Entropy and algorithmic complexity in quantum information theory

A theorem of Brudno says that the entropy production of classical ergodic information sources equals the algorithmic complexity per symbol of almost every sequence emitted by such sources. The recent advances in the theory and technology of quantum information raise the question whether a same relation may hold for ergodic quantum sources. In this paper, we discuss a quantum generalization of Brudno’s result which connects the von Neumann entropy rate and a recently proposed quantum algorithmic complexity.     

Secrecy, Computational Loads and Rates in Practical Quantum Cryptography

Abstract. A number of questions associated with practical implementations of quantum cryptography systems having to do with unconditional secrecy, computational loads and effective secrecy rates in the presence of perfect and imperfect sources are discussed. The different types of unconditional secrecy, and their relationship to general communications security, are discussed in the context of quantum cryptography. In order to carry out a quantum cryptography protocol it is necessary that sufficient computational resources be available to perform the various processing steps, such as sifting, error correction, privacy amplification and authentication. We display the full computer machine instruction requirements needed to support a practical quantum cryptography implementation. We carry out a numerical comparison of system performance characteristics for implementations that make use of either weak coherent sources of light or perfect single photon sources, for eavesdroppers making individual attacks on the quantum channel characterized by different levels of technological capability. We find that, while in some circumstances it is best to employ perfect single photon sources, in other situations it is preferable to utilize weak coherent sources. In either case the secrecy level of the final shared cipher is identical, with the relevant distinguishing figure-of-merit being the effective throughput rate.     

New scheme for measurement-device-independent quantum key distribution

We propose a new scheme for measurement-device-independent quantum key distribution (MDI-QKD) with a two-mode state source. In this scheme, the trigger state is split into different paths and detected at both senders; thus, four types of detection events can be obtained. Based on these events, the signal state is divided into four non-empty sets that can be used for parameter estimation and key extraction. Additionally, we carry out a performance analysis on the scheme with two-intensity (vacuum state and signal state) heralded single-photon sources. We also numerically study the statistical fluctuation in the actual system. Our simulations show that the error rate and the secure transmission distance of our two-intensity scheme are better than those of existing three- and four-intensity MDI-QKD schemes with different light sources. Considering statistical fluctuations, the maximum secure distance of our scheme can reach 344 km when the data length is 10¹³ and remains as long as 250 km when the data length is 10¹⁰. Moreover, our scheme improves the system performance and reduces the challenges of implementing the system.     

Experimental Concatenation of Quantum Error Correction with Decoupling

We experimentally explore the reduction of decoherence via concatenating quantum error correction (QEC) with decoupling in liquid-state NMR quantum information processing. Decoupling provides an efficient means of suppressing decoherence from noise sources with long correlation times, and then QEC can be used more profitably for the remaining noise sources. PACS: 03.67.Lx, 03.65.Bz     

CUGatesDensity—Quantum circuit analyser extended to density matrices

CUGatesDensity is an extension of the original quantum circuit analyser CUGates (Loke and Wang, 2011) [7] to provide explicit support for the use of density matrices. The new package enables simulation of quantum circuits involving statistical ensemble of mixed quantum states. Such analysis is of vital importance in dealing with quantum decoherence, measurements, noise and error correction, and fault tolerant computation. Several examples involving mixed state quantum computation are presented to illustrate the use of this package.     

量子位Bloch坐标的量子人工蜂群优化算法

为了改善人工蜂群(ABC)算法在解决多变量优化问题时存在的收敛速度较慢、容易陷入局部最优的不足,结合量子理论和人工蜂群算法提出一种新的量子优化算法。算法首先采用量子位Bloch坐标对蜂群算法中食物源进行编码,扩展了全局最优解的数量,提高了蜂群算法获得全局最优解的概率;然后用量子旋转门实现搜索过程中的食物源更新。对于量子旋转门的转角关系的确定,提出了一种新的方法。从理论上证明了蜂群算法在Bloch球面每次以等面积搜索时,量子旋转门的两个旋转相位大小近似于反比例关系,避免了固定相位旋转的不均等性,使得搜索呈现规律性。在典型函数优化问题的实验中,所提算法在搜索能力和优化效率两个方面优于普通量子人工蜂群(QABC)算法和单一人工蜂群算法。     

Equation of motion for entanglement

We review an evolution equation for quantum entanglement for 2 × 2 dimensional quantum systems, the smallest system that can exhibit entanglement, and extend it to higher dimensional systems. Furthermore, we provide statistical evidence for the equation’s applicability to the experimentally relevant domain of weakly mixed states.     

Quantum Communication and Information Processing with Quantum Dots

This paper reviews the single photon sources based on semiconductor quantum dots and their applications to quantum information systems. By optically pumping a system consisting of a semiconductor single quantum dot confined in a monolithic microcavity, it is possible to produce a single photon pulse stream at the Fourier transform limit with a negligible jitter. This single photon source is not only useful for BB84 quantum key distribution (QKD), but also find applications in other quantum information systems such as Ekert91/BBM92 QKD and quantum teleportation gate linear optical quantum computers.     

Experimental Demonstration of Quantum Lattice Gas Computation

We report an ensemble nuclear magnetic resonance (NMR) implementation of a quantum lattice gas algorithm for the diffusion equation. The algorithm employs an array of quantum information processors sharing classical information, a novel architecture referred to as a type-II quantum computer. This concrete implementa-tion provides a test example from which to probe the strengths and limitations of this new computation paradigm. The NMR experiment consists of encoding a mass density onto an array of 16 two-qubit quantum information processors and then following the computation through 7 time steps of the algorithm. The results show good agreement with the analytic solution for diffusive dynamics. We also describe numerical simulations of the NMR implementation. The simulations aid in determining sources of experimental errors, and they help define the limits of the implementation. PACS: 03.67.Lx; 47.11.+j; 05.60.-k     

Controlling Spin Qubits in Quantum Dots

We review progress on the spintronics proposal for quantum computing where the quantum bits (qubits) are implemented with electron spins. We calculate the exchange interaction of coupled quantum dots and present experiments, where the exchange coupling is measured via transport. Then, experiments on single spins on dots are described, where long spin relaxation times, on the order of a millisecond, are observed. We consider spin-orbit interaction as sources of spin decoherence and find theoretically that also long decoherence times are expected. Further, we describe the concept of spin filtering using quantum dots and show data of successful experiments. We also show an implementation of a read out scheme for spin qubits and define how qubits can be measured with high precision. Then, we propose new experiments, where the spin decoherence time and the Rabi oscillations of single electrons can be measured via charge transport through quantum dots. Finally, all these achievements have promising applications both in conventional and quantum information processing. PACS: 03.67.Lx, 03.67.Mn, 73.23.Hk, 85.35.Be     

A Bell inequality for a class of multilocal ring networks

Quantum networks with independent sources of entanglement (hidden variables) and nodes that execute joint quantum measurements can create strong quantum correlations spanning the breadth of the network. Understanding of these correlations has to the present been limited to standard Bell experiments with one source of shared randomness, bilocal arrangements having two local sources of shared randomness, and multilocal networks with tree topologies. We introduce here a class of quantum networks with ring topologies comprised of subsystems each with its own internally shared source of randomness. We prove a Bell inequality for these networks, and to demonstrate violations of this inequality, we focus on ring networks with three-qubit subsystems. Three qubits are capable of two non-equivalent types of entanglement, GHZ and W-type. For rings of any number N of three-qubit subsystems, our inequality is violated when the subsystems are each internally GHZ-entangled. This violation is consistently stronger when N is even. This quantitative even-odd difference for GHZ entanglement becomes extreme in the case of W-type entanglement. When the ring size N is even, the presence of W-type entanglement is successfully detected; when N is odd, the inequality consistently fails to detect its presence.     

Linear and Nonlinear Dynamical Chaos

Interrelations between dynamical and statistical laws in physics on the one hand, and between the classical and quantum mechanics on the other hand, are discussed with the emphasis on the new phenomenon of dynamical chaos.The principal results of the studies into chaos in classical mechanics are presented in some detail within the general picture of chaos as a specific case of dynamical behavior. These results include the strong local instability and robustness of motion, continuity of both the phase space as well as the motion spectrum, and time reversibility but nonrecurrency of statistical evolution.The analysis of apparently very deep and challenging contradictions of this picture with the quantum principles is given. The quantum view of dynamical chaos, as an attempt to resolve these contradictions guided by the correspondence principle and based upon the characteristic time scales of quantum evolution, is explained. The picture of the quantum chaos as a new generic dynamical phenomenon is outlined together with a few other examples of such a chaos, including linear (classical) waves and a digital computer.I conclude with the discussion of two fundamental physical problems: the quantum measurement (-collapse) and the causality principle, which both appear to be related to the phenomenon of dynamical chaos.     

Security of Practical Time-Reversed EPR Quantum Key Distribution

Abstract. We propose a proof of the security of a time-reversed EPR quantum key distribution protocol against enemies with unlimited computational power. The considered protocol uses interactive key distillation, and the proof holds for implementations using imperfect photon sources.     

Closed-form analysis of end-to-end network delay with Markov-modulated Poisson and fluid traffic

This paper develops a method for using traffic sources modelled as a Markov-modulated Poisson process (MMPP) and Markov-modulated fluid process (MMFP) in the framework of the bounded-variance network calculus, a novel stochastic network calculus framework for the approximated analysis of end-to-end network delay. The bounded-variance network calculus is an extension to multi-hop end-to-end paths of the Choe’s and Shroff’s Central-Limit-Theorem-based analysis of isolated network nodes. The input of the analysis is the statistical traffic envelope of sources, which is not available for generic MMPP and MMFP sources. The paper provides two statistical traffic envelopes, named two-moment and linear envelope, for general MMPP and MMFP sources, which can be used as an input of Central-Limit-Theorem-based frameworks for the analysis of network delay and, in turn, make it possible to use the rich collection of MMPP and MMFP models of voice, audio, data and video sources available in the literature. In this way, it is possible to avoid the computational complexity of traditional Markov analysis of end-to-end delay with MMPP and MMFP sources. With the linear envelope we can use simple analytical closed-form solutions for many important schedulers.     

Statistical Zero Knowledge and quantum one-way functions

One-way functions are a fundamental notion in cryptography, since they are the necessary condition for the existence of secure encryption schemes. Most examples of such functions, including Factoring, Discrete Logarithm or the RSA function, however, can be inverted with the help of a quantum computer. Hence, it is very important to study the possibility of quantum one-way functions, i.e. functions which are easily computable by a classical algorithm but are hard to invert even by a quantum adversary. In this paper, we provide a set of problems that are good candidates for quantum one-way functions. These problems include Graph Non-Isomorphism, Approximate Closest Lattice Vector and Group Non-Membership. More generally, we show that any hard instance of Circuit Quantum Sampling gives rise to a quantum one-way function. By the work of Aharonov and Ta-Shma [D. Aharonov, A. Ta-Shma, Adiabatic quantum state generation and statistical zero knowledge, in: Proceedings of STOC02 — Symposium on the Theory of Computing, 2001], this implies that any language in Statistical Zero Knowledge which is hard-on-average for quantum computers leads to a quantum one-way function. Moreover, extending the result of Impagliazzo and Luby [R. Impagliazzo, M. Luby, One-way functions are essential for complexity based cryptography, in: Proceedings of FOCS89 — Symposium on Foundations of Computer Science, 1989] to the quantum setting, we prove that quantum distributionally one-way functions are equivalent to quantum one-way functions.     

Data Compression Limit for an Information Source of Interacting Qubits

A system of interacting qubits can be viewed as a non-i.i.d quantum information source. A possible model of such a source is provided by a quantum spin system, in which spin-1/2 particles located at sites of a lattice interact with each other. We establish the limit for the compression of information from such a source and show that asymptotically it is given by the von Neumann entropy rate. Our result can be viewed as a quantum ana-logue of Shannon's noiseless coding theorem for a class of non-i.i.d. quantum informa-tion sources. From the probabilistic point of view it is an analog of the Shannon-McMillan-Breiman theorem considered as a cornerstone of modern Information Theory. PACS: 03.67-a; 03.67.Lx     

High fidelity and flexible quantum state transfer in the atom-coupled cavity hybrid system

We investigate a system composed of N coupled cavities (linear array) and two-level atoms interacting one at a time. Adjusting appropriately the atom-field detuning, and making the hopping rate of photons between neighboring cavities, A, greater than the atom-field coupling g (i.e. A ? N ^3/2 g), we can eliminate the interaction of the atom with the non-resonant normal modes reducing the dynamics to the interaction of the atom with only a single-mode. As an application of this interaction, we propose a two-step protocol for quantum communication of an arbitrary atomic quantum state between distant coupled cavities. In the ideal case, the coupled cavities system acts as a perfect quantum bus and we obtain a flexible and perfect quantum communication for any N. Considering the influence of dissipation, an interesting parity effect emerges and we still obtain a high fidelity quantum state transfer for an appreciable number of cavities with current experimental parameters. We also studied important sources of imperfections during the procedure.     

Inexact and exact quantum searches with a preparation state in a three-dimensional subspace

It is well known that exact quantum searches can be performed by the quantum amplitude amplification algorithm with some phase matching condition. However, recently it was shown that for some preparation states in a three-dimensional subspace, an exact search is impossible to accomplish. We show this impossibility derives from two sources: a problem of state restriction to a cyclic subspace and the solution of a linear system of equations with a \(k\) -potent coefficient matrix. Furthermore, using said system of equations, we introduce a class of preparation states in a three-dimensional space that, even though the quantum amplitude amplification algorithm is unable to find the target state exactly, the same system of equations implies modifications to the quantum amplitude amplification algorithm under which exact solutions in three-dimensional subspaces can be found. We also prove that an inexact quantum search in the 3-potent case can find the target state with high probability if the Grover operator is iterated a number of times inversely proportional to the uncertainty of said 3-potent coefficient matrix as an observable operator.     

Geometry of quantum state space and quantum correlations

Quantum state space is endowed with a metric structure, and Riemannian monotone metric is an important geometric entity defined on such a metric space. Riemannian monotone metrics are very useful for information-theoretic and statistical considerations on the quantum state space. In this article, considering the quantum state space being spanned by \(2\times 2\) density matrices, we determine a particular Riemannian metric for a state \(\rho \) and show that if \(\rho \) gets entangled with another quantum state, the negativity of the generated entangled state is, upto a constant factor, equal to square root of that particular Riemannian metric . Our result clearly relates a geometric quantity to a measure of entanglement. Moreover, the result establishes the possibility of understanding quantum correlations through geometric approach.