It is well known that the unboundedness of operators in Hilbert space entails domain troubles. It is also well known that most domain troubles can be surmounted by extending the Hilbert space to a rigged Hilbert space. In this note, we point out another of such troubles, namely the correspondence between the Schrodinger and the Heisenberg pictures for unbounded operators, and sketch the solution of this problem within the rigged Hilbert space.
The aim of this short Note is to show that the Schrödinger and Heisenberg pictures of quantum mechanics are not equivalent unless one uses a quantization rule clearly stated by Born and Jordan in their famous 1925 paper. This rule is sufficient and necessary to ensure energy conservation in Heisenberg's matrix mechanics. It follows, in particular, that Schrödinger and Heisenberg mechanics are not equivalent if one quantizes observables using the Weyl prescription.
The annihilation-creation operators of the harmonic oscillator, the basic and most important tools in quantum physics, are generalised to most solvable quantum mechanical systems of single degree of freedom including the so-called `discrete' quantum mechanics. They admit exact Heisenberg operator solution. We present unified definition of the annihilation-creation operators (a^{(\pm)}) as the positive/negative frequency parts of the exact Heisenberg operator solution.
The Heisenberg picture and Schrodinger picture are supposed to be equivalent representations of quantum mechanics. However this idea has been challenged by P.A.M. Dirac. Also, it has been recently shown by A. J. Faria et al that this is not necessarily the case. In this article a simple problem will be worked out in quantum field theory in which the Heisenberg picture and Schrodinger picture give different results.
In this paper, we investigate from the framework of generalized electrodynamics the differential cross section of the electron-electron scattering process $e^- e^- \rightarrow e^- e^-$, i.e., Møller scattering, in $(2+1)$ dimensions in the Heisenberg picture. To this goal, one starts within the stable and unitary framework of planar generalized electrodynamics, instead of Maxwell one. We argue the Haag's theorem strongly suggests the study of the differential cross section in the Heisenberg representation. Afterward, we explore the influence of Podolsky mass cutoff and calculate the differenti…
We present a simple operator perturbation theory in the backward Heisenberg picture. Compared with the well-known Heisenberg picture, the revised picture is based on the backward time instead of the forward time. The unique feature of the uncommon picture is that the perturbation expansion becomes very simple and the famous Dyson expansion is not directly involved. Its relationship with the perturbation expansion for the density operator developed by Kubo is also discussed.
The Schrodinger and Heisenberg pictures are equivalent formulations of quantum mechanics. Here we use the Schrodinger picture to calculate the decoherence that occurs when an optical Schrodinger cat state is passed through a beam splitter. We find that an exponentially large amount of decoherence can occur even when there is a negligible change in the quadrature operator x(t) in the Heisenberg picture. Similar results have been observed in the decoherence produced by an optical amplifier, and we suggest that the usual quadrature operators in the Heisenberg picture do not provide a complete des…
In this paper, we attempt to establish quantum measurement theory in the Heisenberg picture. First, we review foundations of quantum measurement theory, that is usually based on the Schrödinger picture. The concept of instrument is introduced there. Next, we define the concept of system of measurement correlations and that of measuring process. The former is the exact counterpart of instrument in the (generalized) Heisenberg picture. In quantum mechanical systems, we then show a one-to-one correspondence between systems of measurement correlations and measuring processes up to complete equival…
The existence of probability in the sense of the frequency interpretation, i.e. probability as "long term relative frequency," is shown to follow from the dynamics and the interpretational rules of Everett quantum mechanics in the Heisenberg picture. This proof is free of the difficulties encountered in applying to the Everett interpretation previous results regarding relative frequency and probability in quantum mechanics. The ontology of the Everett interpretation in the Heisenberg picture is also discussed.
It is generally assumed that quantum field theory (QFT) is gauge invariant. However it is well known that non-gauge invariant terms appear in various calculations. This problem was recently examined in [9] for a "simple" field theory and it was shown that for this case QFT in the Schroedinger picture is not, in fact, gauge invariant. In order to shed further light on this problem we will examine the Heisenberg and Schroedinger formulations of QFT. It is generally assumed that these two "pictures" are equivalent; however we will show that this is not necessarily the case. We shall consider a si…
It is argued that the Heisenberg picture of standard quantum mechanics does not save Einstein locality as claimed in Deutsch and Hayden (2000). In particular, the EPR-type correlations that DH obtain by comparing two qubits in a local manner are shown to exist before that comparison. In view of this result, the local comparison argument would appear to ineffective in supporting their locality claim.
Quantum regression theorem is a very useful result in open quantum system and extensively used for computing multi-point correlation functions. Traditionally it is derived for two-time correlators in the Markovian limit employing the Schrödinger picture. In this paper we make use of the Heisenberg picture to derive quantum regression theorems for multi-time correlation functions which in the special limit reduce to the well known two-time regression theorem. For multi-time correlation function we find that the regression theorem takes the same form as it takes for two-time correlation function…
The effects of the entanglement of the inflaton field which initially entangled with those of another field on observables like power spectrum are known in the context of the Schrödinger field theory. To clarify this effect in Heisenberg picture, there were some attempts to construct the initial entangled state by making use of an entangled transformation (like Bogoliubov transformation) between the Bunch-Davies vacuums and squeezed states. We study the role of the time-dependent entangled transformation in the Schrödinger field theory. We derive the relation between two vacuum states which th…
For $N$-coupled generalized time-dependent oscillators, primary invariants and a generalized invariant are found in terms of classical solutions. Exact quantum motions satisfying the Heisenberg equation of motion are also found. For number states and coherent states of the generalized invariant, the uncertainties in positions and momenta are obtained.
In the framework of the Heisenberg picture, an alternative derivation of the reduced density matrix of a driven dissipative quantum harmonic oscillator as the prototype of an open quantum system is investigated. The reduced density matrix for different initial states of the combined system is obtained from a general formula, and different limiting cases are studied. Exact expressions for the corresponding characteristic function in quantum thermodynamics and Wigner quasi distribution function are found. A possible generalization based on the Magnus expansion of the evolution operator is presen…
In some recent literature the role of non self-adjoint Hamiltonians, $H\neq H^\dagger$, is often considered in connection with gain-loss systems. The dynamics for these systems is, most of the times, given in terms of a Schrödinger equation. In this paper we rather focus on the Heisenberg-like picture of quantum mechanics, stressing the (few) similarities and the (many) differences with respected to the standard Heisenberg picture for systems driven by self-adjoint Hamiltonians. In particular, the role of the symmetries, *-derivations and integrals of motion is discussed.
In the Heisenberg picture, the generalized invariant and exact quantum motions are found for a time-dependent forced harmonic oscillator. We find the eigenstate and the coherent state of the invariant and show that the dispersions of these quantum states do not depend on the external force. Our formalism is applied to several interesting cases.
We describe the quantum interference of a single photon in the Mach-Zehnder interferometer using the Heisenberg picture. Our purpose is to show that the description is local just like in the case of the classical electromagnetic field, the only difference being that the electric and the magnetic fields are, in the quantum case, operators (quantum observables). We then consider a single-electron Mach-Zehnder interferometer and explain what the appropriate Heisenberg picture treatment is in this case. Interestingly, the parity superselection rule forces us to treat the electron differently to th…
Everett's relative-state construction in quantum theory has never been satisfactorily expressed in the Heisenberg picture. What one might have expected to be a straightforward process was impeded by conceptual and technical problems that we solve here. The result is a construction which, unlike Everett's one in the Schrödinger picture, makes manifest the locality of Everettian multiplicity, and its inherently approximative nature, and its origin in certain kinds of entanglement and locally inaccessible information. Our construction also allows us to give a more precise definition of an Everett…
A fast simulation algorithm for the calculation of multitime correlation functions of open quantum systems is presented. It is demonstrated that any stochastic process which ``unravels'' the quantum Master equation can be used for the calculation of matrix elements of reduced Heisenberg picture operators, and thus for the calculation of multitime correlation functions, by extending the stochastic process to a doubled Hilbert space. The numerical performance of the stochastic simulation algorithm is investigated by means of a standard example.
Using quantum-classical analogies, we find that dynamical pictures of quantum mechanics have precise counterparts in classical mechanics. In particular, the Eulerian and Lagrangian descriptions of fluid dynamics in classical mechanics are the analogs of the Schroedinger and Heisenberg pictures in quantum mechanics, respectively. Similarities between classical and quantum dynamical pictures are explored within the framework of the Koopman-von Neumann formalism. These allow for a natural definition of various dynamical pictures in classical mechanics as well as the application of classical conc…
Divisibility of dynamical maps: Schrödinger vs. Heisenberg picture
University of Turku（图尔库大学）；Università degli Studi di Milano（米兰大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Milano（意大利国家核物理研究所米兰分部）；Nicolaus Copernicus University（尼古拉·哥白尼大学）
Quantum simulation in the Heisenberg picture via vectorization
Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）；Center for Quantum Science and Engineering, EPFL（洛桑联邦理工学院量子科学与工程中心）
A stochastic simulation algorithm for the computation of multitime correlation functions which is based on the quantum state diffusion model of open systems is developed. The crucial point of the proposed scheme is a suitable extension of the quantum master equation to a doubled Hilbert space which is then unraveled by a stochastic differential equation.
In quantum theory it is generally assumed that there exists a special state called the vacuum state and that this state is a lower bound to the energy. However it has recently been demonstrated that this is not necessarily the case for some situations [5]. In order to clarify the situation we will consider a "very simple" field theory in the Heisenberg picture consisting of a quanitized fermion field with zero mass pariticles in 1-1D space-time interacting with a classical electrical potential. It will be shown that for this example there is no lower bound to the energy.
Quantum Zeno effect is conventionally interpreted by the assumption of the wave-packet collapse, in which does not involve the duration of measurement. However, we predict duration $τ_m$ of each measurement will appear in quantum Zeno effect by a dynamical approach. Moreover, there exists a model-free critical measurement time, which quantum Zeno effect does not occur when $τ_m$ takes some special values. In order to give these predictions, we first present a description of quantum Zeno effect in the Heisenberg picture, which is based on the expectation value of an observable and its fluctuati…
In this note, we develop a framework to describe open quantum systems in the Heisenberg picture, i.e., via time evolving operator algebras. We point out the incompleteness of the previous proposals in this regard. We argue that a complete Heisenberg picture for an open quantum system involves multiple image Heisenberg operators for each system observable. For a given system observable, the number of such image operators is equal to the dimension of the environment Hilbert space. We derive a perturbative expression, accurate upto arbitrary orders in the system environment coupling, for these im…
The Heisenberg picture for non-Hermitian but $η$-pseudo-Hermitian Hamiltonian systems is suggested. If a non-Hermitian but $η$-pseudo-Hermitian Hamiltonian leads to real second order equations of motion, though their first order Heisenberg equations of motion are complex, we can construct a Hermitian counterpart that gives the same second order equations of motion. In terms of a similarity transformation we verify the iso-spectral property of the Hermitian and non-Hermitian Hamiltonians and obtain the related eigenfunctions. This feature can be used to determine real eigenvalues for such non-H…
We describe an approximate solution to the Heisenberg operator equations of motion for an atom in a laser field. The solution is based on a quantum generalization of the physical picture given by the well-known Simple Man Model (SMM). We provide justification of the plausibility of this generalization and test its validity by applying it for the calculation of the coordinate and velocity autocorrelation functions which, to our knowledge, have not been studied before in the context of the strong field ionization. Both our model and results of the ab initio numerical calculations show distinct t…
We construct a class of Hamiltonians that describe the photodetection process from beginning to end. Our Hamiltonians describe the creation of a photon, how the photon travels to an absorber (such as a molecule), how the molecule absorbs the photon, and how the molecule after irreversibly changing its configuration triggers an amplification process---at a wavelength that may be very different from the photon's wavelength---thus producing a macroscopic signal. We use a simple prototype Hamiltonian to describe the single-photon detection process analytically in the Heisenberg picture, which neat…
In some cases the state of a quantum system with a large number of subsystems can be approximated efficiently by the density matrix renormalization group, which makes use of redundancies in the description of the state. Here we show that the achievable efficiency can be much better when performing density matrix renormalization group calculations in the Heisenberg picture, as only the observable of interest but not the entire state is considered. In some non-trivial cases, this approach can even be exact for finite bond dimensions.
Bell's theorem depends crucially on counterfactual reasoning, and is mistakenly interpreted as ruling out a local explanation for the correlations which can be observed between the results of measurements performed on spatially-separated quantum systems. But in fact the Everett interpretation of quantum mechanics, in the Heisenberg picture, provides an alternative local explanation for such correlations. Measurement-type interactions lead, not to many worlds but, rather, to many local copies of experimental systems and the observers who measure their properties. Transformations of the Heisenbe…
We show that the theory of operator quantum error correction can be naturally generalized by allowing constraints not only on states but also on observables. The resulting theory describes the correction of algebras of observables (and may therefore suitably be called ``operator algebra quantum error correction''). In particular, the approach provides a framework for the correction of hybrid quantum-classical information and it does not require the state to be entirely in one of the corresponding subspaces or subsystems. We discuss applications to quantum teleportation and to the study of info…
Quantum cellular automata (QCA) are discrete models of space and time homogeneous quantum field theories (QFTs) and regarded as natural candidates for quantum simulation. Description of a QCA over the separable Hilbert space of finite, unbounded configurations (UFC Hilbert space) with unitary state evolution is the {\it Schrödinger template}, and over the incomplete infinite tensor product algebra (ITPA) with evolution by algebra automorphism is the {\it Heisenberg template}. Whether every Heisenberg template admits an equivalent Schrödinger template is a foundational question, and one that ha…
In has been recently shown [1] that in Dirac's hole theory the vacuum state is not the minimum energy state but that there exist quantum states with less energy than that of the vacuum state. In this paper we extend this discussion to quantum field theory (QFT) and consider the question of whether or not the vacuum in QFT is the state of minimum energy. It will be shown that for a "simple" field theory, consisting of a quantized fermion field interacting with a classical electric field in 1-1D space-time, there exist quauntum states with less energy than that of the vacuum state.
A new non-perturbative method of solution of the nonlinear Heisenberg equations in the finite-dimensional subspace is illustrated. The method, being a counterpart of the traditional Schrodinger picture method, is based on a finite operator expansion into the elementary processes. It provides us with the insight into the nonlinear quantal interaction from the different point of view. Thus one can investigate the nonlinear system in both pictures of quantum mechanics.
In 2008 Coecke and Duncan proposed the graphical ZX-calculus rewrite system which came to formalize reasoning with quantum circuits, measurements and quantum states. The ZX-calculus is sound for qubit quantum mechanics. Hence, equality of diagrams under ZX-equivalent transformations lifts to an equality of corresponding equations over matrices. Conversely, in 2014 Backens proved completeness, establishing that any derivation done in stabilizer quantum mechanics with matrices can be derived graphically using the ZX-calculus. A graphical rewrite system that is both confluent and also terminates …
We address the framework of analysing quantum metrology in the information-theoretic picture. Firstly we show how to extract the maximum amount of information in general via suitable state initialization of the probes at the beginning and a quantum measurement at the end. Our analysis can apply to both the single-parameter and the multi-parameter estimation procedures as well as to any other quantum information processing procedures. We then establish a direct connection between the information-theoretic picture of quantum metrology and its conventional variance-covariance picture, by showing …
Schrödinger and Heisenberg non-Markovianity in quantum information tasks
University of Turku（图尔库大学）；Università degli Studi di Milano（米兰大学）；Istituto Nazionale di Fisica Nucleare（意大利国家核物理研究所）；Nicolaus Copernicus University（尼古拉·哥白尼大学）
The fact that certain "extraordinary" probabilistic phenomena--in particular, macroscopic violations of the second law of thermodynamics--have never been observed to occur can be accounted for by taking hard preclusion as a basic physical law; i.e. precluding from existence events corresponding to very small but nonzero values of quantum-mechanical weight. This approach is not consistent with the usual ontology of the Everett interpretation, in which outcomes correspond to branches of the state vector, but can be successfully implemented using a Heisenberg-picture-based ontology in which outco…
The numerical simulation of quantum many-body dynamics is typically limited by the linear growth of entanglement with time. Recently numerical studies have shown, however, that for 1D Bethe-integrable models the simulation of local operators in the Heisenberg picture can be efficient as the corresponding operator-space entanglement grows only logarithmically. Using the spin-1/2 XX chain as generic example of an integrabel model that can be mapped to free particles, we here provide a simple explanation for this. We show furthermore that the same reduction of complexity applies to operators that…
We study solutions to the quantum trajectory evolution of $N$-mode open quantum systems possessing a time-independent Hamiltonian, linear Heisenberg-picture dynamics, and Gaussian measurement noise. In terms of the mode annihilation and creation operators, a system will have linear Heisenberg-picture dynamics under two conditions. First, the Hamiltonian must be quadratic. Second, the Lindblad operators describing the coupling to the environment (including those corresponding to the measurement) must be linear. In cases where we can solve the $2N$-degree polynomials that arise in our calculatio…
Stochastic unravelings for Heisenberg picture and trace-nonpreserving dynamics
University of Turku（图尔库大学）；Nicolaus Copernicus University（哥白尼大学）；Università degli Studi di Milano（米兰大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Milano（意大利国家核物理研究所米兰分部）
A background-independent quantization of the Universe near its Big Bang singularity is considered using a drastically simplified toy model. Several conceptual issues are addressed. (1) The observable spatial-geometry characteristics of our empty-space expanding Universe is sampled by the time-dependent operator $Q=Q(t)$ of the distance between two space-attached observers (``Alice and Bob''). (2) For any pre-selected guess of the simple, non-covariant time-dependent observable $Q(t)$ one of the Kato's exceptional points (viz., $t=τ_{(EP)}$) is postulated {\em real-valued}. This enables us to t…
The canonical quantum theory of a free field using arbitrary foliations of a flat two-dimensional spacetime is investigated. It is shown that dynamical evolution along arbitrary spacelike foliations is unitarily implemented on the same Fock space as that associated with inertial foliations. It follows that the Schrodinger picture exists for arbitrary foliations as a unitary image of the Heisenberg picture for the theory. An explicit construction of the Schrodinger picture image of the Heisenberg Fock space states is provided. The results presented here can be interpreted in terms of a Dirac co…
In recent work Hartmann et al [Phys. Rev. Lett. 102, 057202 (2009)] demonstrated that the classical simulation of the dynamics of open 1D quantum systems with matrix product algorithms can often be dramatically improved by performing time evolution in the Heisenberg picture. For a closed system this was exemplified by an exact matrix product operator solution of the time-evolved creation operator of a quadratic fermi chain with a matrix dimension of just two. In this work we show that this exact solution can be significantly generalized to include the case of an open quadratic fermi chain subj…
The vibrational motion equations of both homo and hetero-nuclei diatomic molecules are here derived for the first time. A diatomic molecule is first considered as a one dimensional quantum mechanics oscillator. The second and third-order Hamiltonian operators are then formed by substituting the number operator for the quantum number in the corresponding vibrational energy eigenvalues. The expectation values of relative position and linear momentum operators of two oscillating atoms are calculated by solving Heisenbergs equations of motion. Subsequently, the expectation values of potential and …
A general theory of quantum stochastic processes was formulated by Accardi, Frigerio and Lewis in 1982 within the operator-algebraic framework of quantum probability theory, as a non-commutative extension of the Kolmogorovian classical stochastic processes. More recently, studies on non-Markovian quantum processes have led to the discrete-time process tensor formalism in the Schrödinger picture to describe the outcomes of sequential interventions on open quantum systems. However, there has been no treatment of the relationship of the process tensor formalism to the quantum probabilistic theory…
