Schmidt Basis Research Articles

Mechanism for the quantum natured gravitons to entangle masses

This paper points out the importance of the quantum nature of the gravitational interaction with matter in a linearized theory of quantum gravity induced entanglement of masses. We will show how the quantum interaction entangles the steady states of a closed system (eigenstates) of two test masses placed in the harmonic traps, and how such a quantum matter-matter interaction emerges from an underlying quantum gravitational field. We will rely upon quantum perturbation theory highlighting the critical assumptions for generating a quantum matter-matter interaction and showing that a classical gravitational field does not render such an entanglement. We will consider two distinct examples: one where the two harmonic oscillators are static, and the other where the harmonic oscillators are nonstatic. In both cases it is the quantum nature of the gravitons interacting with the harmonic oscillators that are responsible for creating an entangled state with the ground and the excited states of harmonic oscillators as the Schmidt basis. We will compute the concurrence as a criterion for the above entanglement and compare the two ways of entangling the two harmonic oscillators.

Quantifying Entanglement with Coherence

Quantifying entanglement is a work in progress which is important for the active field of quantum information and computation. A measure of bipartite pure state entanglement is proposed here, named entanglement coherence, which is essentially the normalized coherence of the entangled state in its Schmidt basis. Its value is 1 for maximally entangled states, and 0 for separable states, irrespective of the dimensionality of the Hilbert space. So a maximally entangled state is also the one which is maximally coherent in its Schmidt basis. Quantum entanglement and quantum coherence are thus intimately connected. Entanglement coherence turns out to be closely related to the unified entropy of the reduced state of one of the subsystems. Additionally it is shown that the entanglement coherence is closely connected to the Wigner-Yanase skew information of the reduced density operator of one of the subsystems, in an interesting way.

Low-rank sum-of-products finite-basis-representation (SOP-FBR) of potential energy surfaces.

The sum-of-products finite-basis-representation (SOP-FBR) approach for the automated multidimensional fit of potential energy surfaces (PESs) is presented. In its current implementation, the method yields a PES in the so-called Tucker sum-of-products form, but it is not restricted to this specific ansatz. The novelty of our algorithm lies in the fact that the fit is performed in terms of a direct product of a Schmidt basis, also known as natural potentials. These encode in a non-trivial way all the physics of the problem and, hence, circumvent the usual extra ad hoc and a posteriori adjustments (e.g., damping functions) of the fitted PES. Moreover, we avoid the intermediate refitting stage common to other tensor-decomposition methods, typically used in the context of nuclear quantum dynamics. The resulting SOP-FBR PES is analytical and differentiable ad infinitum. Our ansatz is fully general and can be used in combination with most (molecular) dynamics codes. In particular, it has been interfaced and extensively tested with the Heidelberg implementation of the multiconfiguration time-dependent Hartree quantum dynamical software package.

High-dimensional cryptography with spatial modes of light: tutorial

Fast and secure sharing of information is among the prime concerns of almost any communication system. While commonly used cryptographic algorithms cannot provide unconditional security, high-dimensional (HD) quantum key distribution (QKD) offers an exceptional means to this end. Here, we provide a tutorial to demonstrate that HD QKD protocols can be implemented in an effective way using optical elements that are known to most optics labs. We use spatial modes of light as our HD basis and show how to simulate QKD experiments with bright classical light, fostering its easy implementation for a more general audience including industry laboratories or laboratory classes in university teaching and in advanced laboratories for validation purposes. In particular, we use orbital angular momentum Bessel–Gaussian modes for our HD QKD demonstration to illustrate and highlight the benefits of using spatial modes as their natural Schmidt basis and self-healing feature.

Hydrogenic entanglement

Is there any entanglement in the simplest ubiquitous bound system? We study the solutions to the time-independent Schrödinger equation for a hydrogenic system and devise two entanglement tests for free and localised states. For free hydrogenic systems, we compute the Schmidt basis diagonalisation for general energy eigenstates, and for a hydrogenic system localised to a three-dimensional Gaussian wavepacket, we demonstrate that measuring its second moments is sufficient for detecting entanglement. Our results apply to any system that exhibits hydrogenic structure.

Spectral shaping of the biphoton state from multiplexed thermal atomic ensembles

We theoretically investigate the spectral property of a biphoton state from multiplexed thermal atomic ensembles. This biphoton state originates from the cascade emissions, which can be generated by two weak pump fields under four-wave mixing condition. Under this condition, a signal photon from the upper transition, chosen in a telecommunication bandwidth, can be generated along with a correlated idler photon from the lower infrared transition. We can spectrally shape the biphoton state by multiplexing the atomic ensembles with frequency-shifted emissions, where the entropy of entanglement can be analyzed via Schmidt decompositions. We find that this spectral entanglement increases when more vapor cells are multiplexed with correlated or anti-correlated signal and idler fields. The eigenvalues in Schmidt bases approach degenerate under this multiplexing scheme, and corresponding Schmidt numbers can be larger than the number of the multiplexed vapor cells, showing the enlarged entropy of entanglement and excess correlated modes in continuous frequency spaces. We also investigate the lowest entropy of entanglement allowed in the multiplexing scheme, which is preferential for generating a pure single photon source. This shows the potentiality to spectrally shape the biphoton source, where high-capacity spectral modes can be applied in long-distance quantum communication and multimode quantum information processing.

Differential-Detection Aided Large-Scale Generalized Spatial Modulation is Capable of Operating in High-Mobility Millimeter-Wave Channels

A large-scale differential-detection aided generalized spatial modulation (GSM) system is proposed, which relies on a novel Gram–Schmidt basis set and an adaptive low-complexity detector, and is evidently suitable for high-mobility millimeter-wave (mmWave) channels. We consider nonstationary time-varying mmWave channels and assume that the beam-angles remain relatively fixed, while the channel coefficients vary rapidly. In this scenario, it is a challenging task to find the accurate estimates of channel coefficients for digital beamforming, which becomes an even more severe problem, as the numbers of subarrays and subcarriers increase. Our analog-beamforming-aided nonsquare differentially detected scheme achieves a higher transmission rate than the conventional coherent multiple-input multiple-output schemes because the pilot overhead and the complex-valued feedback are eliminated. Our simulation results following the IEEE 802.11ad specifications show that the performance of our proposed nonsquare differential GSM improved upon increasing the number of subarrays, where the maximum transmission rate of 16 b/s/Hz was considered.

Separated Schmidt modes in the angular spectrum of biphotons

We prepared qudits based on angular multimode biphoton states by modulating the pump angular spectrum. The modes were prepared in the Schmidt basis and their intensity distributions did not overlap in space. This allows one to get rid of filtering losses while addressing single modes and to realize a single-shot qudit readout.

Quantum-Limited Loss Sensing: Multiparameter Estimation and Bures Distance between Loss Channels.

The problem of estimating multiple loss parameters of an optical system using the most general ancilla-assisted parallel strategy is solved under energy constraints. An upper bound on the quantum Fisher information matrix is derived assuming that the environment modes involved in the loss interaction can be accessed. Any pure-state probe that is number diagonal in the modes interacting with the loss elements is shown to exactly achieve this upper bound even if the environment modes are inaccessible, as is usually the case in practice. We explain this surprising phenomenon, and show that measuring the Schmidt bases of the probe is a parameter-independent optimal measurement. Our results imply that multiple copies of two-mode squeezed vacuum probes with an arbitrarily small nonzero degree of squeezing, or probes prepared using single-photon states and linear optics, can achieve quantum-optimal performance in conjunction with on-off detection. We also calculate explicitly the energy-constrained Bures distance between any two product loss channels. Our results are relevant to standoff image sensing, biological imaging, absorption spectroscopy, and photodetector calibration.

Electronic-nuclear entanglement in H2+ : Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets

We compute the entanglement between the electronic and vibrational motions in the simplest molecular system, the hydrogen molecular ion, considering the molecule as a bipartite system, electron and vibrational motion. For that purpose we compute an accurate total non-Born-Oppenheimer wave function in terms of a huge expansion using nonorthogonal B-spline basis sets that expand separately the electronic and nuclear wave functions. According to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, it is possible to find a much shorter but equivalent expansion in terms of the natural orbitals or Schmidt bases for the electronic and nuclear half spaces. Here we extend the Schmidt decomposition theorem to the case in which nonorthogonal bases are used to span the partitioned Hilbert spaces. This extension is first illustrated with two simple coupled systems, the former without an exact solution and the latter exactly solvable. In these model systems of distinguishable coupled particles it is shown that the entanglement content does not increase monotonically with the excitation energy, but only within the manifold of states that belong to an existing excitation mode, if any. In the hydrogen molecular ion the entanglement content for each non-Born-Oppenheimer vibronic state is quantified through the von Neumann and linear entropies and we show that entanglement serves as a witness to distinguish vibronic states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes.

Spectral shaping of cascade emissions from multiplexed cold atomic ensembles

We investigate the spectral properties of the biphoton state from the cascade emissions of cold atomic ensembles, which composes of a telecommunication photon (signal) followed by an infrared one (idler) via four-wave mixing.\ With adiabatic conditions for Gaussian driving pulses of width $\tau$, the spectrum of the biphoton state has the form of a Gaussian that conserves signal and idler photon energies within $\hbar/\tau$ modulated by a Lorentzian with a superradiant linewidth.\ Multiplexing the atomic ensembles with frequency-shifted cascade emissions, we may manipulate and shape the spectrum of the biphoton state.\ The entropy of entanglement is derived from Schmidt decomposition, which can be larger if we multiplex the atomic ensembles in a way that conserves signal and idler photon central energies.\ The eigenvalues in Schmidt bases are degenerate in pairs for symmetric spectral shaping in which the mode probability densities show interference patterns.\ We also demonstrate the excess entropy of entanglement that comes from continuous frequency space, which scales up the total entropy.\ The scheme of multiplexed cascade-emitted biphoton state provides multimode structures that are useful in long-distance quantum communication and multimode quantum information processing.

Quantum random state generation with predefined entanglement constraint

Entanglement plays an important role in quantum communication, algorithms, and error correction. Schmidt coefficients are correlated to the eigenvalues of the reduced density matrix. These eigenvalues are used in von Neumann entropy to quantify the amount of the bipartite entanglement. In this paper, we map the Schmidt basis and the associated coefficients to quantum circuits to generate random quantum states. We also show that it is possible to adjust the entanglement between subsystems by changing the quantum gates corresponding to the Schmidt coefficients. In this manner, random quantum states with predefined bipartite entanglement amounts can be generated using random Schmidt basis. This provides a technique for generating equivalent quantum states for given weighted graph states, which are very useful in the study of entanglement, quantum computing, and quantum error correction.

Self-calibrating tomography for angular Schmidt modes in spontaneous parametric down-conversion

We report an experimental self-calibrating tomography scheme for entanglement characterization in high-dimensional quantum systems using Schmidt decomposition techniques. The self-tomography technique based on maximal likelihood estimation was developed for characterizing nonideal measurements in the Schmidt basis, allowing us to infer both Schmidt eigenvalues and detecting efficiencies.

Cartesian and polar Schmidt bases for down-converted photons

We derive an analytical form of the Schmidt modes of spontaneous parametric down-conversion (SPDC) biphotons in both Cartesian and polar coordinates. We show that these correspond to Hermite-Gauss (HG) or Laguerre-Gauss (LG) modes only for a specific value of their width, and we show how such value depends on the experimental parameters. The Schmidt modes that we explicitly derive allow one to set up an optimised projection basis that maximises the mutual information gained from a joint measurement. The possibility of doing so with LG modes makes it possible to take advantage of the properties of orbital angular momentum eigenmodes. We derive a general entropic entanglement measure using the Rényi entropy as a function of the Schmidt number, K, and then retrieve the von Neumann entropy, S. Using the relation between S and K we show that, for highly entangled states, a non-ideal measurement basis does not degrade the number of shared bits by a large extent. More specifically, given a non-ideal measurement which corresponds to the loss of a fraction of the total number of modes, we can quantify the experimental parameters needed to generate an entangled SPDC state with a sufficiently high dimensionality to retain any given fraction of shared bits.

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.

Projective approach to the entanglement entropy of one-dimensional fermions

The entanglement entropy of two gapless non-interacting fermion subsystems is computed approximately in a way that avoids the introduction of replicas and a geometric interpretation of the reduced density matrix. We exploit the similarity between the Schmidt basis wavefunction and superfluid BCS wavefunction and compute the entropy using the BCS approximation. Within this analogy, the Cooper pairs are particle-hole pairs straddling the boundary and the effective interaction between them is induced by the projection of the Hilbert space onto the incomplete Schmidt basis. The resulting singular interaction may be thought of as "lifting" the degeneracy of the single particle distribution function. For two coupled fermion systems of linear size $L$, we solve the BCS gap equation approximately to find the entropy $S \approx (w^2/t^2)\log{L}$ where $w$ is the hopping amplitude at the boundary of the subsystem and $2t$ is the bandwidth. We further interpret this result based upon the relationship between entanglement spectrum, entropy and number fluctuations.

Rotating stellar core-collapse waveform decomposition: a principal component analysis approach

This paper introduces the use of principal component analysis (PCA) as a method to decompose the catalogues of gravitational waveforms to produce a set of orthonormal basis vectors. We apply PCA to a set of gravitational waveforms produced by rotating stellar core-collapse simulations and compare its basis vectors with those obtained through Gram–Schmidt decomposition. The comparison is made using the match parameter which quantifies how well each waveform is reconstructed by a set of basis vectors. The performance of the two methods is found to be comparable with 14 Gram–Schmidt basis vectors and 12 principal components required if we require all waveforms in the catalogue to be reconstructed with a match of 0.9 or better. Additionally, we observe that the chosen set of waveforms has very similar features, and a match of at least 0.7 can be obtained by decomposing only waveforms generated from simulations with A = 2. We discuss the implications of this observation and the advantages of eigen-decomposing waveform catalogues with PCA.

Maximum observable correlation for a bipartite quantum system

The maximum observable correlation between the two components of a bipartite quantum system is a property of the joint density operator, and is achieved by making particular measurements on the respective components. For pure states it corresponds to making measurements diagonal in a corresponding Schmidt basis. More generally, it is shown that the maximum correlation may be characterised in terms of a `correlation basis' for the joint density operator, which defines the corresponding (nondegenerate) optimal measurements. The maximum coincidence rate for spin measurements on two-qubit systems is determined to be (1+s)/2, where s is the spectral norm of the spin correlation matrix, and upper bounds are obtained for n-valued measurements on general bipartite systems. It is shown that the maximum coincidence rate is never greater than the computable cross norm measure of entanglement, and a much tighter upper bound is conjectured. Connections with optimal state discrimination and entanglement bounds are briefly discussed.

Test of Bell-type theorems based on parity-coherent states of light fields

We propose to test Bell-type theorems of continuous variables based on the pseudospin operator of parity-coherent states of light fields. The proposed Bell-type theorem has a constant, maximal violation in the Schmidt bases for large values of the mean photon number. Continuous-variable systems usually have the advantage of high detection efficiency with the homodyne detection method. The required parity-coherent states are experimentally realizable. Using two-mode light fields entangled both in parity and polarization, we also present a proposal of all-versus-nothing test of local realism.

Unambiguous discrimination of special sets of multipartite states using local measurements and classical communication

We initially consider a quantum system consisting of two qubits, which can be in one of two nonorthogonal states $\ensuremath{\mid}{\ensuremath{\Psi}}_{0}⟩$ and $\ensuremath{\mid}{\ensuremath{\Psi}}_{1}⟩$. We distribute the qubits to two parties, Alice and Bob. They each measure their qubits and then compare their measurement results to determine which state they were sent. This procedure is error-free, which implies that it must sometimes fail. In addition, no quantum memory is required; it is not necessary for one of the qubits to be stored until the result of the measurement on the other is known. We consider the cases in which, should failure occur, both parties receive a failure signal or only one does. In the latter case, if the two states share the same Schmidt basis, the states can be discriminated with the same failure probability that would be obtained if the qubits were measured together. This scheme is sufficiently simple that it can be generalized to multipartite qubit, and qudit, states. Applications to quantum secret sharing are discussed. Finally, we present an optical scheme to experimentally realize the protocol in the case of two qubits.