We show that for general deformations of SU(2) algebra, the dynamics in terms of ladder operators is preserved. This is done for a system of precessing magnetic dipole in magnetic field, using the unitary phase operator which arises in the polar decomposition of SU(2) operators. It is pointed out that there is a single phase operator dynamics underlying the dynamics of usual and deformed ladder operators.
The purpose of this comment is to show that a reinterpretation of the results from the Letter: "Physics and Metaphysics of Wigner's Friends: Even performed pre-measurements have no results" allows for reaching the conclusion "pre-measurements have no resul" [arXiv:2003.07464] based only on postulates of quantum mechanics without additional assumptions on irreversibility. Additionally, with supplementary reasoning based on Bell's theorem, one can show that unitary decoherence cannot be solely responsible for the quantum-to-classical transition, and an additional irreversibility model is require…
A general argument is presented against relativistic, unitary, single-outcome quantum mechanics. This is achieved by combining the Wigner's Friend thought experiment with measurements on a Greenberger-Horne-Zeilinger (GHZ) state, and describing the evolution of the quantum state in various inertial frames. Assuming unitary quantum mechanics and single outcomes, the result is that the Born rule must be violated in some inertial frame: in that frame, outcomes are obtained for which no corresponding term exists in the pre-measurement wavefunction.
We obtain the accuracy limit for estimating the expectation value of the position of a relativistic particle for an observer moving along one direction at a constant velocity. We use a specific model of a relativistic spin-1/2 particle described by a gaussian wave function with a spin down in the rest frame. To derive the state vector of the particle for the moving observer, we use the Wigner rotation that entangles the spin and the momentum of the particle. Based on this wave function for the moving frame, we obtain the symmetric logarithmic derivative (SLD) Cramer-Rao bound that sets the est…
In quantum theory, symmetry has to be defined necessarily in terms of the family of unit rays, the state space. The theorem of Wigner asserts that a symmetry so defined at the level of rays can always be lifted into a linear unitary or an antilinear antiunitary operator acting on the underlying Hilbert space. We present a proof of this theorem which is both elementary and economical. Central to our proof is the recognition that a given Wigner symmetry can, by post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis involves a judicious i…
We present a new treatment for the spin of a massive relativistic particle in the context of quantum information based on a physical interpretation of the Wigner rotations, obtaining different results in relation to the previous works. We are lead to the conclusions that it is not possible to define a reduced density matrix for the particle spin and that the Pauli-Lubanski (or similar) spin operators are not suitable to describe measurements where spin couples to an electromagnetic field in the measuring apparatus. These conclusions contradict the assumptions made by most of the previous paper…
We study the behaviour of the geometric phase under isometries of the ray space. This leads to a better understanding of a theorem first proved by Wigner: isometries of the ray space can always be realised as projections of unitary or anti-unitary transformations on the Hilbert space. We suggest that the construction involved in Wigner's proof is best viewed as an use of the Pancharatnam connection to ``lift'' a ray space isometry to the Hilbert space.
An experiment by Proietti {\it et al} purporting to instantiate the `Wigner's Friend' thought experiment is discussed. It is pointed out that the stated implications of the experiment regarding the alleged irreconcilability of facts attributed to different observers warrant critical review. In particular, violation of a Clauser-Horne-Shimony inequality by the experimental data actually shows that the attribution of measurement outcomes to the ``Friends'' (modeled by internal photons undergoing unitary interactions) is erroneous. An elementary but often overlooked result regarding improper mixt…
The Heisenberg evolution of a given unitary operator corresponds classically to a fixed canonical transformation that is viewed through a moving coordinate system. The operators that form the bases of the Weyl representation and its Fourier transform, the chord representation, are, respectively, unitary reflection and translation operators. Thus, the general semiclassical study of unitary operators allows us to propagate arbitrary operators, including density operators, i.e. the Wigner function. The various propagation kernels are different representations of the superoperators which act on th…
The family of unitary non-equivalent Weyl-Stratonovich kernels determining the Wigner probability distribution function of an arbitrary N-level quantum system is constructed.
The algebra of generalized linear quantum canonical transformations is examined in the prespective of Schwinger's unitary-canonical basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the quantum canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary ope…
Manifestly covariant formulation of discrete-spin, real-mass unitary representations of the Poincaré group is given. We begin with a field of spin-frames associated with 4-mometa p and use them to simplify the eigenvalue problem for the Pauli-Lubanski vector projection in a direction given by a world-vector t. As opposed to the standard treatments where t is a constant time direction, our t is in general p-dependent and timelike, spacelike or null. The corresponding eigenstates play a role of a basis used to define Bargmann-Wigner spinors which form a carrier space of the unitary representatio…
Quantum Wasserstein isometries of the $n$-qubit state space: a Wigner-type result
HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；ELTE Eötvös Loránd University（罗兰大学）
The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. We prove that, in the discrete scenario, this permutation symmetry is equivalent to the symmetry group being a unitary 2-design. Such a highly symmetric representation can only appear in od…
We introduce a new class of unitary transformations based on the su(1,1) Lie algebra that generalizes, for certain particular representations of its generators, well-known squeezing transformations in quantum optics. To illustrate our results, we focus on the two-mode bosonic representation and show how the parametric amplifier model can be modified in order to generate such a generalized squeezing operator. Furthermore, we obtain a general expression for the bipartite Wigner function which allows us to identify two distinct sources of entanglement, here labelled by dynamical and kinematical e…
Does quantum theory apply at all scales, including that of observers? New light on this fundamental question has recently been shed through a resurgence of interest in the long-standing Wigner's friend paradox. This is a thought experiment addressing the quantum measurement problem -- the difficulty of reconciling the (unitary, deterministic) evolution of isolated systems and the (non-unitary, probabilistic) state update after a measurement. Here, by building on a scenario with two separated but entangled friends introduced by Brukner, we prove that if quantum evolution is controllable on the …
The Weyl-Wigner map yields the entire structure of Moyal quantum mechanics directly from the standard operator formulation. The covariant generalization of Moyal theory, also known as Vey quantum mechanics, was presented in the literature many years ago. However, a derivation of the formalism directly from standard operator quantum mechanics, clarifying the relation between the two formulations is still missing. In this paper we present a covariant generalization of the Weyl order prescription and of the Weyl-Wigner map and use them to derive Vey quantum mechanics directly from the standard op…
We present a self-consistent theoretical framework for finite-dimensional discrete phase spaces that leads us to establish a well-grounded mapping scheme between Schwinger unitary operators and generators of the special unitary group $\mathrm{SU(N)}$. This general mathematical construction provides a sound pathway to the formulation of a genuinely discrete Wigner function for arbitrary quantum systems described by finite-dimensional state vector spaces. To illustrate our results, we obtain a general discrete Wigner function for the group $\mathrm{SU(3)}$ and apply this to the study of a partic…
We explicitly establish a unitary correspondence between spherical irreducible tensor operators and cartesian tensor operators of any rank. That unitary relation is implemented by means of a basis of integer-spin wave functions that constitute simultaneously a basis of the spaces of cartesian and spherical irreducible tensors. As a consequence, we extend the Wigner--Eckart theorem to cartesian irreducible tensor operators of any rank, and to totally symmetric reducible ones. We also discuss the tensorial structure of several standard spherical irreducible tensors such as ordinary, bipolar and …
The notorious quantum measurement problem brings out the difficulty to reconcile two quantum postulates: the unitary evolution of closed quantum systems and the wave-function collapse after a measurement. This problematics is particularly highlighted in the Wigner's friend thought experiment, where the mismatch between unitary evolution and measurement collapse leads to conflicting quantum descriptions for different observers. A recent no-go theorem has established that the (quantum) statistics arising from an extended Wigner's friend scenario is incompatible when one try to hold together thre…
An integral of the Wigner function of a wavefunction |psi >, over some region S in classical phase space is identified as a (quasi) probability measure (QPM) of S, and it can be expressed by the |psi > average of an operator referred to as the region operator (RO). Transformation theory is developed which provides the RO for various phase space regions such as point, line, segment, disk and rectangle, and where all those ROs are shown to be interconnected by completely positive trace increasing maps. The latter are realized by means of unitary operators in Fock space extended by 2D vector spac…
In honor of Minkowski's great contribution to Special Relativity, celebrated at this conference, we first review Wigner's theory of the projective irreducible representations of the inhomogeneous Lorentz group. We also sketch those parts of Mackey's mathematical theory on induced representations which are particularly useful for physicists. As an important application of the Wigner-Mackey theory, we shall describe in a unified manner free classical and quantum fields for arbitrary spin, and demonstrate that locality implies the normal spin-statistics connection.
This paper considers the control landscape of quantum transitions in multi-qubit systems driven by unitary transformations with single-qubit interaction terms. The two-qubit case is fully analyzed to reveal the features of the landscape including the nature of the absolute maximum and minimum, the saddle points and the absence of traps. The results permit calculating the Schmidt state starting from an arbitrary two-qubit state following the local gradient flow. The analysis of multi-qubit systems is more challenging, but the generalized Schmidt states may also be located by following the local…
The Lindblad equation governs general markovian evolution of the density operator in an open quantum system. An expression for the rate of change of the Wigner function as a sum of integrals is one of the forms of the Weyl representation for this equation. The semiclassical description of the Wigner function in terms of chords, each with its classically defined amplitude and phase, is thus inserted in the integrals, which leads to an explicit differential equation for the Wigner function. All the Lindblad operators are assumed to be represented by smooth phase space functions corresponding to …
By means of a well-grounded mapping scheme linking Schwinger unitary operators and generators of the special unitary group $\mathrm{SU(N)}$, it is possible to establish a self-consistent theoretical framework for finite-dimensional discrete phase spaces which has the discrete $\mathrm{SU(N)}$ Wigner function as a legitimate by-product. In this paper, we apply these results with the aim of putting forth a detailed study on the discrete $\mathrm{SU(2)} \otimes \mathrm{SU(2)}$ and $\mathrm{SU(4)}$ Wigner functions, in straight connection with experiments involving, among other things, the tomogra…
We give a pedagogical presentation of the irreducible unitary representations of $\mathbb{C}^4\rtimes\mathbf{Spin}(4,\mathbb{C})$, that is, of the universal cover of the complexified Poincaré group $\mathbb{C}^4\rtimes\mathbf{SO}(4,\mathbb{C})$. These representations were first investigated by Roffman in 1967. We provide a modern formulation of his results together with some facts from the general Wigner-Mackey theory which are relevant in this context. Moreover, we discuss different ways to realize these representations and, in the case of a non-zero "complex mass", we give a detailed constru…
Phase space reflection operators lie at the core of the Wigner-Weyl representation of density operators and observables. The role of the corresponding classical reflections is known in the construction of semiclassical approximations to Wigner functions of pure eigenstates and their coarsegrained microcanonical superpositions, which are not restricted to classically integrable systems. In their active role as unitary operators, they generate transitions between pairs of eigenstates specified by transition Wigner functions (or cross-Wigner functions): The square modulus of the transition Wigner…
Operators in quantum mechanics - either observables, density or evolution operators, unitary or not - can be represented by c-numbers in operator bases. The position and momentum bases are in one to one correspondence with lagrangian planes in double phase space, but this is also true for the well known Wigner-Weyl correspondence based on translation and reflection operators. These phase space methods are here extended to the representation of superoperators. We show that the Choi-Jamiolkowsky isomorphism between the dynamical matrix and the linear action of the superoperator constitutes a "do…
Schwinger's finite (D) dimensional periodic Hilbert space representations are studied on the toroidal lattice ${\ee Z}_{D} \times {\ee Z}_{D}$ with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-prese…
Wigner entropy conjecture and the interference formula in quantum phase space
Wyant College of Optical Sciences, The University of Arizona（亚利桑那大学怀特光学科学学院）；DIENS, École Normale Supérieure, PSL University, CNRS, INRIA (QAT)（巴黎高等师范学院，PSL大学，法国国家科学研究中心，INRIA（QAT））
In the report we analyze the indicator/measure of classicality of quantum states defined as the probability to find a state with a positive Wigner function within a unitary invariant random ensemble. The indicators of classicality of three ensembles associated with the Hilbert-Schmidt, Bures and Bogoliubov-Kubo-Mori metrics on the space of quantum states of 3-level system are computed. Their dependence on a moduli parameter of the Wigner function is studied for all strata of a qutrit state space stratified in accordance with the unitary group action.
An explicit construction of all finite-dimensional irreducible representations of the Lie superalgebra gl(1|n) in a Gel'fand-Zetlin basis is given. Particular attention is paid to the so-called star type I representations (``unitary representations''), and to a simple class of representations V(p), with p any positive integer. Then, the notion of Wigner Quantum Oscillators (WQOs) is recalled. In these quantum oscillator models, the unitary representations of gl(1|DN) are physical state spaces of the N-particle D-dimensional oscillator. So far, physical properties of gl(1|DN) WQOs were describe…
A cornerstone of quantum mechanics is the characterisation of symmetries provided by Wigner's theorem. Wigner's theorem establishes that every symmetry of the quantum state space must be either a unitary transformation, or an antiunitary transformation. Here we extend Wigner's theorem from quantum states to quantum evolutions, including both the deterministic evolution associated to the dynamics of closed systems, and the stochastic evolutions associated to the outcomes of quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state spa…
Generation of Wigner functions of Landau levels and determination of their symmetries and generic properties are achieved in the autonomous framework of deformation quantization. Transformation properties of diagonal Wigner functions under space inversion, time reversal and parity transformations are specified and their invariance under a four-parameter subgroup of symplectic transformations are established. A generating function for all Wigner functions is developed and this has been identified as the phase-space coherent state for Landau levels. Integrated forms of generating function are us…
We present the quantum computation of nuclear observables where the operators of interest are first decomposed in terms of the linear combination of unitaries. Then we utilise the Hadamard test and the linear combination of unitaries (LCU) based methods to compute the expectation values. We apply these methods to calculate the electric quadrupole moment of deuteron. The results are compared for the Jordan-Wigner transformation and Gray code encoding. We discuss the versatility of our approach that can be utilized in general to calculate several observables on a quantum computer.
The one-dimensional transverse field Ising model is solved by continuous unitary transformations in the high-field limit. A high accuracy is reached due to the closure of the relevant algebra of operators which we call string operators. The closure is related to the possibility to map the model by Jordan-Wigner transformation to non-interacting fermions. But it is proven without referring to this mapping. The effective model derived by the continuous unitary transformations is used to compute the contributions of one, two, and three elementary excitations to the diagonal dynamic structure fact…
Quantization of a random-walk model is performed by giving a qudit (a multi-component wave function) to a walker at site and by introducing a quantum coin, which is a matrix representation of a unitary transformation. In quantum walks, the qudit of walker is mixed according to the quantum coin at each time step, when the walker hops to other sites. As special cases of the quantum walks driven by high-dimensional quantum coins generally studied by Brun, Carteret, and Ambainis, we study the models obtained by choosing rotation as the unitary transformation, whose matrix representations determine…
The algebra su(2) is derived from two commuting quon algebras for which the parameter q is a root of unity. This leads to a polar decomposition of the shift operators of the group SU(2). The Wigner-Racah algebra of SU(2) is developed in a new basis arising from the simultanenous diagonalization of two commuting operators, viz., the Casimir of SU(2) and a unitary operator which takes its origin in the polar decomposition of the shift operators of SU(2).
We provide an introduction into the formulation of non-relativistic quantum mechanics using the Wigner phase-space distribution function and apply this concept to two physical situations at the interface of quantum theory and general relativity: (i) the motion of an ensemble of cold atoms relevant to tests of the weak equivalence principle, and (ii) the Kasevich-Chu interferometer. In order to lay the foundations for this analysis we first present a representation-free description of the Kasevich-Chu interferometer based on unitary operators.
We pose and resolve a seeming paradox about spontaneous symmetry breaking in the quantum theory of infinite systems. For a symmetry to be spontaneously broken, it must not be implementable by a unitary operator. But Wigner's theorem guarantees that every symmetry is implemented by a unitary operator that preserves transition probabilities between pure states. We show how it is possible for a unitary operator of this sort to connect the folia of unitarily inequivalent representations. This result undermines interpretations of quantum theory that hold unitary equivalence to be necessary for phys…
A symmetry $SU(2,2)$ group in terms of ladder operators is presented for the Jacobi polynomials, $J_{n}^{(α,β)}(x)$, and the Wigner $d_j$-matrices where the spins $j=n+(α+β)/2$ integer and half-integer are considered together. A unitary irreducible representation of $SU(2,2)$ is constructed and subgroups of physical interest are discussed. The Universal Enveloping Algebra of $su(2,2)$ also allows to construct group structures $(SU(1,1), SO(3,2), Spin(3,2))$ whose representations separate integers and half-integers values of the spin $j$. Appropriate $L^2$--functions spaces are realized inside …
Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulati…
The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle and angular momentum is solved. While a key element for the construction of Wigner functions for the planar phase space (q,p) in R^2 is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space (theta,p) in S^1 x R is the Euclidean group E(2) of the plane and its unitary representations. Here the angle theta is replaced by the pair (cos theta, sin theta) which corresponds uniquely to the points on the unit circle. The main structural properties of the Wi…
In many of the approximate functionals in one-body reduced density matrix (1RDM) functional theory, the approximate two-body reduced density matrix (2RDM) in the natural orbital representation only depends on the natural occupation numbers. In Phys. Rev. A 92, 012520 (2015) Wang and Knowles initialised a discussion of to what extent this simplification is valid, by introducing two different H$_4$ geometries with identical natural occupation numbers, but different 2RDMs. Gritsenko has argued that this feature is due symmetry [Phys. Rev. A 97, 026501 (2018)]. This work aims to contribute to the …
The Lie algebra su(2) of the classical group SU(2) is built from two commuting quon algebras for which the deformation parameter is a common root of unity. This construction leads to (i) a not very well-known polar decomposition of the ladder generators of the SU(2) group, in terms of a unitary operator and a Hermitean operator, and (ii) a nonstandard quantization scheme, alternative to the usual quantization scheme correponding to the diagonalization of the Casimir of su(2) and of the z-generator. The representation theory of the SU(2) group can be developed in this nonstandard scheme. The ke…
The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics. We put aside the relativistic …
