Gaussian States Research Articles

The negativity contour: a quasi-local measure of entanglement for mixed states

In this paper, we study the entanglement structure of mixed states in quantum many-body systems using the negativity contour, a local measure of entanglement that determines which real-space degrees of freedom in a subregion are contributing to the logarithmic negativity and with what magnitude. We construct an explicit contour function for Gaussian states using the fermionic partial-transpose. We generalize this contour function to generic many-body systems using a natural combination of derivatives of the logarithmic negativity. Though the latter negativity contour function is not strictly positive for all quantum systems, it is simple to compute and produces reasonable and interesting results. In particular, it rigorously satisfies the positivity condition for all holographic states and those obeying the quasi-particle picture. We apply this formalism to quantum field theories with a Fermi surface, contrasting the entanglement structure of Fermi liquids and holographic (hyperscale violating) non-Fermi liquids. The analysis of non-Fermi liquids show anomalous temperature dependence of the negativity depending on the dynamical critical exponent. We further compute the negativity contour following a quantum quench and discuss how this may clarify certain aspects of thermalization.

Optimal distributed quantum sensing using Gaussian states

We find and investigate the optimal scheme of quantum distributed Gaussian sensing for estimation of the average of independent phase shifts. We show that the ultimate sensitivity is achievable by using an entangled symmetric Gaussian state, which can be generated using a single-mode squeezed vacuum state, a beam-splitter network, and homodyne detection on each output mode in the absence of photon loss. Interestingly, the maximal entanglement of a symmetric Gaussian state is not optimal although the presence of entanglement is advantageous as compared to the case using a product symmetric Gaussian state. It is also demonstrated that when loss occurs, homodyne detection and other types of Gaussian measurements compete for better sensitivity, depending on the amount of loss and properties of a probe state. None of them provide the ultimate sensitivity, indicating that non-Gaussian measurements are required for optimality in lossy cases. Our general results obtained through a full-analytical investigation will offer important perspectives to the future theoretical and experimental study for quantum distributed Gaussian sensing.

Quantum information capsule in multiple-qudit systems and continuous-variable systems

Quantum correlations in an entangled many-body system are capable of storing information. Even when the information is injected by a local unitary operation to the system, the entanglement delocalizes it. In a recent study on multiple-qubit systems, it is shown that a virtual qubit defined in the correlation space plays a role of perfect storage of delocalized information, which is called a quantum information capsule (QIC). To enhance the capacity of quantum information storage, it is crucial to formulate the cases for multiple-qudit systems and continuous-variable (CV) systems. We analytically prove that it is possible to construct a QIC for general write operations of the systems. It turns out that the extension to quantum field theory is achievable. For Gaussian states, we explicitly construct a QIC for shift write operations. We analyze the time-evolution of QIC in a CV system to demonstrate the diffusion of information in entangled pure states.

General superposition states associated to the rotational and inversion symmetries in the phase space

The general quantum superposition states containing the irreducible representation of the n-dimensional groups associated to the rotational symmetry of the n-sided regular polygon i.e. the cyclic group (Cn) and the rotational and inversion symmetries of the polygon, i.e. the dihedral group (Dn) are defined and studied. It is shown that the resulting states form an n-dimensional orthogonal set of states which can lead to the finite representation of specific systems. The correspondence between the symmetric states and the renormalized states, resulting from the selective erasure of photon numbers from an arbitrary, noninvariant initial state, is also established. As an example, the general cyclic Gaussian states are presented. The presence of nonclassical properties in these states as subpoissonian photon statistics is addressed. Also, their use in the calculation of physical quantities as the entanglement in a bipartite system is discussed.

Exact simulation of Gaussian boson sampling in polynomial space and exponential time

We introduce an exact classical algorithm for simulating Gaussian Boson Sampling (GBS). The complexity of the algorithm is exponential in the number of photons detected, which is itself a random variable. For a fixed number of modes, the complexity is in fact equivalent to that of calculating output probabilities, up to constant prefactors. The simulation algorithm can be extended to other models such as GBS with threshold detectors, GBS with displacements, and sampling linear combinations of Gaussian states. In the specific case of encoding non-negative matrices into a GBS device, our method leads to an approximate sampling algorithm with polynomial runtime. We implement the algorithm, making the code publicly available as part of Xanadu's The Walrus library, and benchmark its performance on GBS with random Haar interferometers and with encoded Erd\H{o}s-Renyi graphs.

One-sided device-independent quantum key distribution for two independent parties.

Remote distribution of secret keys is a challenging task in quantum cryptography. A significant step in this direction is the measurement-device independence quantum key distribution (MDI-QKD). For two remote (or independent) parties Alice and Bob who initially no share secret information, the MDI-QKD enables them to share a secret key by the measurement of an untrusted relay. Unfortunately, the MDI-QKD yields the assumption that the devices of both Alice and Bob have to be trusted. Here, we show that QKD between two independent parties can also be realized even if the device of either Alice or Bob is untrusted. We tackle the problem by resorting to the recently developed one-sided device-independent QKD protocol. We derive conditions on the extracted secret key to be unconditionally secure against arbitary attacks in the limit of asymptotic keys. In the presence of Gaussian states and measurements, we theoretically demonstrate our scheme is feasible, which could be an attractive candidate for long distance secret communication.

Connecting quantum steering with extractable work in a two-mode Gaussian state

Quantum steering is an intermediate-level correlation between Bell nonlocality and quantum entanglement. Motivated by the fact that quantum correlation in working medium plays a nontrivial role in extracting work, this paper explores the relationship between the extractable work and quantum steering of the working medium, which is in a two-mode Gaussian state. Measuring one mode with Gaussian operation will produce a back-action on the other mode, which can be used to extract work. It is found that for two special kinds of quantum states with suitable measurements, there is a simple linear relationship between the extractable work and the quantum steering.

Learning stochastically stable Gaussian process state–space models

Control systems are increasingly applied in domains where an analytic description of the system dynamics does not exist or is difficult to obtain. Example applications include autonomous robots in unstructured environments, human behavior modeling for prediction and action recognition in human–machine-interaction, and chemical process industry. In many of these cases, classical system identification is challenging, because a parametric model structure is unknown. Data-driven nonparametric models such as Gaussian process state–space models (GPSSMs) offer a suitable alternative: GPSSMs are known for their data-efficiency and rely on Bayesian principles to include prior knowledge. However, properties like stability or boundedness are often known a priori, but rarely exploited during modeling. We therefore propose a novel approach for learning GPSSMs subject to stability constraints. Our approach enforces the convergence using control Lyapunov functions which are also obtained in a data-driven fashion. We analyze the resulting dynamics with respect to convergence radius and data collection. In simulation, we illustrate the precision of the identified model on a real-world dataset of goal-directed motions.

Circuit complexity for free fermion with a mass quench

By using a recent approach proposed by Hackl et al. to evaluate the complexity of the free fermionic Gaussian state, we compute the complexity of the Dirac vacuum state of the Fermi system with a mass quench. First of all, we review the counting method given by Hackl et al., and demonstrate that the result can be adapted to all of the compact transformation group G. Then, we utilize this result to study the time evolution of the complexity of these states. We show that, for the rotational invariant reference state, the total complexity of the incoming vacuum state will saturate the value of the instantaneous vacuum state at the late times, with a typical timescale to achieve the final stable state. Moreover, we find that the complexity growth under the sudden quench is directly proportional to the mass difference. And all of the results show some similar behaviors with the circuit complexity of Thermofield double states.

Regimes of Classical Simulability for Noisy Gaussian Boson Sampling.

As a promising candidate for exhibiting quantum computational supremacy, Gaussian boson sampling (GBS) is designed to exploit the ease of experimental preparation of Gaussian states. However, sufficiently large and inevitable experimental noise might render GBS classically simulable. In this work, we formalize this intuition by establishing a sufficient condition for approximate polynomial-time classical simulation of noisy GBS-in the form of an inequality between the input squeezing parameter, the overall transmission rate, and the quality of photon detectors. Our result serves as a nonclassicality test that must be passed by any quantum computational supremacy demonstration based on GBS. We show that, for most linear-optical architectures, where photon loss increases exponentially with the circuit depth, noisy GBS loses its quantum advantage in the asymptotic limit. Our results thus delineate intermediate-sized regimes where GBS devices might considerably outperform classical computers for modest noise levels. Finally, we find that increasing the amount of input squeezing is helpful to evade our classical simulation algorithm, which suggests a potential route to mitigate photon loss.

Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes

Encoding a qubit in the continuous degrees of freedom of an oscillator is a promising path to error-corrected quantum computation. One advantageous way to achieve this is through Gottesman-Kitaev-Preskill (GKP) grid states, whose symmetries allow for the correction of any small continuous error on the oscillator. Unfortunately, ideal grid states have infinite energy, so it is important to find finite-energy approximations that are realistic, practical, and useful for applications. In the first half of this work we investigate the impact of imperfect GKP states on computational circuits independently of the physical architecture. To this end, we analyze the behaviour of the physical and logical content of normalizable GKP states through several figures of merit, employing a recently-developed modular subsystem decomposition. By tracking the errors that enter into the computational circuit due to imperfections in the GKP states, we are able to gauge the utility of these states for NISQ (Noisy Intermediate-Scale Quantum) devices. In the second half, we focus on a state preparation approach in the photonic domain wherein photon-number-resolving measurements on some modes of Gaussian states produce non-Gaussian states in others. We produce detailed numerical results for the preparation of GKP states alongside estimating the resource requirements in practical settings and probing the quality of the resulting states with the tools we develop. Our numerical experiments indicate that we can generate any state in the GKP Bloch sphere with nearly equal resources, which has implications for magic state preparation overheads.

Weak measurement amplification based on thermal noise effect

Most studies for postselected weak measurement focus on using the pure Gaussian state as a pointer, which can only give an amplification limit reaching the level of the ground state fluctuation. When the pointer is initialized in a thermal state, we find that the amplification limit after the postselection can reach the level of thermal fluctuation, indicating that the amplification effect achieving the level of thermal fluctuation is also increased with the temperature growth, and also that the amplification mechanism is different from that with pure Gaussian-state pointer. To illustrate these results, we propose two schemes to implement room temperature amplification of the mechanical oscillator’s displacement caused by a single photon in the optomechanical system. The two schemes both enhance the mechanical oscillator’s original displacement by nearly seven orders of magnitude, attaining sensitivity to displacements of ∼0.26 nm. Such amplification effect can be used to observe the impact of a single photon on a room temperature mechanical oscillator, which is hard to detect in traditional measurement.

Continuous variable direct secure quantum communication using Gaussian states

Continuous variable one-way and controlled-two-way secure direct quantum communication schemes have been designed using Gaussian states. Specifically, a scheme for continuous variable quantum secure direct communication and another scheme for continuous variable controlled quantum dialogue are proposed using single-mode squeezed coherent states. The security of the proposed schemes against a set of attacks (e.g., Gaussian quantum cloning machine and intercept resend attacks) has been proved. Further, it is established that the proposed schemes do not require two-mode squeezed states which are essential for a set of existing proposals. The controlled two-way communication scheme is shown to be very general in nature as it can be reduced to schemes for various relatively simpler cryptographic tasks like controlled deterministic secure communication, quantum dialogue, quantum key distribution. In addition, it is briefly discussed that the proposed schemes can provide us tools to design quantum cryptographic solutions for several socioeconomic problems.

Charge transport and dynamic response of organic and polymer transistors

It is shown that the dynamic response of a thin-film transistor (TFT) comprises a polymer or organic semiconductor and can be directly correlated with charge transport parameters. Charge transport is assumed to take place in accordance with the often employed multiple trap and release model. We can simulate the non-quasistatic response of TFTs for devices in which the mobility is not constant, but depends on carrier density and other parameters. The previous model of Burns has been significantly improved and made applicable to organic and polymer transistors. Our approach works for an arbitrary density of states including both a relatively shallow exponential distribution of tail states and deeper Gaussian trap states. We set the gate voltage, VG, in our TFTs to be zero. The source voltage, VS, is varied and, thus, the value of gate-to-source voltage is varied. The predictive power of the model that is presented in this paper will be very useful for accurate device modeling and for charge transport studies.

Complexity of mixed states in QFT and holography

We study the complexity of Gaussian mixed states in a free scalar field theory using the ‘purification complexity’. The latter is defined as the lowest value of the circuit complexity, optimized over all possible purifications of a given mixed state. We argue that the optimal purifications only contain the essential number of ancillary degrees of freedom necessary in order to purify the mixed state. We also introduce the concept of ‘mode-by- mode purifications’ where each mode in the mixed state is purified separately and examine the extent to which such purifications are optimal. We explore the purification complexity for thermal states of a free scalar QFT in any number of dimensions, and for subregions of the vacuum state in two dimensions. We compare our results to those found using the various holographic proposals for the complexity of subregions. We find a number of qualitative similarities between the two in terms of the structure of divergences and the presence of a volume law. We also examine the ‘mutual complexity’ in the various cases studied in this paper.

Geometric perspective on quantum parameter estimation

Quantum metrology holds the promise of an early practical application of quantum technologies, in which measurements of physical quantities can be made with much greater precision than what is achievable with classical technologies. In this Review, the authors collect some of the key theoretical results in quantum parameter estimation by presenting the theory for the quantum estimation of a single parameter, multiple parameters, and optical estimation using Gaussian states. The authors give an overview of results in areas of current research interest, such as Bayesian quantum estimation, noisy quantum metrology, and distributed quantum sensing. The authors address the question of how minimum measurement errors can be achieved using entanglement as well as more general quantum states. This review is presented from a geometric perspective. This has the advantage that it unifies a wide variety of estimation procedures and strategies, thus providing a more intuitive big picture of quantum parameter estimation.

A stable quantum Darmois-Skitovich theorem

The Darmois-Skitovich theorem is a simple characterization of the normal distribution in terms of the independence of linear forms. We present here a non-commutative version of this theorem in the context of Gaussian bosonic states and show that this theorem is strongly stable under small errors in its underlying conditions. An explicit estimate of the stability constants which depend on the physical parameters of the problem is given.

Quantifying non-Gaussianity of bosonic fields via an uncertainty relation

While Gaussian states and associated Gaussian operations are basic ingredients and convenient objects for continuous-variable quantum information, it is also realized that non-Gaussianity is an important resource for quantum information processing. The characterization and quantification of non-Gaussianity have been widely studied in the past decade, with several significant measures for non-Gaussianity introduced. In this work, by exploiting an information-theoretic refinement of the conventional Heisenberg uncertainty relation and a physical characterization of Gaussian states as minimum uncertainty states, we introduce an easily computable measure for non-Gaussianity of bosonic field states in terms of the Wigner-Yanase skew information. Fundamental properties, as well as intuitive meaning, of this measure are unveiled. The concept is illustrated by prototypical non-Gaussian states, and compared with various existent measures for non-Gaussianity. Its merit and physical significance are elucidated.

Ancilla-assisted schemes are beneficial for Gaussian state phase estimation

We study interferometry with Gaussian states and show that an ancilla-assisted scheme outperforms coherent state interferometry for all levels of loss. We also compare the ancilla-assisted scheme to other interferometric schemes involving squeezing, and show that it is the most advantage in the high-loss, high photon-number regime. In fact, in the presence of high loss, it out-performs many other strategies proposed to date. We find the optimal measurement observable for each scheme discussed. We also find that with the appropriate measurement, the achievable precision of the proposal by Caves [Phys. Rev. D 23, 1693 (1981)] can be improved upon, and is less vulnerable to losses than previously thought.

Semi-supervised optimal recursive filtering and smoothing in non-Gaussian Markov switching models

Filtering and smoothing in switching state-space models are important in numerous applications. The classic family of conditionally Gaussian linear state space models (CGLSSMs) is a natural extension of the Gaussian linear system by introducing its dependence on switches. In spite of their simplicity, recursive filtering and smoothing are no longer feasible in CGLSSMs and approximate methods must be used. Conditionally Markov switching hidden linear models (CMSHLMs) are alternative models which allow recursive optimal exact filtering and smoothing. We introduce an original family of CMSHLMs defined with copulas and we address the problem of their identification. The proposed identification method chooses a model in a family of admissible parametric models and estimates the parameters. It is applied to a learning sample containing observations and states, while the switches are unknown. The interest of the proposed ”semi-unsupervised” filtering and smoothing is validated via experiments on simulated data.