Multimode States Research Articles

Geometric characterization of separability and entanglement in pure Gaussian states by single-mode unitary operations

We present a geometric approach to the characterization of separability and entanglement in pure Gaussian states of an arbitrary number of modes. The analysis is performed adapting to continuous variables a formalism based on single subsystem unitary transformations that has been recently introduced to characterize separability and entanglement in pure states of qubits and qutrits [S. M. Giampaolo and F. Illuminati, Phys. Rev. A 76, 042301 (2007)]. In analogy with the finite-dimensional case, we demonstrate that the 1xM bipartite entanglement of a multimode pure Gaussian state can be quantified by the minimum squared Euclidean distance between the state itself and the set of states obtained by transforming it via suitable local symplectic (unitary) operations. This minimum distance, corresponding to a, uniquely determined, extremal local operation, defines an entanglement monotone equivalent to the entropy of entanglement, and amenable to direct experimental measurement with linear optical schemes.

Measure of the non-Gaussian character of a quantum state

We address the issue of quantifying the non-Gaussian character of a bosonic quantum state and introduce a non-Gaussianity measure based on the Hilbert-Schmidt distance between the state under examination and a reference Gaussian state. We analyze in detail the properties of the proposed measure and exploit it to evaluate the non-Gaussianity of some relevant single-mode and multimode quantum states. The evolution of non-Gaussianity is also analyzed for quantum states undergoing the processes of Gaussification by loss and de-Gaussification by photon-subtraction. The suggested measure is easily computable for any state of a bosonic system and allows one to define a corresponding measure for the non-Gaussian character of a quantum operation.

Standard forms and entanglement engineering of multimode Gaussian states under local operations

We investigate the action of local unitary operations on multimode (pure or mixed) Gaussian states and single out the minimal number of locally invariant parameters which completely characterize the covariance matrix of such states. For pure Gaussian states, central resources for continuous-variable quantum information, we investigate separately the parameter reduction due to the additional constraint of global purity, and the one following by the local-unitary freedom. Counting arguments and insights from the phase-space Schmidt decomposition and in general from the framework of symplectic analysis, accompany our description of the standard form of pure n-mode Gaussian states. In particular, we clarify why only in pure states with n ⩽ 3 modes all the direct correlations between position and momentum operators can be set to zero by local unitary operations. For any n, the emerging minimal set of parameters contains complete information about all forms of entanglement in the corresponding states. An efficient state engineering scheme (able to encode direct correlations between position and momentum operators as well) is proposed to produce entangled multimode Gaussian resources, its number of optical elements matching the minimal number of locally invariant degrees of freedom of general pure n-mode Gaussian states. Finally, we demonstrate that so-called ‘block-diagonal’ Gaussian states, without direct correlations between position and momentum, are systematically less entangled, on average, than arbitrary pure Gaussian states.

Gaussian localizable entanglement

We investigate localization of the entanglement of multimode Gaussian states into a pair of modes by local Gaussian measurements on the remaining modes and classical communication. We find that for pure states and for mixed symmetric states maximum entanglement between two modes can be localized by local homodyne detection - i.e. projections onto infinitely squeezed states. We also show that non-Gaussian measurements allow us to localize more entanglement than Gaussian ones.

Quantum telecloning of optical images: Multiuser parallel quantum channel

We introduce and describe in detail a different quantum-information protocol: quantum telecloning of optical images or spatially multimode states of light. This protocol is a natural generalization of a continuous-variable quantum telecloning of single-mode states, investigated earlier theoretically and recently realized experimentally. Our protocol is an example of a multiuser parallel quantum channel. In order to realize this scheme one needs a multimode multipartite entanglement between the original and the target systems. We calculate the fidelity of quantum parallel telecloning in dependence on the spatiotemporal scales of our scheme and on the number of pixels. The fidelity is superior to the best classical fidelity that can be achieved without this kind of entanglement.

Coherent control of single photon states

.We define a class of multi-mode single photon states suitable for quantum information applications. We show how standard amplitude modulation techniques may be used to control the pulse shape of single photon states.

Squeezing in multi-mode nonlinear optical state truncation

In this paper, we show that multi-mode qubit states produced via nonlinear optical state truncation driven by classical external pumpings exhibit squeezing condition. We restrict our discussions to the two and three-mode cases.

PREPARATION OF ENTANGLED STATES OF CAVITY FIELD VIA COUPLING TO SUPERCONDUCTING QUANTUM INTERFERENCE DEVICE

Based on the interaction of a superconducting quantum interference device (SQUID) with the field in high-quality QED and a classical microwave pulse, we propose a scheme for preparing an entangled multipartite W state between different modes of cavity field. In principle, the scheme can be extended to generate a N-party W state. Moreover, the scheme can be used for the generation of multimode entangled coherent states.

Observing a coherent superposition of an atom and a molecule

We demonstrate that it is possible, in principle, to perform a Ramsey-type interference experiment to exhibit a coherent superposition of a single atom and a diatomic molecule. This gedanken experiment, based on the techniques of Aharonov and Susskind [Phys. Rev. 155, 1428 (1967)], explicitly violates the commonly accepted superselection rule that forbids coherent superpositions of eigenstates of differing atom number. A Bose-Einstein condensate plays the role of a reference frame that allows for coherent operations analogous to Ramsey pulses. We also investigate an analogous gedanken experiment to exhibit a coherent superposition of a single boson and a fermion, violating the commonly accepted superselection rule forbidding coherent superpositions of states of differing particle statistics. In this case, the reference frame is realized by a multimode state of many fermions. This latter case reproduces all of the relevant features of Ramsey interferometry, including Ramsey fringes over many repetitions of the experiment. However, the apparent inability of this proposed experiment to produce well-defined relative phases between two distinct systems each described by a coherent superposition of a boson and a fermion demonstrates that there are additional, outstanding requirements to fully ``lift" the univalence superselection rule.

One-dimensional electron transport and thermopower in an individual InSb nanowire

We have measured the electrical conductance and thermopower of a singleInSb nanowire in the temperature range from 5 to 340 K. Below temperature(T) 220 K, the conductance(G) shows a power-lawdependence on T and the current (I)–voltage (V) curve follows a power-law dependence onV at large bias voltages. These features are the characteristics ofone-dimensional Luttinger liquid (LL) transport. The thermopower(S) also shows linear temperature dependence forT below 220 K, in agreement with the theoretical prediction based on the LLmodel. Above 220 K, the power law and linear behaviours respectively in theG–T and S–T curves persist but with different slopes from those at low temperatures. The slope changescan be explained by a transition from a single-mode LL state to a multi-mode LL state.

Conditions for multipartite continuous-variable entanglement

Necessary and sufficient observable conditions for the nonnegativity of all partial transpositions of multi-mode quantum states are derived. The result is a hierarchy of inequalities for minors in terms of moments of the given state. Violations of any inequality is a sufficient condition for entanglement. Full entanglement can be certified for a manifold of multi-mode quantum states. A \textit{Mathematica} package is given for a systematic test of the hierarchy of conditions.

Non-Gaussian states from continuous-wave Gaussian light sources

We present a general analysis of the state obtained by subjecting a continuous-wave (cw) Gaussian field to non-Gaussian measurements. The generic multimode state of a cw Gaussian field is fully characterized by the time dependent mean values and variances and the two-time covariances of the field quadrature variables. We present a general theory to extract from this information the results of detection and quantum state reduction within specific temporal output modes. The formalism is applied to schemes for heralded production of propagating light pulses with single photon and Schr\"odinger kitten states from a cw squeezed beam of light.

ENTANGLEMENT SHARING: FROM QUBITS TO GAUSSIAN STATES

It is a central trait of quantum information theory that there exist limitations to the free sharing of quantum correlations among multiple parties. Such monogamy constraints have been introduced in a landmark paper by Coffman, Kundu and Wootters, who derived a quantitative inequality expressing a trade-off between the couplewise and the genuine tripartite entanglement for states of three qubits. Since then, a lot of efforts have been devoted to the investigation of distributed entanglement in multipartite quantum systems. In this paper we report, in a unifying framework, a bird's eye view of the most relevant results that have been established so far on entanglement sharing in quantum systems. We will take off from the domain of N qubits, graze qudits, and finally land in the almost unexplored territory of multimode Gaussian states of continuous variable systems.

Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence

We present a complete analysis of the multipartite entanglement of three-mode Gaussian states of continuous-variable systems. We derive standard forms which characterize the covariance matrix of pure and mixed three-mode Gaussian states up to local unitary operations, showing that the local entropies of pure Gaussian states are bound to fulfill a relationship which is stricter than the general Araki-Lieb inequality. Quantum correlations can be quantified by a proper convex roof extension of the squared logarithmic negativity, the continuous-variable tangle, or contangle. We review and elucidate in detail the proof that in multimode Gaussian states the contangle satisfies a monogamy inequality constraint [G. Adesso and F. Illuminati, New J. Phys8, 15 (2006)]. The residual contangle, emerging from the monogamy inequality, is an entanglement monotone under Gaussian local operations and classical communications and defines a measure of genuine tripartite entanglements. We determine the analytical expression of the residual contangle for arbitrary pure three-mode Gaussian states and study in detail the distribution of quantum correlations in such states. This analysis yields that pure, symmetric states allow for a promiscuous entanglement sharing, having both maximum tripartite entanglement and maximum couplewise entanglement between any pair of modes. We thus name these states $\mathrm{GHZ}∕W$ states of continuous-variable systems because they are simultaneous continuous-variable counterparts of both the GHZ and the $W$ states of three qubits. We finally consider the effect of decoherence on three-mode Gaussian states, studying the decay of the residual contangle. The $\mathrm{GHZ}∕W$ states are shown to be maximally robust against losses and thermal noise.

Multimode Uncertainty Relations and Separability of Continuous Variable States

A multimode uncertainty relation (generalizing the Robertson-Schrödinger relation) is derived as a necessary constraint on the second moments of n pairs of canonical operators. In turn, necessary conditions for the separability of multimode continuous variable states under (m+n)-mode bipartitions are derived from the uncertainty relation. These conditions are proven to be necessary and sufficient for (1+n)-mode Gaussian states and for (m+n)-mode bisymmetric Gaussian states.

Two-Dimensional Spin-Peierls State With a Multimode Lattice Distortion

The effects of electron–electron (e–e) interactions on the two-dimensional (2D) Peierls state are discussed using the 2D Peierls–Hubbard model (2D PHM) in the limit of U ≫ t 0 , where U is the on-site e–e interaction parameter between up and down spin electrons and t 0 is the electron transfer integral between nearest neighbor sites. In this limit, the 2D PHM can be mapped onto the 2D spin-Peierls model (2D SPM) in which the effect of the lattice distortion is taken into account in the spin exchange interaction of the 2D Heisenberg model. As was demonstrated in a previous paper [J. Phys. Soc. Jpn. 69 (2000) 1769], the ground state of the 2D electron–lattice (e–l) system (the 2D 1/2-filled SSH model) is the so-called multimode Peierls state where the lattice distortion involves Fourier components with the nesting wave vector Q =(π/ a ,π/ a ), a being the lattice constant, and with those paralell to it. Using the exact diagonalization technique it is shown that the spin-Peierls transition with a mutlimode l...

Localized and asynchronous patterns via canards in coupled calcium oscillators

In this paper we consider the mechanism for localized behavior in coupled calcium oscillators described by the canonical two-pool model. Localization occurs when the individual cells oscillate with amplitudes of different orders of magnitude. Our analysis and computations show that a combination of diffusive coupling, heterogeneity, and the underlying canard structure of the oscillators all contribute to the localized behavior. Two key quantities characterize the different states of the system by representing the effects of both the autonomous and the non-autonomous terms which are due to the coupling. By highlighting the influence of the canard phenomenon, these quantities identify stabilizing and destabilizing effects of the coupling on the localized behavior. We compare our analysis with computations, describing multi-mode states and asynchronized large oscillations in addition to the localized states.

Quantum teleportation with a single-photon wave packet

A scheme of energy—time quantum teleportation using an entangled photon pair is proposed. The teleportation of a multimode state of a single-photon wave packet is investigated theoretically.

Teleportation and spin squeezing utilizing multimode entanglement of light with atoms

We present a protocol for the teleportation of the quantum state of a pulse of light onto the collective spin state of an atomic ensemble. The entangled state of light and atoms employed as a resource in this protocol is created by probing the collective atomic spin, Larmor precessing in an external magnetic field, off resonantly with a coherent pulse of light. We take here full account of the effects of Larmor precession and show that it gives rise to a qualitatively different type of multimode entangled state of light and atoms. The protocol is shown to be robust against the dominating sources of noise and can be implemented with an atomic ensemble at room temperature interacting with free-space light. We also provide a scheme to perform the readout of the Larmor precessing spin state enabling the verification of successful teleportation as well as the creation of spin squeezing.

Multimode entanglement and telecloning in a noisy environment

We address the generation, propagation, and application of multipartite continuous variable entanglement in a noisy environment. In particular, we focus our attention on the multimode entangled states achievable by second-order nonlinear crystals---i.e., coherent states of the $\mathrm{SU}(m,1)$ group---which provide a generalization of the twin-beam state of a bipartite system. The full inseparability in the ideal case is shown, whereas thresholds for separability are given for the tripartite case in the presence of noise. We find that entanglement of tripartite states is robust against thermal noise, both in the generation process and during propagation. We then consider coherent states of $\mathrm{SU}(m,1)$ as a resource for multipartite distribution of quantum information and analyze a specific protocol for telecloning, proving its optimality in the case of symmetric cloning of pure Gaussian states. We show that the proposed protocol also provides the first example of a completely asymmetric $1\ensuremath{\rightarrow}m$ telecloning and derive explicitly the optimal relation among the different fidelities of the $m$ clones. The effect of noise in the various stages of the protocol is taken into account, and the fidelities of the clones are analytically obtained as a function of the noise parameters. In turn, this permits the optimization of the telecloning protocol, including its adaptive modifications to the noisy environment. In the optimized scheme the clones' fidelity remains maximal even in the presence of losses (in the absence of thermal noise), for propagation times that diverge as the number of modes increases. In the optimization procedure the prominent role played by the location of the entanglement source is analyzed in details. Our results indicate that, when only losses are present, telecloning is a more effective way to distribute quantum information than direct transmission followed by local cloning.