States Of Continuous Variable Systems Research Articles

QuGIT: A numerical toolbox for Gaussian quantum states

Simulating quantum states on a classical computer is hard, typically requiring prohibitive resources in terms of memory and computational power. Efficient simulation, however, can be achieved for certain classes of quantum states, in particular the so-called Gaussian quantum states of continuous variable systems. In this work we introduce QuGIT - a python numerical toolbox based on symplectic methods specialized in efficiently simulating multimode Gaussian states and operations. QuGIT is exact, requiring no truncation of Hilbert space, and provides a wide range of Gaussian operations on arbitrary Gaussian states, including unitaries, partial traces, tensor products, general-dyne measurements, conditional and unconditional dynamics. To illustrate the toolbox, several examples of usage relevant to quantum optics and optomechanics are described. Program summaryProgram Title: QuGIT: Quantum Gaussian Information ToolboxCPC Library link to program files:https://doi.org/10.17632/tp4s84yc5d.1Developer's repository link:https://github.com/IgorBrandao42/Quantum-Gaussian-Information-ToolboxLicensing provisions: CC By 4.0Programming language: PythonExternal routines/libraries: NumPy, SciPyNature of problem: Conditional and unconditional dynamics of Gaussian quantum systems.Solution method: Numerical simulation of symplectic representation of Gaussian states.Additional comments including restrictions and unusual features: Problems must deal with Gaussian states following Gaussian preserving dynamics.

Steering Witness and Steering Criterion of Gaussian States.

Quantum steering is an important quantum resource, which is intermediate between entanglement and Bell nonlocality. In this paper, we study steering witnesses for Gaussian states in continuous-variable systems. We give a definition of steering witnesses by covariance matrices of Gaussian states, and then obtain a steering criterion by steering witnesses to detect steerability of any -mode Gaussian states. In addition, the conditions for two steering witnesses to be comparable and the optimality of steering witnesses are also discussed.

Non-Gaussian Quantum States and Where to Find Them

Gaussian states have played on important role in the physics of continuous-variable quantum systems. They are appealing for the experimental ease with which they can be produced, and for their compact and elegant mathematical description. Nevertheless, many proposed quantum technologies require us to go beyond the realm of Gaussian states and introduce non-Gaussian elements. In this Tutorial, we provide a roadmap for the physics of non-Gaussian quantum states. We introduce the phase-space representations as a framework to describe the different properties of quantum states in continuous-variable systems. We then use this framework in various ways to explore the structure of the state space. We explain how non-Gaussian states can be characterised not only through the negative values of their Wigner function, but also via other properties such as quantum non-Gaussianity and the related stellar rank. For multimode systems, we are naturally confronted with the question of how non-Gaussian properties behave with respect to quantum correlations. To answer this question, we first show how non-Gaussian states can be created by performing measurements on a subset of modes in a Gaussian state. Then, we highlight that these measured modes must be correlated via specific quantum correlations to the remainder of the system to create quantum non-Gaussian or Wigner-negative states. On the other hand, non-Gaussian operations are also shown to enhance or even create quantum correlations. Finally, we will demonstrate that Wigner negativity is a requirement to violate Bell inequalities and to achieve a quantum computational advantage. At the end of the Tutorial, we also provide an overview of several experimental realisations of non-Gaussian quantum states in quantum optics and beyond.

Stabilizable Gaussian states

The unavoidable interaction of quantum systems with their environment usually results in the loss of desired quantum resources. Suitably chosen system Hamiltonians, however, can, to some extent, counteract such detrimental decay, giving rise to the set of stabilizable states. Here, we discuss the possibility to stabilize Gaussian states in continuous-variable systems. We identify necessary and sufficient conditions for such stabilizability and elaborate these on two benchmark examples, a single, damped mode and two locally damped modes. The obtained stabilizability conditions, which are formulated in terms of the states' covariance matrices, are, more generally, also applicable to non-Gaussian states, where they may similarly help to, e.g., discuss entanglement preservation and/or detection up to the second moments.

Unified framework to determine Gaussian states in continuous-variable systems

Gaussian states are the backbone of quantum information protocols with continuous variable systems, whose power relies fundamentally on the entanglement between the different modes. In the case of global pure states, knowledge of the reduced states in a given bipartition of a multipartite quantum system bears information on the entanglement in such bipartition. For Gaussian states, the reduced states are also Gaussian, so there determination requires essentially the experimental determination of their covariance matrix. Here, we develop strategies to determine the covariance matrix of an arbitrary n-mode bosonic Gaussian state through measurement of the total phase acquired when appropriate metaplectic evolutions, associated with quadratic Hamiltonians, are applied. Simply one-mode metaplectic evolutions, such rotations, squeezing and shear transformations, in addition to a single two-mode rotation, allows to determine all the covariance matrix elements of a n-mode bosonic system. All the single-mode metaplectic evolutions are applied conditionally to a state in which an ancilla qubit is entangled with the n-mode system. The ancillary system provides, after measurement, the value of the total phase of each evolution. The proposed method is experimentally friendly to be implemented in the most currently used continuous variable systems.

Non-Markovianity Hierarchy of Gaussian Processes and Quantum Amplification.

We investigate the dynamics of Gaussian states of continuous variable systems under Gaussianity-preserving channels. We introduce a hierarchy of such evolutions encompassing Markovian and weakly and strongly non-Markovian processes and provide simple criteria to distinguish between the classes, based on the degree of positivity of intermediate Gaussian maps. We present an intuitive classification of all one-mode Gaussian channels according to their non-Markovianity degree and show that weak non-Markovianity has an operational significance, as it leads to a temporary phase-insensitive amplification of Gaussian inputs beyond the fundamental quantum limit. Explicit examples and applications are discussed.

Genuine multipartite nonlocality of permutationally invariant Gaussian states

We investigate genuine multipartite nonlocality of pure permutationally invariant multimode Gaussian states of continuous variable systems, as detected by the violation of Svetlichny inequality. We identify the phase space settings leading to the largest violation of the inequality when using displaced parity measurements, distinguishing our results between the cases of even and odd total number of modes. We further consider pseudospin measurements and show that, for three-mode states with asymptotically large squeezing degree, particular settings of these measurements allow one to approach the maximum violation of Svetlichny inequality allowed by quantum mechanics. This indicates that the strongest manifestation of genuine multipartite quantum nonlocality is in principle verifiable on Gaussian states.

Quantification of Gaussian quantum steering.

Einstein-Podolsky-Rosen steering incarnates a useful nonclassical correlation which sits between entanglement and Bell nonlocality. While a number of qualitative steering criteria exist, very little has been achieved for what concerns quantifying steerability. We introduce a computable measure of steering for arbitrary bipartite Gaussian states of continuous variable systems. For two-mode Gaussian states, the measure reduces to a form of coherent information, which is proven never to exceed entanglement, and to reduce to it on pure states. We provide an operational connection between our measure and the key rate in one-sided device-independent quantum key distribution. We further prove that Peres' conjecture holds in its stronger form within the fully Gaussian regime: namely, steering bound entangled Gaussian states by Gaussian measurements is impossible.

Certifying quantumness: Benchmarks for the optimal processing of generalized coherent and squeezed states

Quantum technology promises revolutionary advantages in information processing and transmission compared to classical technology; however, determining which specific resources are needed to surpass the capabilities of classical machines often remains a nontrivial problem. To address such a problem, one first needs to establish the best classical solutions, which set benchmarks that must be beaten by any implementation claiming to harness quantum features for an enhanced performance. Here we introduce and develop a self-contained formalism to obtain the ultimate, generally probabilistic benchmarks for quantum information protocols including teleportation and approximate cloning, with arbitrary ensembles of input states generated by a group action, so-called Gilmore-Perelomov coherent states. This allows us to construct explicit fidelity thresholds for the transmission of multimode Gaussian and non-Gaussian states of continuous variable systems, as well as qubit and qudit pure states drawn according to nonuniform distributions on the Bloch hypersphere, which accurately model the current laboratory facilities. The performance of deterministic classical procedures such as square-root measurement strategies is further compared with the optimal probabilistic benchmarks, and the state-of-the-art performance of experimental quantum implementations against our newly derived thresholds is discussed. This work provides a comprehensive collection of directly useful criteria for the reliable certification of quantum communication technologies.

Theory of Genuine Tripartite Nonlocality of Gaussian States

We investigate the genuine multipartite nonlocality of three-mode Gaussian states of continuous variable systems. For pure states, we present a simplified procedure to obtain the maximum violation of the Svetlichny inequality based on displaced parity measurements, and we analyze its interplay with genuine tripartite entanglement measured via Rényi-2 entropy. The maximum Svetlichny violation admits tight upper and lower bounds at fixed tripartite entanglement. For mixed states, no violation is possible when the purity falls below 0.86. We also explore a set of recently derived weaker inequalities for three-way nonlocality, finding violations for all tested pure states. Our results provide a strong signature for the nonclassical and nonlocal nature of Gaussian states despite their positive Wigner function, and lead to precise recipes for its experimental verification.

Threshold effect and entanglement enhancement through local squeezing of initial separable states in continuous-variable systems

We investigate the dynamical enhancement of entanglement by means of local squeezing of initial separable Gaussian states in two coupled harmonic oscillator systems. Interestingly, the maximum generation of entanglement between the two modes is not a monotonically increasing function of the squeezing parameter. Instead, there exists a critical degree of squeezing rc for the maximum entanglement to be enhanced. For the coupled oscillator systems considered, we have derived the analytical expressions of rc.

GAUSSIAN GEOMETRIC DISCORD

We extend the geometric measure of quantum discord, introduced and computed for two-qubit states, to quantify non-classical correlations in composite Gaussian states of continuous variable systems. We lay the formalism for the evaluation of a Gaussian geometric discord in two-mode Gaussian states, and present explicit formulas for the class of two-mode squeezed thermal states. In such a case, under physical constraints of bounded mean energy, geometric discord is shown to admit upper and lower bounds for a fixed value of the conventional (entropic) quantum discord. We finally discuss alternative geometric approaches to quantify Gaussian quadrature correlations.

Measurement-induced disturbances and nonclassical correlations of Gaussian states

We study quantum correlations beyond entanglement in two-mode Gaussian states of continuous-variable systems by means of the measurement-induced disturbance (MID) and its ameliorated version (AMID). In analogy with the recent studies of the Gaussian quantum discord, we define a Gaussian AMID by constraining the optimization to all bi-local Gaussian positive operator valued measurements. We solve the optimization explicitly for relevant families of states, including squeezed thermal states. Remarkably, we find that there is a finite subset of two-mode Gaussian states comprising pure states where non-Gaussian measurements such as photon counting are globally optimal for the AMID and realize a strictly smaller state disturbance compared to the best Gaussian measurements. However, for the majority of two-mode Gaussian states the unoptimized MID provides a loose overestimation of the actual content of quantum correlations, as evidenced by its comparison with Gaussian discord. This feature displays strong similarity with the case of two qubits. Upper and lower bounds for the Gaussian AMID at fixed Gaussian discord are identified. We further present a comparison between Gaussian AMID and Gaussian entanglement of formation, and classify families of two-mode states in terms of their Gaussian AMID, Gaussian discord, and Gaussian entanglement of formation. Our findings provide a further confirmation of the genuinely quantum nature of general Gaussian states, yet they reveal that non-Gaussian measurements can play a crucial role for the optimized extraction and potential exploitation of classical and nonclassical correlations in Gaussian states.

Determination of the Schmidt number

Optimized, necessary, and sufficient conditions for the identification of the Schmidt number will be derived in terms of general Hermitian operators. These conditions apply to arbitrary mixed quantum states. The optimization procedure delivers equations similar to the eigenvalue problem of an operator. The properties of the solution of these equations will be studied. We solve these equations for classes of operators. The solutions will be applied to phase randomized two-mode squeezed-vacuum states in continuous variable systems.

Detection of bound entanglement in continuous-variable systems

We present several entanglement conditions in order to detect bound entangled states in continuous-variable systems. Specifically, Werner and Wolf [Phys. Rev. Lett. 86, 3658 (2001)] and Horodecki and Lewenstein [Phys. Rev. Lett. 85, 2657 (2000)] have proposed examples of a bound entangled Gaussian state and a bound entangled non-Gaussian state, respectively, of which entanglement can be detected by using our entanglement conditions.

NON-GAUSSIANITY AND PURITY IN FINITE DIMENSION

We address truncated states of continuous variable systems and analyze their statistical properties numerically by generating random states in finite-dimensional Hilbert spaces. In particular, we focus to the distribution of purity and non-Gaussianity for dimension up to d = 21. We found that both quantities are distributed around typical values with variances that decrease for increasing dimension. Approximate formulas for typical purity and non-Gaussianity as a function of the dimension are derived.

Squeezing the limit: quantum benchmarks for the teleportation and storage of squeezed states

We derive fidelity benchmarks for the quantum storage and teleportation of squeezed states of continuous variable systems, for input ensembles where the degree of squeezing s is fixed, no information about its orientation in phase space is given, and the distribution of phase-space displacements is a Gaussian. In the limit where the latter becomes flat, we prove analytically that the maximal classical achievable fidelity (which is 1/2 without squeezing, for s=1) is given by , vanishing when the degree of squeezing diverges. For mixed states, as well as for general distributions of displacements, we reduce the determination of the benchmarks to the solution of a finite-dimensional semidefinite program, which yields accurate, certifiable bounds thanks to a rigorous analysis of the truncation error. This approach may be easily adapted to more general ensembles of input states.

Genuine multipartite entanglement of symmetric Gaussian states: Strong monogamy, unitary localization, scaling behavior, and molecular sharing structure

We investigate the structural aspects of genuine multipartite entanglement in Gaussian states of continuous variable systems. Generalizing the results of Adesso and Illuminati [Phys. Rev. Lett. 99, 150501 (2007)], we analyze whether the entanglement shared by blocks of modes distributes according to a strong monogamy law. This property, once established, allows us to quantify the genuine $N$-partite entanglement not encoded into $2,\dots{},K,\dots{},(N\ensuremath{-}1)$-partite quantum correlations. Strong monogamy is numerically verified, and the explicit expression of the measure of residual genuine multipartite entanglement is analytically derived, by a recursive formula, for a subclass of Gaussian states. These are fully symmetric (permutation-invariant) states that are multipartitioned into blocks, each consisting of an arbitrarily assigned number of modes. We compute the genuine multipartite entanglement shared by the blocks of modes and investigate its scaling properties with the number and size of the blocks, the total number of modes, the global mixedness of the state, and the squeezed resources needed for state engineering. To achieve the exact computation of the block entanglement, we introduce and prove a general result of symplectic analysis: Correlations among $K$ blocks in $N$-mode multisymmetric and multipartite Gaussian states, which are locally invariant under permutation of modes within each block, can be transformed by a local (with respect to the partition) unitary operation into correlations shared by $K$ single modes, one per block, in effective nonsymmetric states where $N\ensuremath{-}K$ modes are completely uncorrelated. Due to this theorem, the above results, such as the derivation of the explicit expression for the residual multipartite entanglement, its nonnegativity, and its scaling properties, extend to the subclass of non-symmetric Gaussian states that are obtained by the unitary localization of the multipartite entanglement of symmetric states. These findings provide strong numerical evidence that the distributed Gaussian entanglement is strongly monogamous under and possibly beyond specific symmetry constraints, and that the residual continuous-variable tangle is a proper measure of genuine multipartite entanglement for permutation-invariant Gaussian states under any multipartition of the modes.

Quantum filtering of optical coherent states

We propose and experimentally demonstrate nondestructive and noiseless removal (filtering) of vacuum states from an arbitrary set of coherent states of continuous variable systems. Errors, i.e., vacuum states in the quantum information are diagnosed through a weak measurement, and on that basis, probabilistically filtered out. We consider three different filters based on on-off detection, phase stabilized, and phase randomized homodyne detection. We find that on-off detection, optimal in the ideal theoretical setting, is superior to the homodyne strategy also in a practical setting.

Optical implementation and entanglement distribution in Gaussian valence bond states

We study Gaussian valence bond states of continuous variable systems obtained as the outputs of projection operations from an ancillary space of M infinitely entangled bonds connecting neighboring sites applied at each ofN sites of a harmonic chain. The entanglement distribution in Gaussian valence bond states can be controlled by varying the input amount of entanglement engineered in a (2M+ 1)-mode Gaussian state known as the building block, which is isomorphic to the projector applied at a given site. We show how this mechanism can be interpreted in terms of multiple entanglement swapping from the chain of ancillary bonds, through the building blocks. We provide optical schemes to produce bisymmetric three-mode Gaussian building blocks (which correspond to a single bond, M = 1), and study the entanglement structure in the output Gaussian valence bond states. Finally, the usefulness of such states for quantum communication protocols with continuous variables, like telecloning and teleportation networks, is discussed.