Because light is a quantum object, its quantum states can be measured by homodyne tomography, too. Discrete phasespace quantum mechanics on finite fields is treated for completeness and by virtue of its relevance to quantum computing. We begin our analysis by considering a quantum system with an unknown density matrix, inthebasis ofazimuthal angle, further, we choose to work in a finitedimensional state space spanned by the orbitalangularmomentum eigenvectors jli with fjlj. Continuous variable quantum state tomography is one of the most powerful techniques to study.
Wigner functionwf, a quasipro bability distribution in phase space, was. Although our results are general, we concentrate on the simplest nontrivial example of. The nonclassical character of the fock state shows up as a negativity of the wigner function at the origin. In this work we follow the approach of wootters 11 and gibbons et. Sep 03, 2011 wigner functionwf, a quasipro bability distribution in phase space, was. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature. The phenomenological treatment of evolution superoperator and measurements is given to help clarify the general quantum transport theory. It also discusses the generalized pauli group, mutually unbiased bases, and the discrete wigner function. In section 3 we obtain a closed expression for the starproduct operator in terms of difference operators acting in the discrete phase space and give some example of its usage. Towards optomechanical quantum state reconstruction of.
Quantum state tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states. This scheme for reconstruction of the wigner function i. The nonclassical character of the fockstate shows up as a negativity of the wigner function at the origin. Quantum state tomography 3 pi is the probabalistic weighting p ipi 1, a. Characterization of quantum states of light by rob. It is based on convex optimization with semidefinite programs, and also includes a version of the maximum entropy principle, in order to yield unbiased states. Wigner function for a particle in an infinite lattice iopscience.
Mutually unbiased bases and discrete wigner functions. Odd and evendimensional systems angular momenta and spins, bosons, and fermions are considered separately. Reconstruction of single and 2qubit density matrices. Quantum state reconstruction to get a good feel for what we will be occupied with in this report, let us take a look at the title. Reconstruction of single and 2qubit density matrices using.
However, the need for a xed phase reference has so far prevented widespread usage in other elds such as semiconductor spectroscopy. Quantumstate tomography and discrete wigner function, phys. Quantum state tomography of 1d resonance fluorescence. Quantum bits, or qubits, are 2state quantum systems and form the basis of quantum computation and quantum information.
Reply to comment on wigner function for a particle in an. Chapter 7 the discrete wigner function request pdf. Quantum state tomography qst aims to reconstruct the full quantum state from simple measurements, and therefore provides a key tool to obtain reliable analytics. Quantum state tomography qst aims to reconstruct the full quantum state from simple measurements, and therefore provides a key tool to obtain reliable analytics 1,2,3. This page is intended to be useful to a variety of visitors, from experimental research groups setting up quantum tomography systems, to students learning about the theory of characterizing quantum states. Negative wigner functions as a resource for quantum com putation. Phase space tomography estimates correlation functions entirely from snapshots in the evolution of the wave function along a time or space variable. Out of all available choices, one can argue that the wigner function 1 presents the most natural phase space representation of quantum mechanics 2. The state is represented through the wigner function, a quasi. As an example, we can consider the tomography of a singlephoton state 1. Discrete wigner function reconstruction and compressed sensing. Mutually unbiased bases and discrete wigner functions arxiv. Twophoton polarization entanglement in type i spdc experiment, 2009.
In these experiments the wigner functions of a coherent state and a squeezed vacuum state have been reconstructed from tomographic data. Quantum state tomography is the attempt to discover the quantummechanical state of a physical system, or more precisely, of a. Apr 15, 2011 the wigner function is a gaussian function for a and b, but it takes negative values for the strongly quantum states c and d. The source may be any device or system which prepares quantum states either consistently into quantum pure states or otherwise into general mixed states. Quantum bits, or qubits, are 2 state quantum systems and form the basis of quantum computation and quantum information. By analogy with computed tomography, wigner functions for some quantum states can be reconstructed with less measurements utilizing this compressed sensing based method. It has been experimentally demonstrated for a variety of physical systems, among them the quantum state of a light mode 75, the vibrational state of a single ion 76, and the wave packets of atoms of an atomic beam 77. Operator hilbertspace methods and quantum tomography are discussed. Quantumstate tomography and discrete wigner function. The wigner function can be used to fully describe any quantum system of arbitrary dimension or ensemble size. The discrete wigner function formulation of quantum bits. In this subspace, the discrete wigner distribution function reads 31,32 w.
Mubs are central to quantum tomography and state re construction 28, but are also valuable for quantum key distribution 29, 30, 31, 32. Here we show that, for a quantum system whose state or evolution can be modelled on a quantum computer, tomography and spectroscopy can be interpreted as dual forms of quantum computation2. In section 3 we show how to describe teleportation of the quantum state of a system with ann dimensional hilbert space using the wigner function. Efficient quantum tomography of twomode wigner functions. In section iv we describe the procedure by which a direct.
Discrete wigner function and quantumstate tomography. It is shown that measurements used in the inverse radon transform need to be made on j2d lines where j2d is. We introduce an efficient method to reconstruct the wigner function of manymode continuous variable systems. Physical realizations and wigner representation of coherent. And yet, many fundamental questions remain unanswered. The magic of universal quantum computing with permutations. Wigner function is widely used in nonrelativistic quantum mechanics as an alternative to the density matrix of quantum systems 29. Interpretation of tomography and spectroscopy as dual forms. Nonparametric estimation of the purity of a quantum state. Continuous variable quantum state tomography is one of the most powerful techniques to study the properties of light elds in quantum optics. Such a connection becomes more interesting when the hilbert space has a dimension that is a power of a prime ndn, which describes a composite system of n qudits. Mutually unbiased bases and discrete wigner functions are closely, but not uniquely related.
A set of marginal distributions for various q s will provide us with full information about the quantum state and allows us to reconstruct its wigner function and the density matrix. Pdf discrete wigner function reconstruction and compressed. Neergaardnielsen,1 peter van loock,2 and akira furusawa3 1department of physics, technical university of denmark, fysikvej, 2800 kongens lyngby, denmark. The discrete wigner function dwf representation of quantum states is based on a. Here we show that, for a quantum system whose state or evolution can be modelled on a quantum computer, tomography and spectroscopy can be interpreted as dual forms of. Neergaardnielsen,1 peter van loock,2 and akira furusawa3 1department of physics, technical university of denmark, fysikvej, 2800 kongens lyngby, denmark 2institute of physics, johannesgutenberg universit at mainz, staudingerweg 7, 55128 mainz, germany 3department of applied physics, school of. It is worth noting that lrmr came about in the context of compressive quantum state tomography. In contrast, traditional interferometric methods require measurement of multiple twopoint correlations. Pdf classicality in discrete wigner functions researchgate. The instantaneous state of a quantum system encodes the. Hybrid discrete and continuousvariable quantum information. Wigner function of a single photon fock state from lvovsky et al. Discrete wigner functions and the phase space representation.
Thus, by performing measurements of x on many identical copies of the state and varying. Minimax estimation of the wigner function in quantum homodyne tomography with ideal detectors m. Physical realizations and wigner representation of. Examples of quasiprobability representations of quantum states. The wigner function is a way to describe how quantum a light pulse is. A tomographical scheme is proposed to infer the quantum states of finite dimensional systems from experiments. Hybrid quantum information processing with continuous and. Quantum state tomographyof an itinerant squeezed microwave field. Thus, by performing measurements of x on many identical copies of the state and varying, the hidden quantum object. Now, it is routinely carried out in a wide variety of systems.
A new reconstruction method for wigner function is reported for quantum tomography based on compressed sensing. Minimax estimation of the wigner function in quantum homodyne. These discrete versions of the wigner function have a number of peculiar properties which make them rather. Quantum state tomography from sequential measurement of two. In particular, it was done for the discretevalued position and momentum 12, for the hermitegaussian and laguerre. We study the wigner function for a quantum system with a discrete. Quantum state reconstruction in severalparticle systems. The projections or shadows of the wigner function shown on the sides are. In order to perform a subsequent tomography via homodyning this state had to be matched as good as possible to the local oscillator state. The wigner distribution function is not the only quantum phase space distribution. Several articles on the wigner function alone without the. One may represent systems of any number of qubits using the discrete wigner function. It means that our main focus will be on physical systems with more than one particle. Quantum state tomography allows the estimation of an unknown quantum state that is available in many identical copies.
Quantum tomography has come a long way from early reconstructions of wigner functions from projections along quadratures to the full characterization of multipartite systems. Twophoton polarization entanglement in type i spdc experiment, 2009 6k. While any ensemble of pure states can be represented in this way, it is. Also, the steadystate solution of the discrete wigner equation was. Density matrix estimation in quantum homodyne tomography. This process is experimental and the keywords may be updated as the learning algorithm improves. While any ensemble of pure states can be represented in this way, it is also true that any ensemble of singlequbit states can be represented by an. However, exact bruteforce approaches to qst place a high demand on computational resources, making them unfeasible for anything except small systems. The role of permutation gates for universal quantum computing is investigated. The tomograms will all have a doublepeak structure, and from the whole set one restores the wigner function. These negative features vanish very quickly in the presence of decoherence.
A 65 062309 11 zak j 2011 doubling feature of the wigner function. Density operator wigner function quantum communication tomography approach spin particle these keywords were added by machine and not by the authors. Quantum tomography is the art of determining a quantum state from making measurements on multiple copies of the state with multiple modifications of the measurement apparatus. Chapter 7 the discrete wigner function sciencedirect. Extensions to quantum process tomography 31 as given by the experimental reconstruction of entire propagators e. The theory of discrete wigner functions and of discrete quantumstate tomography u.
An excellent introduction into nonclassical light and methods of its characterization is given in the textbook by u. As opposed to projective measurements, another class of generalized measurement is known as the weak measurement. Focus on quantum tomography new journal of physics. Hybrid discrete and continuousvariable quantum information ulrik l. Although the original wigner function applies only to systems with continuous degrees of freedom, it can be generalized for.
Neuralnetwork quantum state tomography nature physics. Dec 27, 2019 we introduce an efficient method to reconstruct the wigner function of manymode continuous variable systems. The wigner quasiprobability distribution also called the wigner function or the wignerville distribution after eugene wigner and jeanandre ville is a quasiprobability distribution. A key ingredient of the proposed approach is the representation of the state in a truncated fock basis. Its wigner function has a negative peak at the origin and a positive ring around fig. Nonparametric estimation of the purity of a quantum state in. State reconstruction via inverse linear transformation 305 1. Apr 09, 2012 phase space tomography estimates correlation functions entirely from snapshots in the evolution of the wave function along a time or space variable. The following picture shows again the wigner function of a single photon fock state. However, as in every tomographic formulation, undersampling poses a severe limitation. Focus on quantum tomography new journal of physics iopscience. It was introduced 1 by eugene wigner in 1932 to study quantum corrections to classical statistical mechanics. It was introduced by eugene wigner in 1932 to study quantum corrections to classical statistical mechanics. Hybrid quantum information processing with continuous and discrete variables of light elds gaia donati linacre college, oxford submitted for the degree of doctor of philosophy hilary term 2015 abstract quantum correlations play a fundamental role in quantum information science.
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