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We have measured probability distributions of quadrature-field amplitude for both vacuum and quadrature-squeezed states of a mode of the electromagnetic field. From these measurements we demonstrate the technique of optical homodyne tomography to determine the Wigner distribution and the density matrix of the mode. This provides a complete quantum mechanical characterization of the measured mode.

Content uploaded by Michael Raymer

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All content in this area was uploaded by Michael Raymer on Mar 22, 2016

Content may be subject to copyright.

... In a complementary way, the fidelity between an evolved state and its initial state provides insights into the periodic behavior of the system [18]. Likewise, the Wigner function [19], a quasiprobability distribution, offers a perspective on the distribution of coherent states in phase space, shedding light on their classical or quantum nature [20][21][22][23]. Initially, the Wigner function arose from a quantum mechanics formulation in phase space [24,25] in which, under certain requirements, it is possible to associate an integrable function defined on n 2 to an operator in a Hilbert space through the so-called Weyl transform [26][27][28]. ...

... To build now the coherent states associated with this system, we need to introduce an appropriate 2 × 2 annihilation operator A − for those eigenfunctions in equation (23). For that reason we will define A − as follows [18]: ...

In this paper, we examine the electron interaction within tilted anisotropic Dirac materials when subjected to external electric and magnetic fields possessing translational symmetry. Specifically, we focus on a distinct non-zero electric field magnitude, enabling the decoupling of the differential equation system inherent in the eigenvalue problem. Subsequently, employing supersymmetric quantum mechanics facilitates the determination of eigenstates and eigenvalues corresponding to the Hamiltonian operator. To delve into a semi-classical analysis of the system, we identify a set of coherent states. Finally, we assess the characteristics of these states using fidelity and the phase-space representation through the Wigner function.

... Determining the DM accurately is a central task in quantum science. Traditionally, this problem is tackled through Quantum State Tomography (QST) [2,3], which involves performing informationally complete (IC) measurements [4,5] and postprocessing the data to estimate the quantum state. In a d-dimensional Hilbert space, a general density matrix has d 2 − 1 independent parameters. ...

... When designing each eigenbasis, orthogonality and completeness are the key constraints, precisely, each pair of elements in an eigenbasis should be orthogonal, and there should be d elements in each basis. Thus we introduce elements {|ϕ − jk ⟩} and {|ψ − jk ⟩} into the set A d in Eq.(2). Consequently, there are 2d 2 − d elements. ...

Efficient understanding of a quantum system fundamentally relies on the selection of observables. Pauli observables and mutually unbiased bases (MUBs) are widely used in practice and are often regarded as theoretically optimal for quantum state tomography (QST). However, Pauli observables require a large number of measurements for complete tomography and do not permit direct measurement of density matrix elements with a constant number of observables. For MUBs, the existence of complete sets of \(d+1\) bases in all dimensions remains unresolved, highlighting the need for alternative observables. In this work, we introduce a novel set of \(2d\) observables specifically designed to enable the complete characterization of any \(d\)-dimensional quantum state. To demonstrate the advantages of these observables, we explore two key applications. First, we show that direct measurement of density matrix elements is feasible without auxiliary systems, with any element extractable using only three selected observables. Second, we demonstrate that QST for unknown rank-\(r\) density matrices, excluding only a negligible subset, can be achieved with \(O(r \log d)\) observables. This significantly reduces the number of unitary operations compared to compressed sensing with Pauli observables, which typically require \(O(r d \log^2 d)\) operations. Each circuit is iteratively generated and can be efficiently decomposed into at most \(O(n^4)\) elementary gates for an \(n\)-qubit system. The proposed observables represent a substantial advancement in the characterization of quantum systems, enhancing both the efficiency and practicality of quantum state learning and offering a promising alternative to traditional methods.

... The Wigner distribution, a quasiprobability distribution for the two quadratures of the harmonic mode, demonstrates energy pumping through an in-spiral analogous that of a classical harmonic oscillator with constant rate of energy loss. Furthermore, the Wigner distribution is routinely measured in circuit QED and cavity QED experiments [34][35][36]. An illustration of this topological in-spiral is in Fig. 4(c)-(h), showing topological pumping that eventually deviates from quantization when the system reaches the topological phase boundary. ...

There is a close theoretical connection between topological Floquet physics and cavity QED, yet this connection has not been realized experimentally due to complicated cavity QED models that often arise. We propose a simple, experimentally viable protocol to realize non-adiabatic topological photon pumping mediated by a single qubit, which we dub the anomalous Floquet photon pump. For both quantized photons and external drive, the system exhibits a non-trivial topological phase across a broad range of parameter space. Transitions out of the topological phase result from frequency-space delocalization. Finally, we argue that the protocol can be implemented in existing experiments via driven qubit non-linearities, with topological pumping witnessed in measurements of the cavity Wigner distribution functions.

... Its potential applications have been discussed in several areas ranging from a quantum key distribution [3][4][5][6], entanglement and resources theory [6][7][8][9][10][11][12], quantum metrology and states discrimination [13][14][15][16][17], to quantum communication and state transfers [13,[18][19][20][21][22][23]. One of the advantages of this program is the ability to encode a qubit into continuous degrees of freedom that are intact to disturbances from the environment, by only using traditional sources of quantum systems and coherence, especially light. ...

The Gottesman-Kitaev-Preskill (GKP) coding is proven to be a good candidate for encoding a qubit on continuous variables (CV) since it is robust under random-shift disturbance. Its preparation in optical systems, however, is challenging to realize in nowadays state-of-the-art experiments. In this article, we propose a simple optical setup for preparing the approximate GKP states by employing a random walk mechanism. We demonstrate this idea by considering the encoding on the transverse position of a single-mode pulse laser. We also discuss generalization and translation to other types of physical CV systems.

... Our approach generalizes the ideas of previous protocols by using a complete characterization of a quantum system. This is done by quantum state tomography, which directly estimates the density matrix of the system 33,34 . The question of QST in error mitigation was also considered in ref. 28, but restricted to only classical errors, i.e., errors that can be described as a stochastic redistribution of the outcome statistics of single basis measurements. ...

Quantum technologies rely heavily on accurate control and reliable readout of quantum systems. Current experiments are limited by numerous sources of noise that can only be partially captured by simple analytical models and additional characterization of the noise sources is required. We test the ability of readout error mitigation to correct noise found in systems composed of quantum two-level objects (qubits). To probe the limit of such methods, we designed a beyond-classical readout error mitigation protocol based on quantum state tomography (QST), which estimates the density matrix of a quantum system, and quantum detector tomography (QDT), which characterizes the measurement procedure. By treating readout error mitigation in the context of state tomography the method becomes largely readout mode-, architecture-, noise source-, and quantum state-independent. We implement this method on a superconducting qubit and evaluate the increase in reconstruction fidelity for QST. We characterize the performance of the method by varying important noise sources, such as suboptimal readout signal amplification, insufficient resonator photon population, off-resonant qubit drive, and effectively shortened T1 and T2 coherence. As a result, we identified noise sources for which readout error mitigation worked well, and observed decreases in readout infidelity by a factor of up to 30.

... However, the distributions of measured quadratures are visibly distinct, just not in their mean values, but instead in their second-order moments or covariances. To estimate such second-order moments (routinely required for example for Gaussian state tomography [45,52,53]), the standard approach is to obtain S shots and estimate the variance of the measurements over the dataset yielding readout features y = F NL [x] = 1 S S s (I 2 1 , Q 2 1 ) (shot s dependence of quadratures I k , Q k is implied). For Task I, we plot distributions of these nonlinear readout features in the second panel of Fig. 2(c). ...

Although linear quantum amplification has proven essential to the processing of weak quantum signals, extracting higher-order quantum features such as correlations in principle demands nonlinear operations. However, nonlinear processing of quantum signals is often associated with non-idealities and excess noise, and absent a general framework to harness nonlinearity, such regimes are typically avoided. Here we present a framework to uncover general quantum signal processing principles of a broad class of bosonic quantum nonlinear processors (QNPs), inspired by a remarkably analogous paradigm in nature: the processing of environmental stimuli by nonlinear, noisy neural ensembles, to enable perception. Using a quantum-coherent description of a QNP monitoring a quantum signal source, we show that quantum nonlinearity can be harnessed to calculate higher-order features of an incident quantum signal, concentrating them into linearly-measurable observables, a transduction not possible using linear amplifiers. Secondly, QNPs provide coherent nonlinear control over quantum fluctuations including their own added noise, enabling noise suppression in an observable without suppressing transduced information, a paradigm that bears striking similarities to optimal neural codings that allow perception even under highly stochastic neural dynamics. Unlike the neural case, we show that QNP-engineered noise distributions can exhibit non-classical correlations, providing a new means to harness resources such as entanglement. Finally, we show that even simple QNPs in realistic measurement chains can provide enhancements of signal-to-noise ratio for practical tasks such as quantum state discrimination. Our work provides pathways to utilize nonlinear quantum systems as general computation devices, and enables a new paradigm for nonlinear quantum information processing.

... Experimental reconstructions of the Wigner function rely on homodyne detection measurements [42] in which the optical state we want to characterize is combined with a well-characterized coherent state with a controllable phase, typically referred to as the local oscillator (LO), using a 50 : 50 beam splitter, as shown at the right of Fig. 1. The intensities of the two output modes of the beam splitter are then measured and subtracted, leaving us with ...

Quantum state engineering of light is of great interest for quantum technologies, particularly generating non-classical states of light, and is often studied through quantum conditioning approaches. Recently, we demonstrated that such approaches can be applied in intense laser-atom interactions to generate optical "cat" states by using intensity measurements and classical post-selection of the measurement data. Post-processing of the sampled data set allows to select specific events corresponding to measurement statistics as if there would be non-classical states of light leading to these measurement outcomes. However, to fully realize the potential of this method for quantum state engineering, it is crucial to thoroughly investigate the role of the involved measurements and the specifications of the post-selection scheme. We illustrate this by analyzing post-selection schemes recently developed for the process of high harmonic generation, which enables generating optical cat states bright enough to induce non-linear phenomena. These findings provide significant guidance for quantum light engineering and the generation of high-quality, intense optical cat states for applications in non-linear optics and quantum information science.

... Such low-dimensional reduced models are particularly well suited for numerical simulation of autonomous quantum error correction schemes developed for bosonic codes. In particular, for cat-qubit systems, around 50 to 100 photons per cat-qubit are required for simulating experimental setups, corresponding to a mean photon number of 10 to 15. Two-qubit quantum process tomography [31,32] is manageable via standard simula-tion methods for a small mean photon number but becomes infeasible when it exceeds 10. In the case of a three-qubit gate with a truncation of 100, standard simulations are impossible as they require storing density matrices of dimension 100 6 and quantum process tomography would present even greater challenges. ...

A numerical method is proposed for simulation of composite open quantum systems. It is based on Lindblad master equations and adiabatic elimination. Each subsystem is assumed to converge exponentially towards a stationary subspace, slightly impacted by some decoherence channels and weakly coupled to the other subsystems. This numerical method is based on a perturbation analysis with an asymptotic expansion. It exploits the formulation of the slow dynamics with reduced dimension. It relies on the invariant operators of the local and nominal dissipative dynamics attached to each subsystem. Second-order expansion can be computed only with local numerical calculations. It avoids computations on the tensor-product Hilbert space attached to the full system. This numerical method is particularly well suited for autonomous quantum error correction schemes. Simulations of such reduced models agree with complete full model simulations for typical gates acting on one and two cat-qubits (Z, ZZ and CNOT) when the mean photon number of each cat-qubit is less than 8. For larger mean photon numbers and gates with three cat-qubits (ZZZ and CCNOT), full model simulations are almost impossible whereas reduced model simulations remain accessible. In particular, they capture both the dominant phase-flip error-rate and the very small bit-flip error-rate with its exponential suppression versus the mean photon number.

The classical shadows protocol is an efficient strategy for estimating properties of an unknown state ρ using a small number of state copies and measurements. In its original form, it involves twirling the state with unitaries from some ensemble and measuring the twirled state in a fixed basis. It was recently shown that for computing local properties, optimal sample complexity (copies of the state required) is remarkably achieved for unitaries drawn from shallow depth circuits composed of local entangling gates, as opposed to purely local (zero depth) or global twirling (infinite depth) ensembles. Here, we consider the sample complexity as a function of the depth of the circuit, in the presence of noise. We find that this noise has important implications for determining the optimal twirling ensemble. Under fairly general conditions, we (i) show that any single-site noise can be accounted for using a depolarizing noise channel with an appropriate damping parameter f, (ii) compute thresholds fth at which optimal twirling reduces to local twirling for Pauli operators, (iii) nth order Renyi entropies (n≥2), and (iv) provide a meaningful upper bound tmax on the optimal circuit depth for any finite noise strength f, which applies to observables and entanglement entropy measurements. These thresholds strongly constrain the search for optimal strategies to implement shadow tomography and are easily tailored to the experimental system at hand.

We present a method for reconstructing intracavity dynamics of an optical parametric oscillator and performing cavity quantum tomography. Our approach involves evaluating the sensitivity of the bistable oscillator’s output to a bias field.

This is the first part of what will be a two-part review of distribution functions in physics. Here we deal with fundamentals and the second part will deal with applications. We discuss in detail the properties of the distribution function defined earlier by one of us (EPW) and we derive some new results. Next, we treat various other distribution functions. Among the latter we emphasize the so-called P distribution, as well as the generalized P distribution, because of their importance in quantum optics.

Quantum-mechanical calculations of the mean-square fluctuation spectra in optical homodyning and heterodyning are made for arbitrary input and local-oscillator quantum states. In addition to the unavoidable quantum fluctuations, it is shown that excess noise from the local oscillator always affects homodyning and, when it is broadband, also heterodyning. Both the quantum and the excess noise of the local oscillator can be eliminated by coherent subtraction of the two outputs of a 50-50 beam splitter. This result also demonstrates the fact that the basic quantum noise in homodyning and heterodyning is signal quantum fluctuation, not local-oscillator shot noise.

The probability of a configuration is given in classical theory by the Boltzmann formula exp[−VhT] where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.

The concept of quantum transition is critically examined from the perspective of the modern quantum theory of measurement. Historically rooted in the famous quantum jump of the Old Quantum Theory, the transition idea survives today in experimental jargon due to (1) the notion of uncontrollable disturbance of a system by measurement operations and (2) the wave-packet reduction hypothesis in several forms. Explicit counterexamples to both (1) and (2) are presented in terms of quantum measurement theory. It is concluded that the idea of transition, or quantum jump, can no longer be rationally comprehended within the framework of contemporary physical theory.

We discuss the profound influence which the Wigner distribution function has had in many areas of physics during its fifty years of existence.

A common approach to quantum physics is enshrouded in a jargon which treats state vectors as attributes of physical systems and the concept of state preparation as a filtration scheme wherein a process involving measurement selects from a primordial assembly of systems those bearing some prescribed vector of interest. By contrast, the empirical experiences with which quantum theory is actually concerned relate measurement and preparation in quite an opposite manner. Reproducible preparation schemes are logically and temporally anterior to measurement acts. Measurement extracts numbers from systems prepared in a specified manner; these data are then regularized by the theory by means of a state concept which is in turn used to characterize succinctly the given mode of preparation. The present paper offers, in a simple spin model, a method for determining the quantum state that represents any reproducible preparation.

In quantum mechanics, the state of an individual particle (or system) is unobservable, i.e., it cannot be determined experimentally, even in principle. However, the notion of measuring a state is meaningful if it refers to anensemble of similarly prepared particles, i.e., the question may be addressed: Is it possible to determine experimentally the state operator (density matrix) into which a given preparation procedure puts particles. After reviewing the previous work on this problem, we give simple procedures, in the line of Lamb's operational interpretation of quantum mechanics, for measuring a translational state operator (whether pure or mixed), via its Wigner function. These procedures closely parallel methods that might be used in classical mechanics to determine a true phase space probability distribution; thus, the Wigner function simulates such a distribution not only formally, but operationally also.
There is no way to determine what the wave function (or state vector) of a system is—if arbitrarily given, there is no way to measure its wave function. Clearly, such a measurement would have to result in afunction of several variables, not in a relatively small set ofnumbers .... In order to verify the [quantum] theory in its generality, at least a succession of two measurements are needed. There is in general no way to determine the original state of the system, but having produced a definite state by a first measurement, the probabilities of the outcomes of a second measurement are then given by the theory.
E. P. Wigner(1)

Quantum-mechanical calculations of the mean-square fluctuation spectra in optical homodyning and heterodyning are made for arbitrary input and local-oscillator quantum states. In addition to the unavoidable quantum fluctuations, it is shown that excess noise from the local oscillator always affects homodyning and, when it is broadband, also heterodyning. Both the quantum and the excess noise of the local oscillator can be eliminated by coherent subtraction of the two outputs of a 50-50 beam splitter. This result also demonstrates the fact that the basic quantum noise in homodyning and heterodyning is signal quantum fluctuation, not local-oscillator shot noise.

Mode-locked lasers provide a train of high-intensity light pulses that can be used to pump nonlinear media in order to produce squeezed light. The squeezed light produced would consist of a train of short-duration pulses. It is shown here that a homodyne detector using a pulsed local oscillator can be used to observe the squeezing even when the response time of the photodetector is much longer than the local oscillator or squeezed light pulse width.