Remote State Preparation in Quadrature Basis

Quantum communication protocols based on nonclassical correlations can be more efficient than known classical methods and offer intrinsic security over direct state transfer. In particular, remote state preparation aims at the creation of a desired and known quantum state at a remote location using classical communication and quantum entanglement. We present an experimental realization of deterministic continuous-variable remote state preparation in the microwave regime over a distance of 35 cm. By employing propagating two-mode squeezed microwave states and feedforward, we achieve the remote preparation of squeezed states with up to 1.6 dB of squeezing below the vacuum level. Finally, security of remote state preparation is investigated by using the concept of the one-time pad and measuring the von Neumann entropies. We find nearly identical values for the entropy of the remotely prepared state and the respective conditional entropy given the classically communicated information and, thus, demonstrate close-to-perfect security.

Using partial entangled states as the quantum channel, two schemes for probabilistic remote preparation of the four-particle cluster-type state with real and complex coefficients are presented. In the first scheme, the sender and the receiver share two partial Bell states and one partial three-qubit GHZ stats as the quantum channel, and the sender can help a remote receiver to prepare a four-particle entangled cluster-type state by using three-qubit projective measurements with certain probability. In the second scheme, the quantum channel is composed of two partial three-qubit GHZ states, the remote state preparation (RSP) can be successfully realized via the positive operator valued measure (POVM), and the two-particle projective measurements are also needed in this process. The total success probability and classical communication cost are calculated.

Remote state preparation (RSP) is the act of preparing a quantum state at a remote location without actually transmitting the state itself. Using at most two classical bits and a single shared maximally entangled state, one can in theory remotely prepare any qubit state with certainty and with perfect fidelity. However, in any experimental implementation the average fidelity between the target and output states cannot be perfect. In order for an RSP experiment to demonstrate genuine quantum advantages, it must surpass the optimal threshold of a comparable classical protocol. Here we study the fidelity achievable by RSP protocols lacking shared entanglement, and determine the optimal value for the average fidelity in several different cases. We implement an experimental scheme for deterministic remote preparation of arbitrary photon polarization qubits, preparing 178 different pure and mixed qubit states with an average fidelity of 0.995. Our experimentally-achieved average fidelities surpass our derived classical thresholds whenever the classical protocol does not trivially allow for perfect RSP.

The wave-particle duality of light has led to two different encodings of optical quantum information. One approach, refered as the discrete-variable (DV) one, relies on finite dimensional quantum systems, while the other one, refered as the continuous-variable (CV) one, is based on wave-like states belonging to an infinite dimensional Hilbert space. These two encodings have historically been separated, but recently, new hybrid protocols which aim at combining the two complementary encodings have emerged [1]. In that prospect our recent demonstration of the measurement-induced generation of hybrid entanglement between discrete and continuous-variable quantum states generated using optical parametric oscillators [2], located at distant places and connected by a lossy channel, has opened the way to the implementation of hybrid protocols and heterogeneous quantum networks. Here we will report the first protocols based on this resource: the remote preparation of continuous-variable qubits [3], and a violation of an Einstein-Podolosky-Rosen steering inequality [4].

In this paper, we present a polynomial-complexity algorithm to construct a special orthogonal matrix for the deterministic remote state preparation (DRSP) of an arbitrary $n$-qubit state, and prove that if $n > 3$, such matrices do not exist. Firstly, the construction problem is split into two sub-problems, i.e., finding a solution of a semi-orthogonal matrix and generating all semi-orthogonal matrices. Through giving the definitions and properties of the matching operators, it is proved that the orthogonality of a special matrix is equivalent to the cooperation of multiple matching operators, and then the construction problem is reduced to the problem of solving an XOR linear equation system, which reduces the construction complexity from exponential to polynomial level. Having proved that each semi-orthogonal matrix can be simplified into a unique form, we use the proposed algorithm to confirm that the unique form does not have any solution when $n > 3$, which means it is infeasible to construct such a special orthogonal matrix for the DRSP of an arbitrary $n$-qubit state.

Recently, Gottesman et al (2001 Phys. Rev. A 64, 012310) showed how to encode a qubit into a continuous variable quantum system. This encoding was realized by using non-normalizable quantum codewords, which therefore can only be approximated in any real physical set-up. Here we show how a neutral atom, falling through an optical cavity and interacting with a single mode of the intracavity electromagnetic field, can be used to safely encode a qubit into its external degrees of freedom. In fact, the localization induced by a homodyne detection of the cavity field is able to project the near-field atomic motional state into an approximate quantum codeword. The performance of this encoding process is then analysed by evaluating the intrinsic errors induced in the recovery process by the approximated form of the generated codeword.

Thermodynamic resources, beyond their well-known usefulness in work extraction and other thermodynamic tasks, are often important also in tasks that are not evidently thermodynamic. Here we develop a framework for identifying such resources in diverse applications of bosonic continuous-variable systems. Introducing the class of bosonic linear thermal operations to model operationally feasible processes, we apply this model to identify uniquely quantum properties of bosonic states that refine classical notions of thermodynamic resourcefulness. Among these are (1) a suite of temperature-like quantities generalizing the equilibrium temperature to quantum, non-equilibrium scenarios; (2) signal-to-noise ratios quantifying a system’s capacity to carry information in phase-space displacement; and (3) well-established non-classicality measures quantifying the resolution in sensing and parameter estimation tasks.

In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there is a Turing machine M which computes f in the sense that, M accepts any rho-name of x as input and outputs a rho-name of f(x) for any x in the domain of f. By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition.

CHAPTER 5 - FUNCTIONS OF A REAL VARIABLE

Cluster state is deemed to be a significant quantum resource, for it can encode much information; its remote preparation is significant to the quantum information processing. In the paper, a novel joint remote quantum state preparation protocol is proposed, in which three participants are included, and they are cooperatively involved for the protocol accomplishment within the configuration constructed by the four-qubit cluster type and EPR pairs. The cluster-type state is successful prepared via the appropriate unitary operations after the relevant participants perform the projection measurement. The presented protocol shows highly secure and could stand against the external eavesdropper.

We study entanglement accumulation in a memory built out of two continuous variable systems interacting with a qubit that mediates their indirect coupling. We show that, in contrast with the case of bidimensional Hilbert spaces, entanglement superior to one ebit can be accumulated in the memory, even though no entangled resource is used. The protocol is immediately implementable and we assess the role of the main imperfections.

We study the quantum communication protocol of remote state preparation (RSP) for pure states of qubits encoded in single photons transmitted through a double slit, the so-called spatial qubits. Two measurement strategies that one can adopt to remotely prepare the states are discussed. The first strategy is the well-known spatial postselection, where a single-pixel detector measures the transverse position of the photon between the focal and the image plane of a lens. The second strategy, proposed by ourselves, is a generalized measurement divided into two steps: the implementation of a two-outcome positive operator-valued measurement (POVM) followed by the spatial postselection at the focal plane of the lens by a two-pixel detector in each output of the POVM. In both cases we analyze the effects of the finite spatial resolution of the detectors over three figures of merit of the protocol, namely, the probability of preparation, the fidelity, and purity of the remotely prepared states. It is shown that our strategy improves these figures compared with spatial postselection, at the expense of increasing the classical communication cost as well as the required experimental resources. In addition, we present a modified version of our strategy for RSP of spatial qudits which is able to prepare arbitrary pure states, unlike spatial postselection alone. We expect that our study may also be extended for RSP of the angular spectrum of a single-photon field as an alternative for quantum teleportation which requires very inefficient nonlinear interactions.

Most of the fundamental characteristics of quantum mechanics, such as non-locality and contextuality, are manifest in discrete, finite-dimensional systems. However, many quantum information tasks that exploit these properties cannot be directly adapted to continuous-variable systems. To access these quantum features, continuous quantum variables can be made discrete by binning together their different values, resulting in observables with a finite number "$d$" of outcomes. While direct measurement indeed confirms their manifestly discrete character, here we employ a salient feature of quantum physics known as mutual unbiasedness to show that such coarse-grained observables are in a sense neither continuous nor discrete. Depending on $d$, the observables can reproduce either the discrete or the continuous behavior, or neither. To illustrate these results, we present an example for the construction of such measurements and employ it in an optical experiment confirming the existence of four mutually unbiased measurements with $d = 3$ outcomes in a continuous variable system, surpassing the number of mutually unbiased continuous variable observables.

We present a numerical method for solving the separability problem of Gaussian quantum states in continuous-variable quantum systems. We show that the separability problem can be cast as an equivalent problem of determining the feasibility of a set of linear matrix inequalities. Thus, it can be efficiently solved using existent numerical solvers. We apply this method to the identification of bound entangled Gaussian states. We show that the proposed method can be used to identify bound entangled Gaussian states that could be simple enough to be producible in quantum optics.

CHAPTER II - Functions