International Journal of Quantum Information

We present a survey on mathematical topics relating to separable states and entanglement witnesses. The convex cone duality between separable states and entanglement witnesses is discussed and later generalized to other families of operators, leading to their characterization via multiplicative properties. The condition for an operator to be an entanglement witness is rephrased as a problem of positivity of a family of real polynomials. By solving the latter in a specific case of a three-parameter family of operators, we obtain explicit description of entanglement witnesses belonging to that family. A related problem of block positivity over real numbers is discussed. We also consider a broad family of block positivity tests and prove that they can never be sufficient, which should be useful in case of future efforts in that direction. Finally, we introduce the concept of length of a separable state and present new results concerning relationships between the length and Schmidt rank. In particular, we prove that separable states of length lower or equal to 3 have Schmidt ranks equal to their lengths. We also give an example of a state which has length 4 and Schmidt rank 3.

We consider the case of a pair of particles initially in a superposition state corresponding to a separated pair of wave packets. In contrast to a previous related work, we avoid a master equation approach and we calculate exactly the time development of this non-Gaussian state due to interaction with an arbitrary heat bath. We find that coherence decays continuously, as expected. We then investigate entanglement and find that at a finite time the system becomes separable (not entangled). Thus, we see that entanglement sudden death is also prevalent in continuous variable systems which should raise concern for the designers of entangled systems.

An encryption-based networking quantum teleportation scheme is proposed for the secure quantum communication network based on the novel networking quantum key distribution (NQKD) with the Greenberger–Horne–Zeilinger (GHZ) state. The security is exactly guaranteed via the entanglement of the GHZ quantum system. In the process of quantum teleportation, the server Charlie prepares the GHZ states and the users exploit the four local unitary operations corresponding to the keys shared aforehand to encrypt/decrpt for the initial messages. One legal user can communicate with another on the network securely as they may perform a quantum privacy application on the encrypted quantum states transmitted in quantum channel. Its intrinsic efficiency is high as almost all the instances are useful and each GHZ state can carry two different encrypted single-qubit states. To reconstruct these transmitted messages, the users exploit Bell measurement and perform local operations with respect to the shared keys in NQKD, which will improve its security in a noise quantum channel.

We present a new non-Gaussian quantum key distribution (QKD) protocol using squeezed states. Compared with a binary modulation, the efficiency can be improved when a four alphabetic letters modulation is chosen. We then analyze the security of the protocol under a collective entangling cloner attack, which is a powerful attack strategy in continuous variable QKD. We also show how the modulation coefficient affects the key rate, then the key rate can be maximized by choosing an optimal coefficient.

We propose a scheme for deterministically teleporting a pure quantum state, making use of the Faraday rotation of photonic polarization in cavity QED system. Based on the single-photon input-output process regarding cavities, the scheme works well even in low-Q cavities. Since only virtual excitation of the atoms is involved, it is also insensitive to atomic spontaneous emission. Besides, the CNOT operation replaced by the Faraday rotation is not needed in the scheme, which could greatly relax the experimental difficulty.

In this paper, we study decoherence in continuous-time quantum walks (CTQWs) on one-dimensional regular networks. For this purpose, we assume that every node is represented by a quantum dot continuously monitored by an individual point contact (Gurvitz's model). This measuring process induces decoherence. We focus on small rates of decoherence and then obtain the mixing time bound of the CTQWs on the one-dimensional regular network, whose distance parameter is l ≥ 2. Our results show that the mixing time is inversely proportional to the rate of decoherence, which is in agreement with the mentioned results for cycles in Refs. 29 and 37. Also, the same result is provided in Ref. 38 for long-range interacting cycles. Moreover, we find that this quantity is independent of the distance parameter l (l ≥ 2) and that the small values of decoherence make short the mixing time on these networks.

To understand quantum gate array complexity, we define a problem named exact non-identity check, which is a decision problem to determine whether a given classical description of a quantum circuit is strictly equivalent to the identity or not. We show that the computational complexity of this problem is non-deterministic quantum polynomial-time (NQP)-complete. As corollaries, it is derived that exact non-equivalence check of two given classical descriptions of quantum circuits is also NQP-complete and that minimizing the number of quantum gates for a given quantum circuit without changing the implemented unitary operation is NQP-hard.

In this paper, we show that any S(S = 1, 2, 3, 4, 5, 6) persons can unite to teleport min(S, 6 - S) qubits to the other 6 - S persons if they a priori share a maximally genuinely entangled six-qubit state introduced by Borras et al. [A. Borras, A. R. Plastino, J. Batle, C. Zander, M. Casas and A. Plastino, J. Phys. A40 (2007) 13407]. Also, explicit protocols for perfectly teleporting arbitrary one-qubit state, two-qubit state as well as three-qubit state between arbitrary split of six partites are constructed. Moreover, to fulfil the protocol, either the senders or the receivers can make local operations individually in distant places.

We present the design and evaluation of a quantum carry-lookahead adder (QCLA) using measurement-based quantum computation (MBQC), called MBQCLA. QCLA was originally designed for an abstract, concurrent architecture supporting long-distance communication, but most realistic architectures heavily constrain communication distances. The quantum carry-lookahead adder is faster than a quantum ripple-carry adder; QCLA has logarithmic depth while ripple adders have linear depth. MBQCLA utilizes MBQC's ability to transfer quantum states in unit time to accelerate addition. MBQCLA breaks the latency limit of addition circuits in nearest neighbor-only architectures: compared to the Θ(n) limit on circuit depth for linear nearest-neighbor architectures, it can reach Θ(log n) depth. MBQCLA is an order of magnitude faster than a ripple-carry adder when adding registers longer than 100 qubits, but requires a cluster state that is an order of magnitude larger. The cluster state resources can be classified as computation and communication; for the unoptimized form, ≈ 88% of the resources are used for communication. Hand optimization of horizontal communication costs results in a ≈ 12% reduction in spatial resources for the in-place MBQCLA circuit. For comparison, a graph state quantum carry-lookahead adder (GSQCLA) uses only ≈ 9% of the spatial resources of the MBQCLA.

So far there is no closed formula for relative entropy of entanglement of arbitrary two-qubit states. In this paper we present a method, which guarantees the derivation of the relative entropy of entanglement for most states that have z-directional Bloch vectors. It is shown that the closest separable states for those states also have z-directional Bloch vectors though there are few exceptions.

Quantum walks play an important role in the area of quantum algorithms. Many interesting problems can be reduced to searching marked states in a quantum Markov chain. In this context, the notion of quantum hitting time is very important, because it quantifies the running time of the algorithms. Markov chain-based algorithms are probabilistic, therefore the calculation of the success probability is also required in the analysis of the computational complexity. Using Szegedy's definition of quantum hitting time, which is a natural extension of the definition of the classical hitting time, we present analytical expressions for the hitting time and success probability of the quantum walk on the complete graph.