On the tomographic picture of quantum mechanics

We formulate necessary and sufficient conditions for a symplectic tomogram of a quantum state to determine the density state. We establish a connection between the (re)construction by means of symplectic tomograms with the construction by means of Naimark positive definite functions on the Weyl-Heisenberg group. This connection is used to formulate properties which guarantee that tomographic probabilities describe quantum states in the probability representation of quantum mechanics.     

Entropic characteristics of photon tomograms

We study the electromagnetic-field tomograms for classical and quantum states. We use the violation of the positivity of entropy for the photon-probability distributions for distinguishing the classical and quantum domains. We show that the photon-probability distribution expressed in terms of optical or symplectic tomograms of the photon quantum state must be a nonnegative function, which yields the nonnegative Shannon entropy. We also show that the optical tomogram of the photon classical state provides the expression for the Shannon entropy, which can be nonpositive.     

Two-mode squeezed vacuum states in tomographic-probability representation

The two-mode quantum electromagnetic field in the vacuum squeezed state is considered in the tomographic-probability representation. The symplectic, center-of-mass, and photon-number tomograms for the two-mode vacuum squeezed state are obtained explicitly. The expressions for photon statistics of the squeezed light are reconsidered using the state tomograms, and some new integral relations are found for one and multimode orthogonal polynomials.     

Evolution Equation for a Joint Tomographic Probability Distribution of Spin-1 Particles

The nine-component positive vector optical tomographic probability portrait of quantum state of spin-1 particles containing full spatial and spin information about the state without redundancy is constructed. Also the suggested approach is expanded to symplectic tomography representation and to representations with quasidistributions like Wigner function, Husimi Q?function, and Glauber-Sudarshan P?function. The evolution equations for constructed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are found. The evolution equations are also obtained in special case of the quantum system of charged spin-1 particle in arbitrary electro-magnetic field, which are analogs of non-relativistic Proca equation in appropriate representations. The generalization of proposed approach to the cases of arbitrary spin is discussed. The possibility of formulation of quantum mechanics of the systems with spins in terms of joint probability distributions without the use of wave functions or density matrices is explicitly demonstrated.     

Robustness of raw quantum tomography

We scrutinize the effects of non-ideal data acquisition on the tomograms of quantum states. The presence of a weight function, schematizing the effects of a finite window or equivalently noise, only affects the state reconstruction procedure by a normalization constant. The results are extended to a discrete mesh and show that quantum tomography is robust under incomplete and approximate knowledge of tomograms.     

Tomography of Nonclassical States

A review of the symplectic tomography method is presented. Superpositions of different types of photon states are considered within the framework of the tomography approach. Such nonclassical photon states as even and odd coherent states, crystallized Schrödinger cat states, and other superposition states are studied using the construction of symplectic tomograms (tomographic symbols) and the star-product formalism for tomograms.     

Optical Gaussian States Carrying Orbital Angular Momentum

Within the framework of the tomographic probability representation, we introduce specific optical Gaussian states, which were recently proved to carry the orbital angular momentum. We obtain the symplectic and optical tomograms defining uniquely both quantum and classical states for the rotating Gaussian states of light. This approach needs to be developed and applied to the mentioned states due to the convenience of using in the state reconstructions and measurements. Having in mind this aim, we obtain the mean values and variances of the amplitude quadratures directly measurable in the homodyne optical-tomography experiments. Also we consider the time evolution of the rotating Gaussian states in terms of the tomograms and obtain the corresponding tomographic propagator.     

Classical Mechanics Is not the ħ, → 0 Limit of Quantum Mechanics

Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied. It is shown that the sets have a common part but there exist tomograms of classical states which are not admissible in quantum mechanics and, vice versa, there exist tomograms of quantum states which are not admissible in classical mechanics. The role of different transformations of reference frames in the phase space of classical and quantum systems (scaling and rotation) determining the admissibility of tomograms as well as the role of quantum uncertainty relations are elucidated. The union of all admissible tomograms of both quantum and classical states is discussed in the context of interaction of quantum and classical systems. Negative probabilities in classical and quantum mechanics corresponding to tomograms of classical and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively. The role of the semigroup of scaling transforms of the Planck's constant is discussed.     

Probability representation of the quantum evolution and energy-level equations for optical tomograms

The von Neumann evolution equation for the density matrix and the Moyal equation for the Wigner function are mapped onto the evolution equation for the optical tomogram of the quantum state. The connection with the known evolution equation for the symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for optical tomograms. The classical Liouville equation for optical tomogram is obtained. An example of the parametric oscillator is considered in detail.     

Positive distribution description for spin states

We investigate a possibility of describing spin states in terms of a positive distribution function depending on continuous variables like Euler's angles. A spin state reconstruction procedure similar to the symplectic tomography is considered. A quantum evolution equation for the classical-like positive distribution function is found. Generalization to arbitrary values of angular momentum is discussed.     

Charged particle coherent states in tomographic probability representation

The symplectic tomograms of coherent states of a charged particle moving in a constant uniform magnetic field are obtained in explicit form. The tomograms are shown to coincide with normal probability distributions of two random variables. The means and dispersions of the variables are found and expressed in terms of means and dispersions of the charged particle coordinates and momenta. The characteristic function of the tomographic probability distribution is found. The center of mass tomogram of the coherent state of charge in magnetic field is also found and the relation of the symplectic tomogram and the center of mass tomogram is established.     

Tomograms for open quantum systems: In(finite) dimensional optical and spin systems

Tomograms are obtained as probability distributions and are used to reconstruct a quantum state from experimentally measured values. We study the evolution of tomograms for different quantum systems, both finite and infinite dimensional. In realistic experimental conditions, quantum states are exposed to the ambient environment and hence subject to effects like decoherence and dissipation, which are dealt with here, consistently, using the formalism of open quantum systems. This is extremely relevant from the perspective of experimental implementation and issues related to state reconstruction in quantum computation and communication. These considerations are also expected to affect the quasiprobability distribution obtained from experimentally generated tomograms and nonclassicality observed from them.     

The driven-oscillator evolution in the tomographic-probability representation

We consider the problem of the driven harmonic oscillator in the probability representation of quantum mechanics, where the oscillator states are described by fair nonnegative probability distributions of position measured in rotated and squeezed reference frames in the system??s phase space. For some specific oscillator states like coherent states and nth excited states, the tomographic-probability distributions (called the state tomograms) are found in an explicit form. The evolution equation for the tomograms is discussed for the classical and quantum driven oscillators, and the tomographic propagator for this equation is studied.     

The Curious Nonexistence of Gaussian 2-Designs

Ensembles of pure quantum states whose 2nd moments equal those of the unitarily uniform Haar ensemble—2-designs—are optimal solutions for several tasks in quantum information science, especially state and process tomography. We show that Gaussian states cannot form a 2-design for the continuous-variable (quantum optical) Hilbert space ${L^2(\mathbb{R})}$ . This is surprising because the affine symplectic group HWSp (the natural symmetry group of Gaussian states) is irreducible on the symmetric subspace of two copies. In finite dimensional Hilbert spaces, irreducibility guarantees that HWSp-covariant ensembles (such as mutually unbiased bases in prime dimensions) are always 2-designs. This property is violated by continuous variables for a subtle reason: the (well-defined) HWSp-invariant ensemble of Gaussian states does not have a density matrix because its defining integral does not converge. In fact, no Gaussian ensemble is even close (in a precise sense) to being a 2-design. This surprising difference between discrete and continuous quantum mechanics has important implications for optical state and process tomography.     

Inequalities for nonnegative numbers and information properties of qudit tomograms

We discuss some inequalities for N nonnegative numbers. We use these inequalities to obtain known inequalities for probability distributions and new entropic and information inequalities for quantum tomograms of qudit states. The inequalities characterize the degree of quantum correlations in addition to noncontextuality and quantum discord. We use the subadditivity and strong subadditivity conditions for qudit tomographic-probability distributions depending on the unitary-group parameters in order to derive new inequalities for Shannon, Rényi, and Tsallis entropies of spin states.     

Statistical reconstruction of mixed states of polarization qubits

The method of estimating the adequacy, completeness, and accuracy of quantum tomography protocols is generalized to the case of mixed states of polarization qubits. The efficiency of the method is illustrated based on mathematical modeling and experimental investigation of some practically important quantum tomography protocols.     

Projectivities of Informationally Complete Measurements

The physical problem behind informationally complete (IC) measurements is determining an unknown state statistically by measurement outcomes, known as state tomography. It is of central importance in quantum information processing such as channel estimating, device testing, quantum key distribution, etc. However, constructing such measurements with good properties is a long-standing problem. In this work, projective realizations of IC measurements are investigated. Conditions of informational completeness are presented with proofs first. Then the projective realizations of IC measurements, including proposing the first general construction of minimal projective IC measurements (MPICM) in no prime power dimensional systems, as well as determining an unknown state in $C^n$ via a single projective measurement with some kinds of optimalities in a larger system, are investigated. Finally, The results can be extended to local state tomography. Some discussions on employing several kinds of optimalities are also provided.     

Spin state tomography

A scheme for measuring the quantum state for an arbitrary spin is proposed that is analogous to the symplectic tomography scheme used to measure quantum states associated with continuous observables such as position and momentum. An invariant form for the spin state density operator is derived in terms of an integral, over the angles which specify the quantization axis, of a product of the measured probability of the values of the spin along a chosen direction and spherical harmonics summed with Clebsch-Gordan functions. Zh. éksp. Teor. Fiz. 112, 796–804 (September 1997)     

Bidirectional Information Flow Quantum State Tomography

The exact reconstruction of many-body quantum systems is one of the major challenges in modern physics,because it is impractical to overcome the exponential complexity problem brought by high-dimensional quantum manybody systems.Recently,machine learning techniques are well used to promote quantum information research and quantum state tomography has also been developed by neural network generative models.We propose a quantum state tomography method,which is based on a bidirectional gated recurrent unit neural network,to learn and reconstruct both easy quantum states and hard quantum states in this study.We are able to use fewer measurement samples in our method to reconstruct these quantum states and to obtain high fidelity.