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 information flows in quantum interactions.
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 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…
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…
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）
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…
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…
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 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…
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 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）
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…
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…
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…
In many-body quantum systems, the quantum Fisher information an observer can obtain is susceptible to decoherence. Consequently, quantum enhanced metrology, such as Heisenberg scaling, cannot usually be achieved. We show, via two distinct methods, that by applying periodic quantum error corrections, we can achieve the Heisenberg scaling for an extended period of time on a three-qubit Tavis-Cumming Model, where three two-level atoms interact with a single cavity mode, under certain approximations. The generalization to arbitrary number of atoms case is also discussed.
In quantum theory, physical systems are usually assumed to evolve relative to a c-number time. This c-number time is unphysical and has turned out to be unnecessary for explaining dynamics: in the timeless approach to quantum theory developed by Page & Wootters (1983), subsystems of a stationary universe can instead evolve relative to a 'clock', which is a quantum system with a q-number time observable. Page & Wootters formulated their construction in the Schrödinger picture and left open the possibility that the c-number time still plays an explanatory role in the Heisenberg picture. I formul…
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…
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…
Quantum metrology has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on measurement precision, called the Heisenberg limit, which can be achieved for noiseless quantum systems, but is not achievable in general for systems subject to noise. Here we study how measurement precision can be enhanced through quantum error correction, a general method for protecting a quantum system from the damaging effects of noise. We find a necessary and sufficient condition for achieving the…
Revivals of Bell nonlocality require Schrödinger and Heisenberg non-Markovianity
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）
