We investigate the unambiguous comparison of quantum states in a scenario that is more general than the one that was originally suggested by Barnett et al. First, we find the optimal solution for the comparison of two states taken from a set of two pure states with arbitrary a priori probabilities. We show that the optimal coherent measurement is always superior to the optimal incoherent measurement. Second, we develop a strategy for the comparison of two states from a set of N pure states, and find an optimal solution for some parameter range when N=3. In both cases we use the reduction method for the corresponding problem of mixed state discrimination, as introduced by Raynal et al., which reduces the problem to the discrimination of two pure states only for N=2. Finally, we provide a necessary and sufficient condition for unambiguous comparison of mixed states to be possible.
Semi-quantum protocols construct connections between quantum users and ``classical'' users who can only perform certain ``classical'' operations. In this paper, we present a new semi-quantum private comparison protocol based on entangled states and single particles, which does not require pre-shared keys between the ``classical'' users to guarantee the security of their private data. By utilizing multi-particle entangled states and single particles, our protocol can be easily extended to multi-party scenarios to meet the requirements of multiple ``classical'' users who want to compare their pr…
The original definition of quantum discord of bipartite states was defined over projective measurements, in this paper we discuss some generalizations of it. These generalizations are defined over general measurements, rank-one general measurements or Neumark extension measurements. We investigate the nonnegativity, zero-discord sets of all these quantum discords and some properties about them.
A motivation is given for expressing classical mechanics in terms of diagonal projection matrices and diagonal density matrices. Then quantum mechanics is seen to be a simple generalization in which one replaces the diagonal real matrices with suitable Hermitian matrices.
We review generalizations of quantum statistics, including parabose, parafermi, and quon statistics, but not including anyon statistics, which is special to two dimensions.
An extension of the Liouville-von Neumann dynamics to a Nambu-type dynamics is proposed. The resulting theory is the first version of nonlinear QM which is free from internal inconsistencies.
We analyze quantum superdense coding as would be seen in Everett's many worlds interpretation of measurement, and compare it to Everettian teleportation.
We introduce the set of quantum channels with constant Frobenius norm, the set of diagonal channels and the notion of equivalence of one-parameter families of channels. First, we show that all diagonal $2$-dimensional channels with constant Frobenius norm are equivalent. Next, we generalize four one-parameter families of $2$-dimensional diagonal channels with constant Frobenius norm to an arbitrary dimension $n$. Finally, we prove that the generalizations are not equivalent in any dimension $n \ge 3$.
The recent article "Entropic Updating of Probability and Density Matrices" [1] derives and demonstrates the inferential origins of both the standard and quantum relative entropies in unison. Operationally, the standard and quantum relative entropies are shown to be designed for the purpose of inferentially updating probability distributions and density matrices, respectively, when faced with incomplete information. We call the inferential updating procedure for density matrices the "quantum maximum entropy method". Standard inference techniques in probability theory can be criticized for lacki…
We generalize the problem of coin flipping to more than two outcomes and parties. We term this problem dice rolling, and study both its weak and strong variants. We prove by construction that in quantum settings (i) weak N-sided dice rolling admits an arbitrarily small bias for any value of N, and (ii) two-party strong N-sided dice rolling saturates the corresponding generalization of Kitaev's bound for any value of N. In addition, we make use of this last result to introduce a family of optimal 2m-party strong n^m-sided dice rolling protocols for any value of m and n.
A systematic approach to the non-Markovian quantum dynamics of open systems is given by the projection operator techniques of nonequilibrium statistical mechanics. Combining these methods with concepts from quantum information theory and from the theory of positive maps, we derive a class of correlated projection superoperators that take into account in an efficient way statistical correlations between the open system and its environment. The result is used to develop a generalization of the Lindblad theory to the regime of highly non-Markovian quantum processes in structured environments.
Quantum Fourier transformation is important in many quantum algorithms. In this paper, we generalize quantum Fourier transformation over the Abelian group $\mathbb{Z}_N$ from two different points to get more efficient unitary transformations. The obtained unitary transformations are given in concise and explicit formula which can be used directly. A relationship between the generalized quantum Fourier transformation and the dihedral hidden subgroup problem are discussed in this paper. This may lead a way to solve the dihedral hidden subgroup problem. The second goal of this paper is to give an…
It is shown that the stochastic model of Fenyes and Nelson can be generalized in such a way that the diffusion constant of the Markov theory becomes a free parameter. This extra freedom allows one to identify quantum mechanics with a class of Markov processes with diffusion constants varying from zero to infinity.
Quantum private comparison (QPC) aims to accomplish the equality comparison of the secrets from different users without disclosing their genuine contents by using the principles of quantum mechanics. In this paper, we summarize eight modes of quantum state preparation and transmission existing in current QPC protocols first. Then, by using the mode of scattered preparation and one-way convergent transmission, we construct a new multi-user quantum private comparison (MQPC) protocol with two-particle maximally entangled states, which can accomplish arbitrary pair's comparison of equality among K…
This study presents the first semi-quantum private comparison protocol under an almost-dishonest third party. The proposed protocol allows two classical participants to compare their secret information without compromising it's privacy. The security analyses indicate that the protocol is free from several well-known attacks.
We show that Coecke's compositionality theorem for quantum information flow follows by the universal property of tensor products from the case in which all relevant states are totally disentangled, for which the proof is almost trivial. With the same technique we deduce a PROP structure behind general multipartite quantum information processing and show that all such are equivalent to a canonical teleportation-type form. Some philosophical issues concerning quantum information are also touched upon.
There are major advantages in a newer version of Grover's quantum algorithm utilizing a general unitary transformation in the search of a single object in a large unsorted database. In this paper, we generalize this algorithm to multiobject search. We show the techniques to achieve the reduction of the problem to one on an invariant subspace of dimension just equal to two.
Quantum parallelism is the main feature of quantum computation. In 1985 D. Deutsch showed that a single quantum computation may be sufficient to state whether a two-valued function of a two-valued variable is constant or not. Though the generalized problem with unconstrained domain and range size admits no deterministic quantum solution, a fully probabilistic quantum algorithm is presented in which quantum parallelism is harnessed to achieve a quicker exploration of the domain with respect to the classical ``sampling'' strategy.
This study points out a semi-quantum protocol for private comparison using Bell states (SQPC) suffering from the double C-NOT attack and the malicious agent attack. The attacker can easily obtain information through these attacks. An improved protocol is proposed, which can effectively resist both of these attacks.
This paper demonstrates the use of entanglement resources in quantum speedup by presenting an algorithm which is the generalization of an algorithm proposed by Goswami and Panigrahi [arXiv:1706.09489 (2017)]. We generalize the algorithm and show that it provides deterministic solutions having an advantage over classical algorithm. The algorithm answers the question of whether a given function is constant or balanced and whether two functions are equal or unequal. Finally, we experimentally verify the algorithm by using IBM's five-qubit quantum computer with a high fidelity.
We analyze the security of the efficient two-party quantum private comparison protocol with decoy photons and two-photon entanglement. It is shown that the compromised third party (TP) can obtain the final comparison result without introducing any detectable errors in the standard semi-honest model. The attack strategy is discussed in detail and an improvement of this protocol is demonstrated. The idea of our attack might be instructive for the cryptanalysis of quantum cryptographic schemes.
L. K. Grover's search algorithm in quantum computing gives an optimal, quadratic speedup in the search for a single object in a large unsorted database. In this paper, we generalize Grover's algorithm in a Hilbert-space framework for both continuous and discrete time cases that isolates its geometrical essence to the case where more than one object satisfies the search criterion.
We show that the theory of operator quantum error correction can be naturally generalized by allowing constraints not only on states but also on observables. The resulting theory describes the correction of algebras of observables (and may therefore suitably be called ``operator algebra quantum error correction''). In particular, the approach provides a framework for the correction of hybrid quantum-classical information and it does not require the state to be entirely in one of the corresponding subspaces or subsystems. We discuss applications to quantum teleportation and to the study of info…
Based on a system-reservoir model and an appropriate choice of nonlinear coupling, we have explored the microscopic quantum generalization of classical Liénard systems. Making use of oscillator coherent states and canonical thermal distributions of the associated c-numbers, we have derived the quantum Langevin equation of the reduced system which admits of single or multiple limit cycles. It has been shown that detailed balance in the form of fluctuation-dissipation relation preserves the dynamical stability of the attractors even in case of vacuum excitation. The quantum versions of Rayleigh,…
Experimental realization of automated error correction is demonstrated through IBM Quantum Experience for Bell and GHZ states using a measurement based approach upon ancilla qubits. The measurement automatically activates error correcting unitary operations to restore the system to its original entangled state. We illustrate the algorithm for the maximally entangled qudit case by applying appropriate Hadamard and Controlled-NOT gates.
We prove a generalization of the quantum Markovian equation for observables. In this generalized equation, we use superoperators that are fractional powers of completely dissipative superoperators. We prove that the suggested superoperators are infinitesimal generators of completely positive semigroups and describe the properties of this semigroup. We solve the proposed fractional quantum Markovian equation for the harmonic oscillator with linear friction. A fractional power of the Markovian superoperator can be considered a parameter describing a measure of "screening" of the environment of t…
We investigate the extent to which we can establish whether or not two quantum systems have been prepared in the same state. We investigate the possibility of universal unambiguous state comparison. We show that it is impossible to conclusively identify two pure unknown states as being identical, and construct the optimal measurement for conclusively identifying them as being different. We then derive optimal strategies for state comparison when the state of each system is one of two known states.
When comparing quantum states to each other, it is possible to obtain an unambiguous answer, indicating that the states are definitely different, already after a single measurement. In this paper we investigate comparison of coherent states, which is the simplest example of quantum state comparison for continuous variables. The method we present has a high success probability, and is experimentally feasible to realize as the only required components are beam splitters and photon detectors. An easily realizable method for quantum state comparison could be important for real applications. As exa…
In this paper, a multi-party quantum private comparison (MQPC) scheme is suggested based on entanglement swapping of Bell entangled states within d-level quantum system, which can accomplish the equality comparison of secret binary sequences from n users via one execution of scheme. Detailed security analysis shows that both the outside attack and the participant attack are ineffective. The suggested scheme needn't establish a private key among n users beforehand through the quantum key distribution (QKD) method to encrypt the secret binary sequences. Compared with previous MQPC scheme based o…
Quantum Private Comparison (QPC) allows us to protect private information during its comparison. In the past various three-party quantum protocols have been proposed that claim to work well under noisy conditions. Here we tackle the problem of QPC under noise. We analyze the EPR-based protocol under depolarizing noise, bit flip and phase flip noise. We show how noise affects the robustness of the EPR-based protocol. We then present a straightforward protocol based on CSS codes to perform QPC which is robust against noise and secure under general attacks.
Possible generalizations of quantum theory permitting to describe in a unique way the development of the quantum system and the measurement process are discussed. The approach to the problem based on the Lindblad's equation for the statistical operator is reviewed. The Tomonaga-Schwinger like equation of this type is introduced to establish Lorentz invariance. The application of tachyonic field to overcome divergences arising in this equation is analyzed. Other approaches to the problem are shortly discussed.
We investigate the counterparts of random walk in universal quantum computing and their implementation using standard quantum circuits. Quantum walk have been recently well investigated for traversing graphs with certain oracles. We focus our study on traversing a 1-D graph, namely a circle, and show how to implement discrete circular quantum walk in quantum circuits built with universal CNOT and single quit gates. We review elementary quantum gates and circuit decomposition and propose a a generalized version of the all CNOT based quantum discrete circular walk. We simulated these circuits on…
To evade the well-known impossibility of unconditionally secure quantum two-party computations, previous quantum private comparison protocols have to adopt a third party. Here we study how far we can go with two parties only. We propose a very feasible and efficient protocol. Intriguingly, although the average amount of information leaked cannot be made arbitrarily small, we find that it never exceeds 14 bits for any length of the bit-string being compared.
It is a fundamental principle of quantum theory that an unknown state cannot be copied or, as a consequence, an unknown optical signal cannot be amplified deterministically and perfectly. Here we describe a protocol that provides nondeterministic quantum optical amplification in the coherent state basis with high gain, high fidelity and which does not use quantum resources. The scheme is based on two mature quantum optical technologies, coherent state comparison and photon subtraction. The method compares favourably with all previous nondeterministic amplifiers in terms of fidelity and success…
We introduce the quantum stochastic walk (QSW), which determines the evolution of generalized quantum mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all possible quantum, classical and quantum-stochastic transitions from a vertex as defined by its connectivity. We show how the family of possible QSWs encompasses both the classical random walk (CRW) and the quantum walk (QW) as special cases, but also includes more general probability distributions. As an example, we study the QSW on a line, the QW to CRW tran…
We present a solution of the problem of the optimal unambiguous comparison of two ensembles of unknown quantum states (psi_1)^k and (psi_2)^l. We consider two cases: 1) The two unknown states psi_1 and psi_2 are arbitrary states of qudits. 2) Alternatively, they are coherent states of a harmonic oscillator. For the case of coherent states we propose a simple experimental realization of the optimal "comparison" machine composed of a finite number of beam-splitters and a single photodetector.
In this work we study an inverse dynamical problem for a bipartite quantum system governed by the time local master equation: to find the class of generators which give rise to a certain time evolution with the constraint of fixed reduced states (marginals). The compatibility of such choice with a global unitary evolution is considered. For the non unitary case we propose a systematic method to reconstruct examples of master equations and address them to different physical scenarios.
We present a comparison between the Quantum Approximate Optimization Algorithm (QAOA) and two widely studied competing methods, Quantum Annealing (QA) and Simulated Annealing (SA). To achieve this, we define a class of optimization problems with respect to their spectral properties which are exactly solvable with QAOA. In this class, we identify instances for which QA and SA have an exponentially small probability to find the solution. Consequently, our results define a first demarcation line between QAOA, Simulated Annealing and Quantum Annealing, and highlight the fundamental differences bet…
Since unconditionally secure quantum two-party computations are known to be impossible, most existing quantum private comparison (QPC) protocols adopted a third party. Recently, we proposed a QPC protocol which involves two parties only, and showed that although it is not unconditionally secure, it only leaks an extremely small amount of information to the other party. Here we further propose the device-independent version of the protocol, so that it can be more convenient and dependable in practical applications.
We investigate the performance of discrimination strategy in the comparison task of known quantum states. In the discrimination strategy, one infers whether or not two quantum systems are in the same state on the basis of the outcomes of separate discrimination measurements on each system. In some cases with more than two possible states, the optimal strategy in minimum-error comparison is that one should infer the two systems are in different states without any measurement, implying that the discrimination strategy performs worse than the trivial "no-measurement" strategy. We present a suffic…
Schur-Weyl duality is a powerful tool in representation theory which has many applications to quantum information theory. We provide a generalization of this duality and demonstrate some of its applications. In particular, we use it to develop a general framework for the study of a family of quantum estimation problems wherein one is given n copies of an unknown quantum state according to some prior and the goal is to estimate certain parameters of the given state. In particular, we are interested to know whether collective measurements are useful and if so to find an upper bound on the amount…
It is always possible to decide, with one-sided error, whether two quantum states are the same under a specific unitary transformation. However we show here that it is {\em impossible} to do so if the transformation is anti-linear and non-singular. This result implies that unitary and anti-unitary operations exist on an unequal footing in quantum information theory.
The study of quantum correlations in High-dimensional bipartite systems is crucial for the development of quantum computing. We propose relative entropy as a distance measure of correlations may be measured by means of the distance from the quantum state to the closest classical-classical state. In particular, We establish relations between relative entropy and quantum discord quantifiers obtained by means of orthogonal projective measurements. We show that for symmetrical X-states density matrices the quantum discord is equal to relative entropy. At the end of paper, various examples of X-sta…
We study the amount of classical communication needed for distributed quantum information processing. In particular, we introduce the concept of "remote preparation" of a quantum state. Given an ensemble of states, Alice's task is to help Bob in a distant laboratory to prepare a state of her choice. We find several examples of an ensemble with an entropy S where the remote preparation can be done with a communication cost lower than the amount (2S) required by standard teleportation. We conjecture that, for an arbitrary N-dimensional pure state, its remote preparation requires 2log_2 N bits of…
In this paper, we consider quantum correlations (quantum discord and local quantum uncertainty) in a vertical quantum dot. Their dependencies on magnetic field and temperature are presented in detail. It is noticeable that, quantum discord and local quantum uncertainty behavior is similar to a large extent. In addition, the time evolution of quantum discord and local quantum uncertainty under dephasing and amplitude damping channels is investigated. It has been found that, for some Belldiagonal states quantum discord is invariant under some decoherence in a finite time interval [Phys. Rev. Let…
The structure of Bell-type inequalities detecting genuine multipartite non-locality, and hence detecting genuine multipartite entanglement, is investigated. We first present a simple and intuitive approach to Svetlichny's original inequality, which provides a clear understanding of its structure and of its violation in quantum mechanics. Based on this approach, we then derive a family of Bell-type inequalities for detecting genuine multipartite non-locality in scenarios involving an arbitrary number of parties and systems of arbitrary dimension. Finally we discuss the thightness and quantum me…
A quantum mechanical generalization of superstatistics is presented here based on the positive operator valued measure transformation property of the system density matrix. This procedure reveals that the origin of the fluctuating factors occurring in the derivation of the superstatistics lies in the choice of the transformation operators governing the dynamics of fluctuations. This generalization addresses situations such as nanosystems based on quantum devices (e.g., superconducting devices, single electron transistors, etc,) operating at low temperatures where cognizance of quantum fluctuat…
In this paper, we generalize the secure quantum information exchange (SQIE) protocol, originally proposed by the authors [J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 115504] for secure exchange of one qubit information with each of Alice and Bob, to the case of secure exchange of quantum information of arbitrary qubits with Alice and Bob. We also discuss security of the original and generalized SQIE protocols with respect to the number of qubits with controller, Charlie.
The Kullback-Leibler divergence offers an information-theoretic basis for measuring the difference between two given distributions. Its quantum analog, however, fails to play a corresponding role for comparing two density matrices, if the reference states are pure states. Here, it is shown that nonadditive (nonextensive) generalization of quantum information theory is free from such a difficulty and the associated quantity, termed the quantum q-divergence, can in fact be a good information-theoretic measure of the degree of purification. The correspondence relation between the ordinary diverge…
We study various annealing dynamics, both classical and quantum, for simple mean-field models and explain how to describe their behavior in the thermodynamic limit in terms of differential equations. In particular we emphasize the differences between quantum annealing (i.e. evolution with Schrödinger equation) and simulated quantum annealing (i.e. annealing of a Quantum Monte Carlo simulation).
