:1703.05876v3Quantum mechanics imposes a fundamental tradeoff between the accuracy of time measurements and the size of the systems used as clocks. When the measurements of different time intervals are combined, the errors due to the finite clock size accumulate, resulting in an overall inaccuracy that grows with the complexity of the setup. Here we introduce a method that eludes the accumulation of errors, by coherently transferring information from a quantum clock to a quantum memory of the smallest possible size. Our method can be used to measure the total duration of a sequence of events with enhanced accuracy, and to reduce the amount of quantum communication needed to stabilize clocks in a quantum network.
- Jan 23 2017 quant-ph arXiv:1701.05688v1Hypergraph states are generalizations of graph states where controlled-$Z$ gates on edges are replaced with generalized controlled-$Z$ gates on hyperedges. Hypergraph states have several advantages over graph states. For example, certain hypergraph states, such as the Union Jack states, are universal resource states for measurement-based quantum computing with only Pauli measurements, while graph state measurement-based quantum computing needs non-Clifford basis measurements. Furthermore, it is impossible to classically efficiently sample measurement results on hypergraph states with a constant $L$1-norm error unless the polynomial hierarchy collapses to the third level. Although several protocols have been proposed to verify graph states with only sequential single-qubit Pauli measurements, there was no verification method for hypergraph states. In this paper, we propose a method for verifying hypergraph states with only sequential single-qubit Pauli measurements. As applications, we consider verified blind quantum computing with hypergraph states, and quantum supremacy demonstrations with hypergraph states.
- We study the compression of $n$ quantum systems, each prepared in the same state belonging to a given parametric family of quantum states. For a family of states with $f$ independent parameters, we devise an asymptotically faithful protocol that requires a hybrid memory of size $(f/2)\log n$, including both quantum and classical bits. Our construction uses a quantum version of local asymptotic normality and, as an intermediate step, solves the problem of compressing displaced thermal states of $n$ identically prepared modes. In both cases, we show that $(f/2)\log n$ is the minimum amount of memory needed to achieve asymptotic faithfulness. In addition, we analyze how much of the memory needs to be quantum. We find that the ratio between quantum and classical bits can be made arbitrarily small, but cannot reach zero: unless all the quantum states in the family commute, no protocol using only classical bits can be faithful, even if it uses an arbitrarily large number of classical bits.
- In this paper we obtain a lower bound of exponent of average probability of error for classical quantum multiple access channel, which implies that for all rate pairs in the capacity region is achievable by a code with exponential probability of error. Thus we re-obtain the direct coding theorem.
- Dec 14 2016 quant-ph arXiv:1612.04047v4In quantum thermodynamics, effects of finiteness of the baths have been less considered. In particular, there is no general theory which focuses on finiteness of the baths of multiple conserved quantities. Then, we investigate how the optimal performance of generalized heat engines with multiple conserved quantities alters in response to the size of the baths. In the context of general theories of quantum thermodynamics, the size of the baths has been given in terms of the number of identical copies of a system, which does not cover even such a natural scaling as the volume. In consideration of the asymptotic extensivity, we deal with a generic scaling of the baths to naturally include the volume scaling. Based on it, we derive a bound for the performance of generalized heat engines reflecting finite-size effects of the baths, which we call fine-grained generalized Carnot bound. We also construct a protocol to achieve the optimal performance of the engine given by this bound. Finally, applying the obtained general theory, we deal with simple examples of generalized heat engines. As for an example of non-i.i.d.~scaling and multiple conserved quantities, we investigate a heat engine with two baths composed of an ideal gas exchanging particles, where the volume scaling is applied. The result implies that the mass of the particle explicitly affects the performance of this engine with finite-size baths.
- We investigate the ability of a quantum measurement device to discriminate two states or, generically, two hypothesis. In full generality, the measurement can be performed a number $n$ of times, and arbitrary pre-processing of the states and post-processing of the obtained data is allowed. Even if the two hypothesis correspond to orthogonal states, perfect discrimination is not always possible. There is thus an intrinsic error associated to the measurement device, which we aim to quantify, that limits its discrimination power. We minimize various error probabilities (averaged or constrained) over all pairs of $n$-partite input states. These probabilities, or their exponential rates of decrease in the case of large $n$, give measures of the discrimination power of the device. For the asymptotic rate of the averaged error probability, we obtain a Chernoff-type bound, dual to the standard Chernoff bound for which the state pair is fixed and the optimization is over all measurements. The key point in the derivation is that i.i.d. states become optimal in asymptotic settings. Minimum asymptotic rates are also obtained for constrained error probabilities, dual to Stein's Lemma and Hoeffding's bound. We further show that adaptive protocols where the state preparer gets feedback from the measurer do not improve the asymptotic rates. These rates thus quantify the ultimate discrimination power of a measurement device.
- Oct 18 2016 quant-ph arXiv:1610.05216v1Quantum systems, in general, output data that cannot be simulated efficiently by a classical computer, and hence is useful for solving certain mathematical problems and simulating quantum many-body systems. This also implies, unfortunately, that verification of the output of the quantum systems is not so trivial, since predicting the output is exponentially hard. As another problem, quantum system is very delicate for noise and thus needs error correction. Here we propose a framework for verification of the output of fault-tolerant quantum computation in the measurement-based model. Contrast to existing analyses on fault-tolerance, we do not assume any noise model on the resource state, but an arbitrary resource state is tested by using only single-qubit measurements to verify whether the output of measurement-based quantum computation on it is correct or not. The overhead for verification including classical processing is linear in the size of quantum computation. Since full characterization of quantum noise is exponentially hard for large-scale quantum computing systems, our framework provides an efficient way of practical verification of experimental quantum error correction. Moreover, the proposed verification scheme is also compatible to measurement-only blind quantum computation, where a client can accept the delegated quantum computation even when a quantum sever makes deviation, as long as the output is correct.
- We establish the ultimate limits to the compression of sequences of identically prepared qubits. The limits are determined by Holevo's information quantity and are attained through use of the optimal universal cloning machine, which finds here a novel application to quantum Shannon theory.
- Coding technology is used in several information processing tasks. In particular, when noise during transmission disturbs communications, coding technology is employed to protect the information. However, there are two types of coding technology: coding in classical information theory and coding in quantum information theory. Although the physical media used to transmit information ultimately obey quantum mechanics, we need to choose the type of coding depending on the kind of information device, classical or quantum, that is being used. In both branches of information theory, there are many elegant theoretical results under the ideal assumption that an infinitely large system is available. In a realistic situation, we need to account for finite size effects. The present paper reviews finite size effects in classical and quantum information theory with respect to various topics, including applied aspects.
- Mar 08 2016 quant-ph arXiv:1603.02195v3In order to guarantee the output of a quantum computation, we usually assume that the component devices are trusted. However, when the total computation process is large, it is not easy to guarantee the whole system when we have scaling effects, unexpected noise, or unaccounted correlations between several subsystems. If we do not trust the measurement basis nor the prepared entangled state, we do need to be worried about such uncertainties. To this end, we proposes a "self-guaranteed" protocol for verification of quantum computation under the scheme of measurement-based quantum computation where no prior-trusted devices (measurement basis nor entangled state) are needed. The approach we present enables the implementation of verifiable quantum computation using the measurement-based model in the context of a particular instance of delegated quantum computation where the server prepares the initial computational resource and sends it to the client who drives the computation by single-qubit measurements. Applying self-testing procedures we are able to verify the initial resource as well as the operation of the quantum devices, and hence the computation itself. The overhead of our protocol scales as the size of the initial resource state to the power of 4 times the natural logarithm of the initial state's size.
- Feb 24 2016 quant-ph arXiv:1602.07131v2In quantum metrology, it is widely believed that the quantum Cramer-Rao bound is attainable bound while it is not true. In order to clarify this point, we explain why the quantum Cramer-Rao bound cannot be attained geometrically. In this manuscript, we investigate noiseless channel estimation under energy constraint for states, using a physically reasonable error function, and present the optimal state and the attainable bound. We propose the experimental generation of the optimal states for enhanced metrology using squeezing transformations. This makes the estimation of unitary channels physically implementable, while existing unitary estimation protocols do not work
- We revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis. We introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms. In this case we find the threshold rate, the optimal error and strong converse exponents (at large deviations from the threshold) and the second order asymptotics (at small deviations from the threshold). We apply our results to find operational interpretations of various Renyi information measures. In case the alternative hypothesis is comprised of bipartite product distributions, we find that the optimal error and strong converse exponents are determined by variations of Renyi mutual information. In case the alternative hypothesis consists of tripartite distributions satisfying the Markov property, we find that the optimal exponents are determined by variations of Renyi conditional mutual information. In either case the relevant notion of Renyi mutual information depends on the precise choice of the alternative hypothesis. As such, our work also strengthens the view that different definitions of Renyi mutual information, conditional entropy and conditional mutual information are adequate depending on the context in which the measures are used.
- Oct 19 2015 quant-ph arXiv:1510.04711v1We propose a general framework for constructing universal steering criteria that are applicable to arbitrary bipartite states and measurement settings of the steering party. The same framework is also useful for studying the joint measurement problem. Based on the data-processing inequality for an extended Rényi relative entropy, we then introduce a family of universal steering inequalities, which detect steering much more efficiently than those inequalities known before. As illustrations, we show unbounded violation of a steering inequality for assemblages constructed from mutually unbiased bases and establish an interesting connection between maximally steerable assemblages and complete sets of mutually unbiased bases. We also provide a single steering inequality that can detect all bipartite pure states of full Schmidt rank. In the course of study, we generalize a number of results intimately connected to data-processing inequalities, which are of independent interest.
- Jun 23 2015 quant-ph arXiv:1506.06447v1We show that the class QMA does not change even if we restrict Arthur's computing ability to only Clifford gate operations (plus classical XOR gate). The idea is to use the fact that the preparation of certain single-qubit states, so called magic states, plus any Clifford gate operations are universal for quantum computing. If Merlin is honest, he sends the witness plus magic states to Arthur. If Merlin is malicious, he might send other states to Arthur, but Arthur can verify the correctness of magic states by himself. We also generalize the result to QIP[3]: we show that the class QIP[3] does not change even if the computational power of the verifier is restricted to only Clifford gate operations (plus classical XOR gate).
- May 29 2015 quant-ph arXiv:1505.07535v2We introduce a simple protocol for verifiable measurement-only blind quantum computing. Alice, a client, can perform only single-qubit measurements, whereas Bob, a server, can generate and store entangled many-qubit states. Bob generates copies of a graph state, which is a universal resource state for measurement-based quantum computing, and sends Alice each qubit of them one by one. Alice adaptively measures each qubit according to her program. If Bob is honest, he generates the correct graph state, and therefore Alice can obtain the correct computation result. Regarding the security, whatever Bob does, Bob cannot learn any information about Alice's computation because of the no-signaling principle. Furthermore, evil Bob does not necessarily send the copies of the correct graph state, but Alice can check the correctness of Bob's state by directly verifying stabilizers of some copies.
- Apr 24 2015 quant-ph cond-mat.stat-mech arXiv:1504.06150v5There exist two formulations for quantum heat engine that models an energy transfer between two microscopic systems. One is semi-classical scenario, and the other is full quantum scenario. The former is formulated as a unitary evolution for the internal system, and is adopted by the community of statistical mechanics. In the latter, the whole process is formulated as unitary, and is adopted by the community of quantum information. This paper proposes a model for quantum heat engine that transfers energy from a collection of microscopic systems to a macroscopic system like a fuel cell. In such a situation, the amount of extracted work is visible for a human. For this purpose, we formulate quantum heat engine as the measurement process whose measurement outcome is the amount of extracted work. Under this model, we derive a suitable energy conservation law and propose a more concrete submodel. Then, we derive a novel trade-off relation between the measurability of the amount of work extraction and the coherence of the internal system, which examines the application of the semi-classical scenario to a heat engine transferring an energy from a collection of microscopic systems to a macroscopic system.
- Recently, entanglement concentration was explicitly shown to be irreversible. However, it is still not clear what kind of states can be reversibly converted in the asymptotic setting by LOCC when neither the initial nor the target state is maximally entangled. We derive the necessary and sufficient condition for the reversibility of LOCC conversions between two bipartite pure entangled states in the asymptotic setting. In addition, we show that conversion can be achieved perfectly with only local unitary operation under such condition except for special cases. Interestingly, our result implies that an error-free reversible conversion is asymptotically possible even between states whose copies can never be locally unitarily equivalent with any finite numbers of copies, although such a conversion is impossible in the finite setting. In fact, we show such an example. Moreover, we establish how to overcome the irreversibility of LOCC conversion in two ways. As for the first method, we evaluate how many copies of the initial state is to be lost to overcome the irreversibility of LOCC conversion. The second method is to add a supplementary state appropriately, which also works for LU conversion unlike the first method. Especially, for the qubit system, any non-maximally pure entangled state can be a universal resource for the asymptotic reversibility when copies of the state is sufficiently many. More interestingly, our analysis implies that far-from-maximally entangled states can be better than nearly maximally entangled states as this type of resource. This fact brings new insight to the resource theory of state conversion.
- Sep 16 2014 quant-ph arXiv:1409.3897v4We consider asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state $\ket{\Psi}$ and the completely mixed state under one-way LOCC (local operations and classical communications), two-way LOCC, and separable POVMs. As a result, we derive the Hoeffding bounds under two-way LOCC POVMs and separable POVMs. Further, we derive a Stein's lemma type of optimal error exponents under one-way LOCC, two-way LOCC, and separable POVMs up to the third order, which clarifies the difference between one-way and two-way LOCC POVM. Our study gives a very rare example in which the optimal performance under the infinite-round two-way LOCC is also equal to that under separable operations and can be attained with two-round communication, but not attained with the one-way LOCC.
- A variety of new measures of quantum Renyi mutual information and quantum Renyi conditional entropy have recently been proposed, and some of their mathematical properties explored. Here, we show that the Renyi mutual information attains operational meaning in the context of composite hypothesis testing, when the null hypothesis is a fixed bipartite state and the alternate hypothesis consists of all product states that share one marginal with the null hypothesis. This hypothesis testing problem occurs naturally in channel coding, where it corresponds to testing whether a state is the output of a given quantum channel or of a 'useless' channel whose output is decoupled from the environment. Similarly, we establish an operational interpretation of Renyi conditional entropy by choosing an alternative hypothesis that consists of product states that are maximally mixed on one system. Specialized to classical probability distributions, our results also establish an operational interpretation of Renyi mutual information and Renyi conditional entropy.
- May 27 2014 quant-ph cond-mat.stat-mech arXiv:1405.6457v3The optimal efficiency of quantum (or classical) heat engines whose heat baths are $n$-particle systems is given by the information geometry and the strong large deviation. We give the optimal work extraction process as a concrete energy-preserving unitary time evolution among the heat baths and the work storage. We show that our optimal work extraction turns the disordered energy of the heat baths to the ordered energy of the work storage, by evaluating the ratio of the entropy difference to the energy difference in the heat baths and the work storage, respectively. By comparing the statistical mechanical optimal efficiency with the macroscopic thermodynamic bound, we evaluate the accuracy of the macroscopic thermodynamics with finite-size heat baths from the statistical mechanical viewpoint. We also evaluate the quantum coherence effect on the optimal efficiency of the cycle processes without restricting their cycle time, by comparing the classical and quantum optimal efficiencies.
- We consider random number conversion (RNC) through random number storage with restricted size. We clarify the relation between the performance of RNC and the size of storage in the framework of first- and second- order asymptotics, and derive their rate regions. Then, we show that the results for RNC with restricted storage recover those for conventional RNC without storage in the limit of storage size. To treat RNC via restricted storage, we introduce a new kind of probability distributions named generalized Rayleigh-normal distributions. Using the generalized Rayleigh-normal distributions, we can describe the second-order asymptotic behaviour of RNC via restricted storage in a unified manner. As an application to quantum information theory, we analyze LOCC conversion via entanglement storage with restricted size. Moreover, we derive the optimal LOCC compression rate under a constraint of conversion accuracy.
- We explicitly construct random hash functions for privacy amplification (extractors) that require smaller random seed lengths than the previous literature, and still allow efficient implementations with complexity $O(n\log n)$ for input length $n$. The key idea is the concept of dual universal$_2$ hash function introduced recently. We also use a new method for constructing extractors by concatenating $\delta$-almost dual universal$_2$ hash functions with other extractors. Besides minimizing seed lengths, we also introduce methods that allow one to use non-uniform random seeds for extractors. These methods can be applied to a wide class of extractors, including dual universal$_2$ hash function, as well as to conventional universal$_2$ hash functions.
- Recently a new quantum generalization of the Renyi divergence and the corresponding conditional Renyi entropies was proposed. Here we report on a surprising relation between conditional Renyi entropies based on this new generalization and conditional Renyi entropies based on the quantum relative Renyi entropy that was used in previous literature. Our result generalizes the well-known duality relation H(A|B) + H(A|C) = 0 of the conditional von Neumann entropy for tripartite pure states to Renyi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional Renyi entropies and derive a new entropic uncertainty relation.
- Nov 14 2013 quant-ph arXiv:1311.3003v2In the decoy quantum key distribution, we show that a smaller decoy intensity gives a better key generation rate in the asymptotic setting when we employ only one decoy intensity and the vacuum pulse. In particular, the counting rate of single photon can be perfectly estimated when the decoy intensity is infinitesimal. The same property holds even when the intensities cannot be perfectly identified. Further, we propose a protocol to improve the key generation rate over the existing protocol under the same decoy intensity.
- We discuss the asymptotic behavior of conversions between two independent and identical distributions up to the second-order conversion rate when the conversion is produced by a deterministic function from the input probability space to the output probability space. To derive the second-order conversion rate, we introduce new probability distributions named Rayleigh-normal distributions. The family of Rayleigh-normal distributions includes a Rayleigh distribution and coincides with the standard normal distribution in the limit case. Using this family of probability distributions, we represent the asymptotic second-order rates for the distribution conversion. As an application, we also consider the asymptotic behavior of conversions between the multiple copies of two pure entangled states in quantum systems when only local operations and classical communications (LOCC) are allowed. This problem contains entanglement concentration, entanglement dilution and a kind of cloning problem with LOCC restriction as special cases.
- May 28 2013 quant-ph arXiv:1305.6250v3In quantum information theory, it is widely believed that entanglement concentration for bipartite pure states is asymptotically reversible. In order to examine this, we give a precise formulation of the problem, and show a trade-off relation between performance and reversibility, which implies the irreversibility of entanglement concentration. Then, we regard entanglement concentration as entangled state compression in an entanglement storage with lower dimension. Because of the irreversibility of entanglement concentration, an initial state can not be completely recovered after the compression process and a loss inevitably arises in the process. We numerically calculate this loss and also derive for it a highly accurate analytical approximation.
- Security analysis of the decoy method with the Bennett-Brassard 1984 protocol for finite key lengthsFeb 19 2013 quant-ph arXiv:1302.4139v4This paper provides a formula for the sacrifice bit-length for privacy amplification with the Bennett-Brassard 1984 protocol for finite key lengths when we employ the decoy method. Using the formula, we can guarantee the security parameter for realizable quantum key distribution system. The key generation rates with finite key lengths are numerically evaluated. The proposed method improves the existing key generation rate even in the asymptotic setting.
- Sep 18 2012 quant-ph arXiv:1209.3463v3This article proposes a unified method to estimation of group action by using the inverse Fourier transform of the input state. The method provides optimal estimation for commutative and non-commutative group with/without energy constraint. The proposed method can be applied to projective representations of non-compact groups as well as of compact groups. This paper addresses the optimal estimation of R, U(1), SU(2), SO(3), and R^2 with Heisenberg representation under a suitable energy constraint.
- Aug 08 2012 quant-ph arXiv:1208.1478v3We consider two fundamental tasks in quantum information theory, data compression with quantum side information as well as randomness extraction against quantum side information. We characterize these tasks for general sources using so-called one-shot entropies. We show that these characterizations - in contrast to earlier results - enable us to derive tight second order asymptotics for these tasks in the i.i.d. limit. More generally, our derivation establishes a hierarchy of information quantities that can be used to investigate information theoretic tasks in the quantum domain: The one-shot entropies most accurately describe an operational quantity, yet they tend to be difficult to calculate for large systems. We show that they asymptotically agree up to logarithmic terms with entropies related to the quantum and classical information spectrum, which are easier to calculate in the i.i.d. limit. Our techniques also naturally yields bounds on operational quantities for finite block lengths.
- May 22 2012 quant-ph arXiv:1205.4370v1For a pure state $\psi$ on a composite system $\mathcal{H}_A\otimes\mathcal{H}_B$, both the entanglement cost $E_C(\psi)$ and the distillable entanglement $E_D(\psi)$ coincide with the von Neumann entropy $H(\mathrm{Tr}_{B}\psi)$. Therefore, the entanglement concentration from the multiple state $\psi^{\otimes n}$ of a pure state $\psi$ to the multiple state $\Phi^{\otimes L_n}$ of the EPR state $\Phi$ seems to be able to be reversibly performed with an asymptotically infinitesimal error when the rate ${L_n}/{n}$ goes to $H(\mathrm{Tr}_{B}\psi)$. In this paper, we show that it is impossible to reversibly perform the entanglement concentration for a multiple pure state even in asymptotic situation. In addition, in the case when we recover the multiple state $\psi^{\otimes M_n}$ after the concentration for $\psi^{\otimes n}$, we evaluate the asymptotic behavior of the loss number $n-M_n$ of $\psi$. This evaluation is thought to be closely related to the entanglement compression in distant parties.
- We treat secret key extraction when the eavesdropper has correlated quantum states. We propose quantum privacy amplification theorems different from Renner's, which are based on quantum conditional Rényi entropy of order 1+s. Using those theorems, we derive an exponential decreasing rate for leaked information and the asymptotic equivocation rate, which have not been derived hitherto in the quantum setting.
- A usual code for quantum wiretap channel requires an auxiliary random variable subject to the perfect uniform distribution. However, it is difficult to prepare such an auxiliary random variable. We propose a code that requires only an auxiliary random variable subject to a non-uniform distribution instead of the perfect uniform distribution. Further, we evaluate the exponential decreasing rate of leaked information and derive its equivocation rate. For practical constructions, we also discuss the security when our code consists of a linear error correcting code.
- It is known that the security evaluation can be done by smoothing of Rényi entropy of order 2 in the classical and quantum settings when we apply universal$_2$ hash functions. Using the smoothing of Renyi entropy of order 2, we derive security bounds for $L_1$ distinguishability and modified mutual information criterion under the classical and quantum setting, and have derived these exponential decreasing rates. These results are extended to the case when we apply $\varepsilon$-almost dual universal$_2$ hash functions. Further, we apply this analysis to the secret key generation with error correction.
- We treat quantum counterparts of testing problems whose optimal tests are given by chi-square, t and F tests. These quantum counterparts are formulated as quantum hypothesis testing problems concerning quantum Gaussian states families, and contain disturbance parameters, which have group symmetry. Quantum Hunt-Stein Theorem removes a part of these disturbance parameters, but other types of difficulty still remain. In order to remove them, combining quantum Hunt-Stein theorem and other reduction methods, we establish a general reduction theorem that reduces a complicated quantum hypothesis testing problem to a fundamental quantum hypothesis testing problem. Using these methods, we derive quantum counterparts of chi-square, t and F tests as optimal tests in the respective settings.
- Sep 08 2011 quant-ph arXiv:1109.1349v3The monogamy of entanglement is one of the basic quantum mechanical features, which says that when two partners Alice and Bob are more entangled then either of them has to be less entangled with the third party. Here we qualitatively present the converse monogamy of entanglement: given a tripartite pure system and when Alice and Bob are entangled and non-distillable, then either of them is distillable with the third party. Our result leads to the classification of tripartite pure states based on bipartite reduced density operators, which is a novel and effective way to this long-standing problem compared to the means by stochastic local operations and classical communications. Furthermore we systematically indicate the structure of the classified states and generate them. We also extend our results to multipartite states.
- Jul 05 2011 quant-ph arXiv:1107.0589v2We present a tight security analysis of the Bennett-Brassard 1984 protocol taking into account the finite size effect of key distillation, and achieving unconditional security. We begin by presenting a concise analysis utilizing the normal approximation of the hypergeometric function. Then next we show that a similarly tight bound can also be obtained by a rigorous argument without relying on any approximation. In particular, for the convenience of experimentalists who wish to evaluate the security of their QKD systems, we also give explicit procedures of our key distillation, and also show how to calculate the secret key rate and the security parameter from a given set of experimental parameters. Besides the exact values of key rates and security parameters, we also present how to obtain their rough estimates using the normal approximation.
- May 20 2011 quant-ph arXiv:1105.3789v2In this paper, we treat an asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state and the completely mixed state by one-way LOCC, two-way LOCC, and separable POVMs. As a result, we derive single-letterized formulas for the Stein's lemma type of optimal error exponents under one-way LOCC, two-way LOCC and separable POVMs, the Chernoff bounds under one-way LOCC POVMs and separable POVMs, and the Hoeffding bounds under one-way LOCC POVMs in the whole region of a parameter and under separable POVMs on a restricted region of a parameter. We also numerically calculate the Chernoff and the Hoeffding bounds under a class of three-step LOCC protocols in low-dimensional systems and show that these bounds not only outperform the bounds for one-way LOCC POVMs but also almost approximates the bounds for separable POVMs in the parameter region where analytical bounds for separable POVMs are derived.
- Feb 15 2011 quant-ph arXiv:1102.2555v1In the changepoint problem, we determine when the distribution observed has changed to another one. We expand this problem to the quantum case where copies of an unknown pure state are being distributed. We study the fundamental case, which has only two candidates to choose. This problem is equal to identifying a given state with one of the two unknown states when multiple copies of the states are provided. In this paper, we assume that two candidate states are distributed independently and uniformly in the space of the whole pure states. The minimum of the averaged error probability is given and the optimal POVM is defined as to obtain it. Using this POVM, we also compute the error probability which depends on the inner product. These analytical results allow us to calculate the value in the asymptotic case, where this problem approaches to the usual discrimination problem.
- Feb 03 2011 quant-ph arXiv:1102.0439v3The monogamy of entanglement is one of the basic quantum mechanical features, which says that when two partners Alice and Bob are more entangled then either of them has to be less entangled with the third party. Here we qualitatively present the converse monogamy of entanglement: given a tripartite pure system and when Alice and Bob are weakly entangled, then either of them is generally strongly entangled with the third party. Our result leads to the classification of tripartite pure states based on bipartite reduced density operators, which is a novel and effective way to this long-standing problem compared to the means by stochastic local operations and classical communications. We also systematically indicate the structure of the classified states and generate them.
- In this paper, we introduce the concept of dual universality of hash functions and present its applications to quantum cryptography. We begin by establishing the one-to-one correspondence between a linear function family \cal F and a code family \cal C, and thereby defining \varepsilon-almost dual universal_2 hash functions, as a generalization of the conventional universal_2 hash functions. Then we show that this generalized (and thus broader) class of hash functions is in fact sufficient for the security of quantum cryptography. This result can be explained in two different formalisms. First, by noting its relation to the \delta-biased family introduced by Dodis and Smith, we demonstrate that Renner's two-universal hashing lemma is generalized to our class of hash functions. Next, we prove that the proof technique by Shor and Preskill can be applied to quantum key distribution (QKD) systems that use our generalized class of hash functions for privacy amplification. While Shor-Preskill formalism requires an implementer of a QKD system to explicitly construct a linear code of the Calderbank-Shor-Steane type, this result removes the existing difficulty of the construction a linear code of CSS code by replacing it by the combination of an ordinary classical error correcting code and our proposed hash function. We also show that a similar result applies to the quantum wire-tap channel. Finally we compare our results in the two formalisms and show that, in typical QKD scenarios, the Shor-Preskill--type argument gives better security bounds in terms of the trace distance and Holevo information, than the method based on the \delta-biased family.
- Dec 17 2010 quant-ph arXiv:1012.3564v2Characterizing the transformation and classification of multipartite entangled states is a basic problem in quantum information. We study the problem under two most common environments, local operations and classical communications (LOCC), stochastic LOCC and two more general environments, multi-copy LOCC (MCLOCC) and multi-copy SLOCC (MCSLOCC). We show that two transformable multipartite states under LOCC or SLOCC are also transformable under MCLOCC and MCSLOCC. What's more, these two environments are equivalent in the sense that two transformable states under MCLOCC are also transformable under MCSLOCC, and vice versa. Based on these environments we classify the multipartite pure states into a few inequivalent sets and orbits, between which we build the partial order to decide their transformation. In particular, we investigate the structure of SLOCC-equivalent states in terms of tensor rank, which is known as the generalized Schmidt rank. Given the tensor rank, we show that GHZ states can be used to generate all states with a smaller or equivalent tensor rank under SLOCC, and all reduced separable states with a cardinality smaller or equivalent than the tensor rank under LOCC. Using these concepts, we extended the concept of "maximally entangled state" in the multi-partite system.
- Nov 12 2010 quant-ph arXiv:1011.2546v2Many researches proposed the use of the noon state as the input state for phase estimation, which is one topic of quantum metrology. This is because the input noon state provides the maximum Fisher information at the specific point. However, the Fisher information does not necessarily give the attainable bound for estimation error. In this paper, we adopt the local asymptotic mini-max criterion as well as the mini-max criterion, and show that the maximum Fisher information does not give the attainable bound for estimation error under these criteria in the phase estimation. We also propose the optimal input state under the constraints for photon number of the input state instead of the noon state.
- Jun 15 2010 quant-ph arXiv:1006.2744v1In this paper, we treat a local discrimination problem in the framework of asymmetric hypothesis testing. We choose a known bipartite pure state $\ket{\Psi}$ as an alternative hypothesis, and the completely mixed state as a null hypothesis. As a result, we analytically derive an optimal type 2 error and an optimal POVM for one-way LOCC POVM and Separable POVM. For two-way LOCC POVM, we study a family of simple three-step LOCC protocols, and show that the best protocol in this family has strictly better performance than any one-way LOCC protocol in all the cases where there may exist difference between two-way LOCC POVM and one-way LOCC POVM.
- Mar 25 2010 quant-ph arXiv:1003.4575v3In a unified viewpoint in quantum channel estimation, we compare the Cramer-Rao and the mini-max approaches, which gives the Bayesian bound in the group covariant model. For this purpose, we introduce the local asymptotic mini-max bound, whose maximum is shown to be equal to the asymptotic limit of the mini-max bound. It is shown that the local asymptotic mini-max bound is strictly larger than the Cramer-Rao bound in the phase estimation case while the both bounds coincide when the minimum mean square error decreases with the order O(1/n). We also derive a sufficient condition for that the minimum mean square error decreases with the order O(1/n).
- Feb 15 2010 quant-ph arXiv:1002.2511v5We study the additivity property of three multipartite entanglement measures, i.e. the geometric measure of entanglement (GM), the relative entropy of entanglement and the logarithmic global robustness. First, we show the additivity of GM of multipartite states with real and non-negative entries in the computational basis. Many states of experimental and theoretical interests have this property, e.g. Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states, the Smolin state, and the generalization of Dür's multipartite bound entangled states. We also prove the additivity of other two measures for some of these examples. Second, we show the non-additivity of GM of all antisymmetric states of three or more parties, and provide a unified explanation of the non-additivity of the three measures of the antisymmetric projector states. In particular, we derive analytical formulae of the three measures of one copy and two copies of the antisymmetric projector states respectively. Third, we show, with a statistical approach, that almost all multipartite pure states with sufficiently large number of parties are nearly maximally entangled with respect to GM and relative entropy of entanglement. However, their GM is not strong additive; what's more surprising, for generic pure states with real entries in the computational basis, GM of one copy and two copies, respectively, are almost equal. Hence, more states may be suitable for universal quantum computation, if measurements can be performed on two copies of the resource states. We also show that almost all multipartite pure states cannot be produced reversibly with the combination multipartite GHZ states under asymptotic LOCC, unless relative entropy of entanglement is non-additive for generic multipartite pure states.
- Dec 15 2009 quant-ph arXiv:0912.2610v3We investigate a discrimination scheme between unitary processes. By introducing a margin for the probability of erroneous guess, this scheme interpolates the two standard discrimination schemes: minimum-error and unambiguous discrimination. We present solutions for two cases. One is the case of two unitary processes with general prior probabilities. The other is the case with a group symmetry: the processes comprise a projective representation of a finite group. In the latter case, we found that unambiguous discrimination is a kind of "all or nothing": the maximum success probability is either 0 or 1. We also closely analyze how entanglement with an auxiliary system improves discrimination performance.
- Quantum cryptographic technology (QCT) is expected to be a fundamental technology for realizing long-term information security even against as-yet-unknown future technologies. More advanced security could be achieved using QCT together with contemporary cryptographic technologies. To develop and spread the use of QCT, it is necessary to standardize devices, protocols, and security requirements and thus enable interoperability in a multi-vendor, multi-network, and multi-service environment. This report is a technical summary of QCT and related topics from the viewpoints of 1) consensual establishment of specifications and requirements of QCT for standardization and commercialization and 2) the promotion of research and design to realize New-Generation Quantum Cryptography.
- May 01 2009 quant-ph arXiv:0905.0010v2In this paper for a class of symmetric multiparty pure states we consider a conjecture related to the geometric measure of entanglement: 'for a symmetric pure state, the closest product state in terms of the fidelity can be chosen as a symmetric product state'. We show that this conjecture is true for symmetric pure states whose amplitudes are all non-negative in a computational basis. The more general conjecture is still open.
- Apr 07 2009 quant-ph arXiv:0904.0704v2The asymptotic discrimination problem of two quantum states is studied in the setting where measurements are required to be invariant under some symmetry group of the system. We consider various asymptotic error exponents in connection with the problems of the Chernoff bound, the Hoeffding bound and Stein's lemma, and derive bounds on these quantities in terms of their corresponding statistical distance measures. A special emphasis is put on the comparison of the performances of group-invariant and unrestricted measurements.
- Apr 03 2009 quant-ph arXiv:0904.0307v1We consider two kind of energy constraints when the output state is a coherent state. One is a constraint on the total energy during a fixed period; the other is a constraint on the total energy for a single code. The first setting can be easily dealt with by using the conventional capacity formula. The second setting requires the general capacity formula for a classical-quantum channel.
- Nov 03 2008 quant-ph arXiv:0810.5602v1For a unified analysis on the phase estimation, we focus on the limiting distribution. It is shown that the limiting distribution can be given by the absolute square of the Fourier transform of $L^2$ function whose support belongs to $[-1,1]$. Using this relation, we study the relation between the variance of the limiting distribution and its tail probability. As our result, we prove that the protocol minimizing the asymptotic variance does not minimize the tail probability. Depending on the width of interval, we derive the estimation protocol minimizing the tail probability out of a given interval. Such an optimal protocol is given by a prolate spheroidal wave function which often appears in wavelet or time-limited Fourier analysis. Also, the minimum confidence interval is derived with the framework of interval estimation that assures a given confidence coefficient.
- Oct 21 2008 quant-ph arXiv:0810.3381v1Group symmetric LOCC measurement for detecting maximally entangled state is considered. Usually, this type measurement has continuous-valued outcomes. However, any realizable measurement has finite-valued outcomes. This paper proposes discrete realizations of such a group symmetric LOCC measurement.
- Oct 21 2008 quant-ph arXiv:0810.3380v1In the asymptotic setting, the optimal test for hypotheses testing of the maximally entangled state is derived under several locality conditions for measurements. The optimal test is obtained in several cases with the asymptotic framework as well as the finite-sample framework. In addition, the experimental scheme for the optimal test is presented.
- Jun 09 2008 quant-ph arXiv:0806.1091v2We have proven that there exists a quantum state approximating any multi-copy state universally when we measure the error by means of the normalized relative entropy. While the qubit case was proven by Krattenthaler and Slater (IEEE Trans. IT, 46, 801-819 (2000); quant-ph/9612043), the general case has been open for more than ten years. For a deeper analysis, we have solved the mini-max problem concerning `approximation error' up to the second order. Furthermore, we have applied this result to quantum lossless data compression, and have constructed a universal quantum lossless data compression.
- May 28 2008 quant-ph arXiv:0805.4092v2We construct a universal code for stationary and memoryless classical-quantum channel as a quantum version of the universal coding by Csiszár and Körner. Our code is constructed by the combination of irreducible representation, the decoder introduced through quantum information spectrum, and the packing lemma.
- May 22 2008 quant-ph arXiv:0805.3190v2In the original BB84 protocol, the bit basis and the phase basis are used with equal probability. Lo et al (J. of Cryptology, 18, 133-165 (2005)) proposed to modify the ratio between the two bases by increasing the final key generation rate. However, the optimum ratio has not been derived. In this letter, in order to examine this problem, the ratio between the two bases is optimized for exponential constraints given Eve's information distinguishability and the final error probability.
- The optimal exponential error rate for adaptive discrimination of two channels is discussed. In this problem, adaptive choice of input signal is allowed. This problem is discussed in various settings. It is proved that adaptive choice does not improve the exponential error rate in these settings. These results are applied to quantum state discrimination.
- Jan 31 2008 quant-ph arXiv:0801.4604v1Quantum optical Gaussian states are a type of important robust quantum states which are manipulatable by the existing technologies. So far, most of the important quantum information experiments are done with such states, including bright Gaussian light and weak Gaussian light. Extending the existing results of quantum information with discrete quantum states to the case of continuous variable quantum states is an interesting theoretical job. The quantum Gaussian states play a central role in such a case. We review the properties and applications of Gaussian states in quantum information with emphasis on the fundamental concepts, the calculation techniques and the effects of imperfections of the real-life experimental setups. Topics here include the elementary properties of Gaussian states and relevant quantum information device, entanglement-based quantum tasks such as quantum teleportation, quantum cryptography with weak and strong Gaussian states and the quantum channel capacity, mathematical theory of quantum entanglement and state estimation for Gaussian states.
- Oct 05 2007 quant-ph arXiv:0710.1056v2We study various distance-like entanglement measures of multipartite states under certain symmetries. Using group averaging techniques we provide conditions under which the relative entropy of entanglement, the geometric measure of entanglement and the logarithmic robustness are equivalent. We consider important classes of multiparty states, and in particular show that these measures are equivalent for all stabilizer states, symmetric basis and antisymmetric basis states. We rigorously prove a conjecture that the closest product state of permutation symmetric states can always be chosen to be permutation symmetric. This allows us to calculate the explicit values of various entanglement measures for symmetric and antisymmetric basis states, observing that antisymmetric states are generally more entangled. We use these results to obtain a variety of interesting ensembles of quantum states for which the optimal LOCC discrimination probability may be explicitly determined and achieved. We also discuss applications to the construction of optimal entanglement witnesses.