In this paper, we formulate the basic Maxwell-Dirac equations for the electromagnetic field of photons interacting with the Dirac field of electrons and positrons taking into account noisy components of the electromagnetic field modeled in the context of the Hudson-Parthasarathy quantum stochastic calculus. These equations are formulated within a waveguide and result in Helmholtz differential equations for the z component of the em fields with Dirac current source. We demonstrate how quantum noise perturbs these fields and how one can calculate the moments of the resulting perturbed quantum fields due to interaction and noise in a coherent state of the photons and Fermions. We also discuss perturbations to these fields produced by quantum gravitational fields using the covariant Maxwell equations of general relativity. Specifically, using elementary perturbation theory, we derive Helmholtz equations for the z components of the perturbed electromagnetic fields with current sources being functions of the unperturbed electromagnetic and Dirac fields and also the gravitational metric perturbations. We indicate how these calculations can be used in quantum guide parameter estimation problems. We also indicate how by expressing the Hamiltonian of the Maxwell-Dirac field in terms of creation and annihilation operators and creation and annihilation quantum stochastic processes for the noise, we can derive the Hudson-Parthasarathy noisy Schrodinger equation for the unitary evolution in second quantized field space and how by measuring non-demolition noise, we can estimate the evolving state of the Maxwell-Dirac field on a real time basis. The paper provides a self-contained discussion of quantum noise in field theory. Some aspects of supersymmetry, string gauge field theory and non-Abelian gauge field theory in the presence of noise are also discussed here. The paper provides a self-contained account of how to treat quantum Bosonic and Fermionic noisy fields in the context of quantum field theory by highlighting the fact that quantum noisy fields have an additional time-component of quantum randomness as compared with that of conventional quantum fields.
This paper explores some of the applications of quantum stochastic calculus (qsc) as founded by R.L.Hudson and K.R.Parthasarathy and the theory of the quantum filter established by V.P.Belavkin based on qsc which was simplified considerably by John Gough, Claude Kostler and others to quantum field theory. Specifically, we first derive some general versions of the Belavkin quantum filter for the case when the Hamiltonian and Lindblad operators in the Hudson-Parthasarathy noisy Schrodinger equation (HPSDE) are functions of a classical Markovian time varying parameter process whose statistics is determined by an infinitesimal generator. Using this generalized Belavkin filter, we show how we can estimate on real time basis not only the evolving mixed state of the system but also the probability density of the Markovian parameter. Our generalized filter enables us to obtain a function
ρ(t,θ) that represents an estimate of the state of the quantum system at time
t as well as the probability density of the parameter
θ at time
t. Specifically, when
X(θ)=∑k=1nfk(θ)Xk is a system observable dependent upon the Markovian
θ, then,
∫Tr(ρ(t,θ)X(θ))dθ would represent the conditional expectation of
X(t,θ(t))=U(t)∗X(θ(t))U(t) given non-demolition measurements at time
tThis paper explores some of the applications of quantum stochastic calculus (qsc) as founded by R.L.Hudson and K.R.Parthasarathy and the theory of the quantum filter established by V.P.Belavkin based on qsc which was simplified considerably by John Gough, Claude Kostler and others to quantum field theory. Specifically, we first derive some general versions of the Belavkin quantum filter for the case when the Hamiltonian and Lindblad operators in the Hudson-Parthasarathy noisy Schrodinger equation (HPSDE) are functions of a classical Markovian time varying parameter process whose statistics is determined by an infinitesimal generator. Using this generalized Belavkin filter, we show how we can estimate on real time basis not only the evolving mixed state of the system but also the probability density of the Markovian parameter. Our generalized filter enables us to obtain a function
ρ(t,θ) that represents an estimate of the state of the quantum system at time
t as well as the probability density of the parameter
θ at time
t. Specifically, when
X(θ)=∑k=1nfk(θ)Xk is a system observable dependent upon the Markovian
θ, then,
∫Tr(ρ(t,θ)X(θ))dθ would represent the conditional expectation of
X(t,θ(t))=U(t)∗X(θ(t))U(t) given non-demolition measurements at time
t. After that, we show how the Weyl-Moyal-Wigner transform can be used to formulate noisy quantum evolutions as well as Belavkin filter equations in a classical language wherein intepretations of quantum observables and states are given in terms of classical complex functions of positions and momenta. Marginals of the Wigner distribution of the state for example represent probabiity densities of positions and momenta but the Wigner distribution of the state does not represent any joint probability density of the positions and momenta, indeed the Heisenberg uncertainty does not allow any such interpretation. We then illustrate how a quantum mechanical theory taking into account quantum noise can be constructed in a natural way by perturbing the electromagnetic field with quantum super-symmetric noise introducing even and odd system operators. We explain how the interaction between the matter supercurrent and the metric superfield in supergravity theories can be perturbed by quantum noise by adding supersymmetric quantum noise to the matter supercurrent. Further, with the hope that Feynman path integrals can be used in quantum noisy field theories to evaluate transition amplitudes of scattering, we propose a Feynnman path integral approach to the solution of the HPQSDE when the system can be described by canonical position and momentum operators. We then describe quantum scattering in the presence of quantum noise in the HPQSDE framework. Here either one or both of the Hamiltonians contains noise and the corresponding joint unitary evolution of system and bath is described by a HPQSDE and we illustrate some methods for computing the scattering wave operators and analyzing their existence using standard quantum scattering inequalities. Finally, we develop some formulae for computing the rate of change of Von-Neumann entropy of the Belavkin filtered quantum state based on combining Lie algebraic methods for evaluating the differential of the matrix exponential map with Ito’s formula for Brownian motion, Poisson and Levy processes and even for Martingales. These formulae that describe rate of entropy increase have potential applications to quantum information and quantum commmunication when we require to measure how much information has been transmitted as a function of time across a quantum channel after filtering out the noise on a real time basis. In the context of quantum communication theory, we illustrate the situation of an optical fibre wherein the information carrying quantum electromagnetic field interacts with a classical electromagnetic field that has been scattered by the phonon lattice and hence contains unknown Markovian parameters. By estimating the Markovian paremeters using our generalized Belvakin filter on a real time basis, we are therefore able to generate a counter Hamiltonian via causing a simulated counter classical field using this estimated parameter to interact with the propagating quantum field and hence mitigate the effect of the interaction of the classical field with the quantum field. One of the novel results of this paper is the exact representation of the Weyl-Moyal-Wigner transform of products of quantum operators that includes Lie brackets between operators to all orders in Planck’s constant. We use this idea to obtain a classical representation of the quantum Boltzmann equation in terms of Wigner densities that are complex valued functions of position and momenta representing the system state. Future work on Wigner transforming the quantum Boltzmann equation for all r-particle marginal states of an N-particle system is in progress. Work on interpreting the Belavkin filter and quantum Boltzmann equation in the Wigner domain in terms of corrections to the classical Fujisaki-Kushner-Kallianpur stochastic filter and the classical Boltzmann equation is also in progress.
The aim of this article is to introduce into quantum field theory,
Zn×Zn graded quantum stochastic calculus with the aim of generalizing supersymmetric quantum stochastic calculus to situations where rather than just two kinds of particles, Bosons and Fermions, we can have particles of
n2 kinds graded by
Zn×Zn rather than by
Z2. Following the suggestions at the end of the book [1] by Timothy Eyre, we introduce the
Zn×Zn graded tensor product by means of a bi-character constructed from a primitive
nth root of unity and then proceed to grade both system and noise operators in Boson Fock space. This enables us to construct consistent Lie algebra supercommutation relations graded by
Zn×Zn for appropriately defined graded quantum stochastic processes. Using the
Zn×Zn graded quantum noise for driving the Hudson-Parthasarathy noisy Schrodinger equation combined with an appropriate non-demolition counting measurement process, we formulate the problem of
Zn×Zn graded quantum filtering as a generalization of the Belavkin filter. After that, we discuss how to construct Bosonic quantum noise out of these graded quantum noise processes by tensoring with appropriately graded system operators and then how to add such noise to Bosonic field theories including quantum gravity. We also discuss quantum noise in string theories and a little bit of string field theories wherein the action for the string field is chosen based on the BRST charge quantization process that defines the condition for a state to be physical in the absence of ghosts. The addition of higher degree terms in the string field action is then based on the condition that the action should satisfy the quantum master equation for invariance of the matrix elements under the choice of the gauge fixing functional. We suggest methods by which quantum noise fields can also be taken into account in string field theories. The entire aim of adding quantum noise to a quantum field theory is motivated by the fact that the resulting noisy Hamiltonian should describe a Hudson-Parthasarathy noisy Schrodinger equation with unitary evolution on the joint System-Bath Hilbert space so that after tracing out over the bath, one obtains a quantum dynamical semigroup of TPCP maps describing system evolution alone for open quantum systems. For noisy quantum field theories, finally, we explain how to compute propagator corrections caused by noise.
A model, based on a noncommutative geometry, unifying general relativity and quantum mechanics, is developed. It is shown that the dynamics in this model can be described in terms of one-parameter groups of random operators, and that the noncommutative counterparts of the concept of state and that of probability measure coincide. We also demonstrate that the equation describing noncommutative dynamics in the quantum mechanical approximation gives the standard unitary evolution of observables, and in the “space–time limit” it leads to the state vector reduction. The cases of the spin and position operators are discussed.
It is known that non-unital noise such as the amplitude damping can sometimes increase quantum correlations, while unital noise such as the dephasing usually decreases quantum correlations. It is, therefore, important to delineate the conditions, when noise can enhance the quantumness of the system. Here, we show that if the noise acting on the system is unital (non-unital), then the noise acting on the environment must also be unital (non-unital), for the evolution to be unitary in the joint system-environment space. For example, if the first two qubits are treated as system and the third qubit is treated as environment, then both the system and the environment evolve unitally in case of a three-qubit GHZ state, and both of them evolve non-unitally in case of a three-qubit W state. Our result may be of interest in quantum information, and we anticipate it to be useful in various contexts, such as to better tackle noise in quantum computing and quantum information processing.
Abstract The dynamics of classical and quantum systems in the presence of noise is usually described in the language of stochastic differential equations. When the system observables comprise a C*-algebra, stochastic evolutions are obtained by solving such equations driven by creation, preservation and annihilation processes on Fock space, with linear maps on the algebra as coefficients. *-Homomorphic evolutions are obtained precisely when the collection of maps has a certain structure; this structure admits a cohomological description. Here we consider equations governing the joint evolution of the system and noise (from input to output) by supposing that the characteristics of the (input) noise processes are given by a group representation. The structure required for *-homomorphic evolution is determined.