Schmidt Decomposition Research Articles

Experimental Schmidt Decomposition and State Independent Entanglement Detection

We introduce an experimental procedure for the detection of quantum entanglement of an unknown quantum state with a small number of measurements. The method requires neither apriori knowledge of the state nor a shared reference frame between the observers and can thus be regarded as a perfectly state-independent entanglement witness. The scheme starts with local measurements, possibly supplemented with suitable filtering, which essentially establishes the Schmidt decomposition for pure states. Alternatively we develop a decision tree that reveals entanglement within few steps. These methods are illustrated and verified experimentally for various entangled states of two and three qubits.

Wigner separability entropy and complexity of quantum dynamics

We propose the Wigner separability entropy as a measure of complexity of a quantum state. This quantity measures the number of terms that effectively contribute to the Schmidt decomposition of the Wigner function with respect to a chosen phase-space decomposition. We prove that the Wigner separability entropy is equal to the operator space entanglement entropy, measuring entanglement in the space of operators, and, for pure states, twice the entropy of entanglement. The quantum to classical correspondence between the Wigner separability entropy and the separability entropy of the classical phase-space Liouville density is illustrated by means of numerical simulations of chaotic maps. In this way, the separability entropy emerges as an extremely broad complexity quantifier in both the classical and quantum realms.

Robustness of tripartite entanglement transfer from bosonic modes to localized qubits

We address entanglement transfer from a three-mode bosonic system to a tripartite systems of spatially separated flying or fixed qubits through the interaction with their local environments. We focus on the robustness of entanglement transfer against several effects, including off-resonant interactions for both qubit-local environment and local environment-bosonic mode subsystems, and also exploring the effect of changing the coupling constants, with the possibility to have different values for each qubit-local environment interaction. For the entangled bosonic modes we consider both Gaussian states and qubit-like states, comparing three different Generalized Schmidt Decompositions forms widely used in the literature and analyzing how the deviation from qubit-like approximation influences entanglement transfer. Finally, we investigate the multimode coupling between bosonic modes and each local environment showing a comparison between various qubit-like initial states and discussing how to improve the efficiency of entanglement transfer.

Real-space entanglement spectrum of quantum Hall systems

We study the real-space entanglement spectrum for fractional quantum Hall systems, which maintains locality along the spatial cut, and provide evidence that it possesses a scaling property. We also consider the closely related particle entanglement spectrum and carry out the Schmidt decomposition of the Laughlin state analytically at large size.

Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime

In this paper we consider frequency translation enabled by Bragg scattering, a four-wave mixing process. First we introduce the theoretical background of the Green function formalism and the Schmidt decomposition. Next the Green functions for the low-conversion regime are derived perturbatively in the frequency domain, using the methods developed for three-wave mixing, then transformed to the time domain. These results are also derived and verified using an alternative time-domain method, the results of which are more general. For the first time we include the effects of convecting pumps, a more realistic assumption, and show that separability and arbitrary reshaping is possible. This is confirmed numerically for Gaussian pumps as well as higher-order Hermite-Gaussian pumps.

The physical significances and applications of Schmidt decompositions of continuum bipartile entangled state representation

According to the continuum bipartite entangled state representation |η> we derive the Schmidt decomposition of |η>, respectively in the coordinate representation, momentum representation and the particle number representation, and explain their physical significances. As the applications of the Schmidt decomposition, we directly derive the action of the single-mode squeeze operator on |η>, the two-mode squeeze operator's entangled state representation, as well as the matrix element of displacement operator in the Fock space. Generalization of our discussion in this paper to the multipartite entangled state cases is feasible.

Entanglement-based perturbation theory for highly anisotropic Bose-Einstein condensates

We investigate the emergence of three-dimensional behavior in a reduced-dimension Bose-Einstein condensate trapped by a highly anisotropic potential. We handle the problem analytically by performing a perturbative Schmidt decomposition of the condensate wave function between the tightly confined (transverse) direction(s) and the loosely confined (longitudinal) direction(s). The perturbation theory is valid when the nonlinear scattering energy is small compared to the transverse energy scales. Our approach provides a straightforward way, first, to derive corrections to the transverse and longitudinal wave functions of the reduced-dimension approximation and, second, to calculate the amount of entanglement that arises between the transverse and longitudinal spatial directions. Numerical integration of the three-dimensional Gross-Pitaevskii equation for different cigar-shaped potentials and experimentally accessible parameters reveals good agreement with our analytical model even for relatively high nonlinearities. In particular, we show that even for such stronger nonlinearities the entanglement remains remarkably small, which allows the condensate to be well described by a product wave function that corresponds to a single Schmidt term.

SINGULAR VALUE DECOMPOSITION AND MATRIX REORDERINGS IN QUANTUM INFORMATION THEORY

We review Schmidt and Kraus decompositions in the form of singular value decomposition using operations of reshaping, vectorization and reshuffling. We use the introduced notation to analyze the correspondence between quantum states and operations with the help of Jamiołkowski isomorphism. The presented matrix reorderings allow us to obtain simple formulae for the composition of quantum channels and partial operations used in quantum information theory. To provide examples of the discussed operations, we utilize a package for the Mathematica computing system implementing basic functions used in the calculations related to quantum information theory.

ALTERNATIVE DECOMPOSITION OF TWO-QUTRIT PURE STATES AND ITS RELATION WITH ENTANGLEMENT INVARIANTS

Based on maximally entangled states in the full- and sub-spaces of two-qutrits, we present an alternative decomposition of two-qutrit pure states in a form [Formula: see text]. Similar to the Schmidt decomposition, all two-qutrit pure states can be transformed into the alternative decomposition under local unitary transformations, and the parameter p1 is shown to be an entanglement invariant.

Geometrical entanglement of highly symmetric multipartite states and the Schmidt decomposition

In a previous paper, we examined a geometric measure of entanglement based on the minimum distance between the entangled target state of interest and the space of unnormalized product states. Here, we present a detailed study of this entanglement measure for n-qubit target states that are invariant under the permutation of any two qubits. We analytically obtain the permutation-invariant unnormalized direct product states that extremize the distance function. We then solve for the Hessian to show that, up to the action of trivial symmetries, the solutions correspond to local minima of the distance function. In addition, we show that the conditions that determine the extremal solutions for general target states can be obtained directly by parametrizing the product states via their Schmidt decomposition.

Observables can be tailored to change the entanglement of any pure state

We show that, for a finite-dimensional Hilbert space, there exist observables that induce a tensor product structure such that the entanglement properties of any pure state can be tailored. In particular, we provide an explicit, finite method for constructing observables in an unstructured $d$-dimensional system so that an arbitrary known pure state has any Schmidt decomposition with respect to an induced bipartite tensor product structure. In effect, this article demonstrates that, in a finite-dimensional Hilbert space, entanglement properties can always be shifted from the state to the observables and all pure states are equivalent as entanglement resources in the ideal case of complete control of observables.

Angular Schmidt modes in spontaneous parametric down-conversion

We report a proof-of-principle experiment demonstrating that appropriately chosen set of Hermite-Gaussian modes constitutes a Schmidt decomposition for transverse momentum states of biphotons generated in the process of spontaneous parametric down conversion. We experimentally realize projective measurements in Schmidt basis and observe correlations between appropriate pairs of modes. We perform tomographical state reconstruction in the Schmidt basis, by direct measurement of single-photon density matrix eigenvalues.

THE APPLICATION AND CHARACTERISTICS OF THE COHERENT ENTANGLED STATE |β, x〉

The coherent entangled state |β, x〉 is proposed in Fock space, which exhibits both the properties of the coherent and entangled states. The |β, x〉 makes up a new quantum mechanical representation, and the completeness relation of |β, x〉 is proved by virtue of the technique of integral within an ordered product of operators. The corresponding Schmidt decomposition of |β, x〉 is investigated. Furthermore, a feasible experimental scheme of |β, x〉 is presented, and generalized P-representation is constructed in the coherent entangled state |β, x〉.

Complexity and nonseparability of classical Liouvillian dynamics

We propose a simple complexity indicator of classical Liouvillian dynamics, namely the separability entropy, which determines the logarithm of an effective number of terms in a Schmidt decomposition of phase space density with respect to an arbitrary fixed product basis. We show that linear growth of separability entropy provides a stricter criterion of complexity than Kolmogorov-Sinai entropy, namely it requires that the dynamics be exponentially unstable, nonlinear, and non-Markovian.

A new three-mode entangled state representation generated by beamsplitter and polarizer

We constructed a new three-mode entangled state representation in the three-mode Fock space, which is complete and can make up a new quantum representation. This state can be generated experimentally by combining beamsplitter with polarizer. The entanglement properties are analyzed by obtaining its Schmidt decomposition. As an important application of this state, we consider a single-mode state teleportation by using this new state as a quantum channel, and present the corresponding teleportation scheme.

Application of Generalized EPR Entangled State in Quantum Teleportation

We show how the generalized EPR entangled state |η,θ〉 generated by an asymmetric beamsplitter can be directly applied to quantum teleportation to make detailed analysis and calculation. In the whole process of teleportation, the entanglement source is more flexible with the change of transmission and reflection coefficients of the beam splitter. When the squeezed state of |η,θ〉 is used as quantum channel, the fidelity depends on the degrees of squeezing for entanglement and the reflection coefficient. Our calculation has been greatly simplified by using the Schmidt decomposition of |η,θ〉.

A family of norms with applications in quantum information theory

We consider a family of vector and operator norms defined by the Schmidt decomposition theorem for quantum states. We use these norms to tackle two fundamental problems in quantum information theory: the classification problem for k-positive linear maps and entanglement witnesses, and the existence problem for nonpositive partial transpose bound entangled states. We begin with an analysis of the norms, showing that the vector norms can be explicitly calculated, and we derive several inequalities in order to bound the operator norms and compute them in special cases. We then use the norms to establish what appears to be the most general spectral test for k-positivity currently available, showing how it implies several other known tests as well as some new ones. Building on this work, we frame the nonpositive partial transpose bound entangled problem as a concrete problem on a specific limit, specifically that a particular entangled Werner state is bound entangled if and only if a certain norm inequality holds on a given family of projections.

Geometric measures of entanglement and the Schmidt decomposition

In the standard geometric approach, the entanglement of a pure state is sin 2θ, where θ is the angle between the entangled state and the closest separable state of products of normalized qubit states. We consider here a generalization of this notion by considering separable states that consist of products of unnormalized states of different dimensions. The distance between the target-entangled state and the closest unnormalized product state can be interpreted as a measure of the entanglement of the target state. The components of the closest product state and its norm have an interpretation in terms of, respectively, the eigenvectors and eigenvalues of the reduced density matrices arising in the Schmidt decomposition of the state vector. For several cases where the target state has a large degree of symmetry, we solve the system of equations analytically and look specifically at the limit where the number of qubits is large.

A Kind Tripartite Entangled State Representation and Its Application in Quantum Teleportation

We discuss entanglement for a tripartite system by setting up a new state vector representation |p,χ 1,χ 2〉 in three-mode Fock space. The Schmidt decomposition of |p,χ 1,χ 2〉 is presented and its application in teleporting a bipartite entangled state or a two-mode squeezed state to a pair of receivers is analyzed.

Comment on “Entanglement of two interacting bosons in a two-dimensional isotropic harmonic trap” [Phys. Lett. A 373 (2009) 3833

The correct form of the Schmidt decomposition of the stationary wave functions for a system of two interacting particles trapped in a two-dimensional harmonic potential is given.