QIC Abstracts

 Vol.12 No.3&4, March 1, 2012

Research Articles:

Quantum McEliece public-key cryptosystem (pp0181-0203)
Hachiro Fujita
The McEliece cryptosystem is one of the best-known (classical) public-key cryptosystems, which is based on algebraic coding theory. In this paper, we present a quantum analogue of the classical McEliece cryptosystem. Our quantum McEliece public-key cryptosystem is based on the theory of stabilizer codes and has the key generation, encryption and decryption algorithms similar to those in
the classical McEliece cryptosystem. We present an explicit construction of the quantum McEliece public-key cryptosystem using Calderbank-Shor-Steane codes based on generalized Reed-Solomon codes. We examine the security of our quantum McEliece cryptosystem and compare it with alternative systems.

Improved Data Post-Processing in Quantum Key Distribution and Application to Loss Thresholds in device independent QKD (pp0203-0214)
Xiongfeng Ma and Norbert Lutkenhaus
Security proofs of quantum key distribution (QKD) often require post-processing schemes to simplify the data structure, and hence the security proof. We show a generic method to improve resulting secure key rates by partially reversing the simplifying post-processing for error correction purposes. We apply our method to the security analysis of device-independent QKD schemes and of detection-device-independent QKD schemes, where in both cases one is typically required to assign binary values even to lost signals. In the device-independent case, the loss tolerance threshold is cut down by our method from 92.4% to 90.9%. The lowest tolerable transmittance of the detection-device-independent scheme can be improved from 78.0% to 65.9%

One-step implementation of the Fredkin gate via quantum Zeno dynamics (pp0215-0230)
Zhi-Cheng Shi, Yan Xia, Jie Song, and He-Shan Song
We study one-step implementation of the Fredkin gate in a bi-modal cavity under both resonant and large detuning conditions based on quantum Zeno dynamics, which reduces the complexity of experiment operations. The influence of cavity decay and atomic spontaneous emission is discussed by numerical calculation. The results demonstrate that the fidelity and the success probability are robust against cavity decay in both models and they are also insensitive to atomic spontaneous emission in the large detuning model. In addition, the interaction time is rather short in the resonant model compared to the large detuning model.

Steady-state correlations of two atoms interacting with a reservoir (pp0231-0252)
Luis Octavio Castanos
We consider two two-level atoms fixed at different positions, driven by a resonant monochromatic laser field, and interacting collectively with the quantum electromagnetic field. A Born-Markov-secular master equation is used to describe the dynamics of the two atoms and the steady-state is obtained analytically for a configuration of the atoms. The steady-state populations of the energy levels of the free atoms, entanglement, quantum and geometric discords and degree of mixedness are calculated analytically as a function of the laser field intensity and the distance between the two atoms. It is found that there is a possibility of considerable steady-state entanglement and left/right quantum discord and that these can be controlled either by increasing/decreasing the intensity of the laser field or by increasing/decreasing the distance between atoms. It is shown that the system of two atoms can be prepared in a separable mixed state with non-zero quantum discord that turns into an $X$-state for high laser field intensities. The behavior and relationships between the different correlations are studied and several limiting cases are investigated.

Probabilistic secret sharing through noise quantum channel (pp0253-0261)
Satyabrata Adhikari, Indranil Chakrabarty, and Pankaj Agrawal
In a realistic situation, the secret sharing of classical or quantum information will involve the transmission of this information through noisy channels. We consider a three qubit pure state. This state becomes a mixed-state when the qubits are distributed over noisy channels. We focus on a specific noisy channel, the phase-damping channel. We propose a protocol for secret sharing of classical information with this and related noisy channels. This protocol can also be thought of as cooperative superdense coding. We also discuss other noisy channels to examine the possibility of secret sharing of classical information.

Decomposition of orthogonal matrix and synthesis of two-qubit and three-qubit orthogonal gates (pp0262-0270)
Hai-Rui Wei and Yao-Min Di
The decomposition of matrices associated to two-qubit and three-qubit orthogonal gates is studied, and based on the decomposition the synthesis of these gates is investigated. The optimal synthesis of general two-qubit orthogonal gate is obtained. For two-qubit unimodular orthogonal gate, it requires at most 2 CNOT gates and 6 one-qubit R_y gates. For the general three-qubit unimodular orthogonal gate, it can be synthesized by 16 CNOT gates and 36 one-qubit R_y and R_z gates in the worst case.

Locally unextendible non-maximally entangled basis (pp0271-0282)
Indranil Chakrabarty, Pankaj Agrawal, and Arun K. Pati

We introduce the concept of the locally unextendible non-maximally entangled basis (LUNMEB) in H^d \bigotimes H^d. It is shown that such a basis consists of d orthogonal vectors for a non-maximally entangled state. However, there can be a maximum of (d-1)^2 orthogonal vectors for non-maximally entangled state if it is maximally entangled in (d-1) dimensional subspace. Such a basis plays an important role in determining the number of classical bits that one can send in a superdense coding protocol using a non-maximally entangled state as a resource. By constructing appropriate POVM operators, we find that the number of classical bits one can transmit using a non-maximally entangled state as a resource is (1+p_0\frac{d}{d-1})\log d, where p_0 is the smallest Schmidt coefficient. However, when the state is maximally entangled in its subspace then one can send up to 2\log (d-1) bits. We also find that for d= 3, former may be more suitable for the superdense coding.

An exact tensor network for the 3SAT problem (pp0283-0292)
Artur Garcia-Saez and Jose I. Latorre

We construct a tensor network that delivers an unnormalized quantum state whose coefficients are the solutions to a given instance of 3SAT, an NP-complete problem. The tensor network contraction that corresponds to the norm of the state counts the number of solutions to the instance. It follows that exact contractions of this tensor network are in the \#P-complete computational complexity class, thus believed to be a hard task. Furthermore, we show that for a 3SAT instance with $n$ bits, it is enough to perform a polynomial number of contractions of the tensor network structure associated to the computation of local observables to obtain one of the explicit solutions to the problem, if any. Physical realization of a state described by a generic tensor network is equivalent to finding the satisfying assignment of a 3SAT instance and, consequently, this experimental task is expected to be hard.

Perfect state transfer on quotient graphs (pp0293-0313)
Rachel Bachman, Eric Fredette, Jessica Fuller, Michael Landry, Michael Opperman, Christino Tamon, and Andrew Tollefson
We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph G has perfect state transfer if and only if its quotient G/\pi, under any equitable partition \pi, has perfect state transfer, we exhibit graphs with perfect state transfer between two vertices but which lack automorphism swapping them. This answers a question of Godsil (Discrete Mathematics  312(1):129-147, 2011). We also show that the Cartesian product of quotient graphs \Box_{k} G_{k}/\pi_{k} is isomorphic to the quotient graph \Box_{k} G_{k}/\pi, for some equitable partition \pi. This provides an algebraic description of a construction due to Feder (Physical Review Letters 97, 180502, 2006) which is based on many-boson quantum walk.

Limit theorems for the discrete-time quantum walk on a graph with joined half lines (pp0314-0333)
Kota Chisaki, Norio Konno, and Etsuo Segawa
We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_{t,\kappa}. On each half line, W_{t,\kappa} converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.

Realization of an economical phase-covariant telecloning in separate cavities (pp0334-0345)
Bao-Long Fang, Tao Wu, and Liu Ye
We present a scheme to realize an economical $1 \to 2$ phase-covariant telecloning in separate cavities. In the scheme, an entangled state between a photon pulse and the tapped atoms will be prepared through cavity-assisted interaction. Next, the Bell state measurement is execute on the photon pulse and a trapped atom or another photon pulse on which the quantum information is encoded. In this way, the system can provide symmetric (asymmetric) economical $1 \to 2$ phase-covariant telecloning.

On the geometry of tensor network states (pp0346-0354)
          Joseph M. Landsburg, Yang Qi, and Ke Ye

We answer a question of L. Grasedyck that arose in quantum information theory, showing that the limit of tensors in a space of tensor network states need not be a tensor network state. We also give geometric descriptions of spaces of tensor networks states corresponding to trees and loops. Grasedyck's question has a surprising connection to the area of Geometric Complexity Theory, in that the result is equivalent to the statement that the boundary of the Mulmuley-Sohoni type variety
associated to matrix multiplication is strictly larger than the projections of matrix multiplication (and re-expressions of matrix multiplication and its projections after changes of bases). Tensor Network States are also related to graphical models in algebraic statistics.

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