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The Series Product for Gaussian Quantum Input Processes

John E. Gough, Matthew R. James

arXiv:1602.01991v1quant-ph

TL;DR

The paper addresses how to connect quantum Markov components when the input fields are Gaussian rather than vacuum, without making the physical rules depend on a chosen representation. It uses fictitious vacuum fields and generalized Wick calculations, then introduces Wick-Stratonovich equations to recover covariance-independent dynamics. The resulting theory provides consistent, representation-free series and feedback-network connections for general Gaussian inputs.

  • Problem

    Connection rules for Gaussian quantum inputs can appear to depend on covariance parameters because mathematically convenient vacuum representations may be unitarily inequivalent.

    Source: §I · p. 1

  • Method

    The paper combines generalized Wick calculations in vacuum-field representations with a Wick-Stratonovich formulation of the quantum stochastic equations.

    Source: §I · pp. 3–4

  • Results

    The connection rules and dynamical equations can be expressed independently of the Gaussian input state, yielding a consistent theory for quantum systems in series.

    Source: §I · p. 4, §VIII · p. 21

  • Takeaways & Limitations

    Quantum feedback-network theory extends to thermal, squeezed, and general Gaussian inputs while preserving universal, representation-free interconnection rules.

    Source: §I · p. 1, §VIII · p. 21

Abstract

from arXiv · show

We present the theory for connecting quantum Markov components into a network with quantum input processes in a Gaussian state (including thermal and squeezed), not necessarily vacuum fields.One would expect on physical grounds that the connection rules should be independent of the state of the input to the network. To compute statistical properties, we use a version of Wicks' Theorem involving fictitious vacuum fields (Fock space based representation of the fields) and while this aids computation, and gives a rigorous formulation, the various representations need not be unitarily equivalent. In particular, a naive application of the connection rules would lead to the wrong answer. We establish the correct interconnection rules, and show that while the quantum stochastic differential equations of motion display explicitly the covariances (thermal and squeezing parameters) of the Gaussian input fields We introduce the Wick-Stratonovich form which leads to a way of writing these equations that does not depend on these covariances and so corresponds to the universal equations written in terms of formal quantum input processes. We show that a wholly consistent theory of quantum open systems in series can be developed in this way, and as required physically, is universal and in particular representation-free.

I. INTRODUCTION

The paper develops state-independent connection rules for quantum networks driven by Gaussian input fields, addressing the mismatch between physical universality and representation-dependent vacuum-field calculations. It introduces Wick-Stratonovich equations to recover representation-free dynamics and consistent series connections.

Scope

  • The theory extends quantum feedback-network interconnections beyond vacuum fields while retaining physically state-independent series and scattering rules.The development builds on earlier series-product and quantum-feedback-network work, including cascading, feedback, beam splitting, and general scattering,.Source: §I · pp. 1, 4

Gaussian representations

  • The framework covers thermal, squeezed, and more general Gaussian input fields, including non-Fock representations realized through Bogoliubov transformations of vacuum fields.A collection of d Gaussian fields can be represented using 2d vacuum fields, but the resulting representations may be unitarily inequivalent.Source: §I · pp. 1, 3

Motivation

  • Gaussian quantum networks require connection rules that remain intrinsic to the network rather than depending on the noise-field state, paralleling impedance-based circuit rules.Naive application of rules to vacuum representations can introduce spurious dependence on Gaussian covariance parameters.Source: §I · p. 2, Figure 1 · p. 2

Gaussian stochastic calculus

  • Wick’s Theorem is generalized to compute chronological n-point functions of Gaussian fields and derive reduced dynamics from the input processes.The resulting stochastic equations contain damping and Lindblad terms determined by the Gaussian correlations.Source: §I · pp. 2–3

Representation issue

  • Vacuum-field representations reproduce both internal dynamics and input-output relations, but their doubled inputs must be restricted to the combinations representing the original thermal field.Without this constraint, the representation has a larger algebra and appears to have more inputs and outputs than the physical model.Source: §I · pp. 3–4

Wick-Stratonovich formulation

  • The paper introduces the Wick-Stratonovich form, in which noncommutative Stratonovich ordering yields equations independent of the Gaussian covariance parameters.This form is presented as the representation-free counterpart of equations written using formal quantum input processes.Source: §I · p. 4

A. Statement of the Problem

The section shows why Gaussian-state series connections cannot be obtained by naively applying vacuum-field rules, and motivates Wick-ordered Stratonovich equations as a state-independent alternative.

A. Statement of the Problem

  • The series product combines two systems in series through coupling and Hamiltonian parameters, extending quantum feedback-network interconnections.The equivalent model uses coupling LA + LB and a Hamiltonian correction involving L∗BLA.Source: A. Statement of the Problem · p. 4
  • Applying vacuum representations to two systems driven by the same thermal field introduces an unphysical n-dependent Hamiltonian term.The coupling rule remains LA + LB, but the Hamiltonian acquires an additional term proportional to the thermal parameter n.Source: A. Statement of the Problem · p. 5
  • The naive construction fails because Bogoliubov representations mix creation and annihilation operators, breaking the Wick ordering required by the original equations.A complete description would also include the b′(t) fields and corresponding Bogoliubov boxes.Source: A. Statement of the Problem · p. 5

B. Motivating Example

  • In the motivating feedback example, direct Itō substitutions are ambiguous and inconsistent for thermal inputs because field increments need not commute with system operators.The vacuum case avoids the ambiguity through the series product, whereas thermal equations contain the field parameter n.Source: B. Motivating Example · pp. 5–6
  • Recasting the dynamics in Wick-ordered Stratonovich form removes explicit dependence on field parameters and recovers the same intrinsic feedback parameters as the vacuum case.The two equivalent procedures are direct substitution into Wick-ordered Stratonovich equations or use of the intrinsic series product.Source: B. Motivating Example · p. 6
  • The models initially omit scattering, and the paper states that subsequent text will examine the extent to which scattering can be included.Source: B. Motivating Example · p. 6

II. GAUSSIAN NOISE WITH A FLAT SPECTRUM

This section defines Gaussian oscillator states and Bogoliubov transformations, then characterizes how covariances transform and when vacuum transformations can be unitarily implemented.

A. Notation

B. Finite Dimensional Gaussian States

  • Gaussian oscillator states are described by covariance matrices involving N and M, subject to positivity, symmetry, and range constraints.N is Hermitian, M is symmetric, and the covariance conditions ensure mathematically valid variances.Source: B. Finite Dimensional Gaussian States · p. 7
  • Bogoliubov transformations map the covariance matrix through the doubled-up transformation, producing transformed parameters N′ and M′.The transformed covariance is expressed using W = ∆(U,V).Source: B. Finite Dimensional Gaussian States · p. 7
  • Vacuum modes can generate Gaussian covariances through Bogoliubov transformations, but a single target mode generally requires two vacuum modes.The determinant of the covariance matrix is preserved, so the target covariance is typically a sub-block of a larger transformed covariance.Source: B. Finite Dimensional Gaussian States · pp. 7–8
  • Finite-dimensional Bogoliubov transformations admit unitary implementations under the finite-dimensional version of Shale’s theorem.The construction includes a unitary R implementing the transformation for finite-dimensional invertible Hermitian matrices.Source: B. Finite Dimensional Gaussian States · p. 8

C. Gaussian Noise with Flat Power Spectrum

This section specifies flat-spectrum Gaussian quantum noise, including vacuum and thermal cases, and introduces the Fock-space vacuum representation used for field processes.

C. Gaussian Noise with Flat Power Spectrum

  • Flat-spectrum Gaussian noise is characterized by delta-correlated covariances and means βk(t) for the input processes bk(t).The covariance structure is specified by the matrices N and M.Source: C. Gaussian Noise with Flat Power… · p. 8
  • Quadrature processes have well-defined Gaussian distributions, but the number operator can fail to have a well-defined distribution because δ(0) is undefined.This marks a distributional limitation of the flat-spectrum idealization.Source: C. Gaussian Noise with Flat Power… · p. 8
  • Vacuum noise has N = M = 0, while thermal noise has N = nI with n > 0 and M = 0.The associated covariance matrices inherit the positivity properties established for finite-dimensional Gaussian states.Source: C. Gaussian Noise with Flat Power… · p. 8

1. The Vacuum Representation.

  • The Fock space is built from symmetrized tensor powers, with the n = 0 sector generated by the vacuum vector Ω.Creation and annihilation fields, exponential vectors, and Weyl unitaries provide the basic representation tools.Source: §1 · pp. 8–9
  • The vacuum representation realizes the noise processes on Fock space with h = C^n ⊗ L2(R), and vacuum expectations are taken in Ω.This gives the reference Fock representation for the subsequent non-vacuum constructions.Source: §1 · p. 9

2. Non-vacuum State Representations

The paper represents non-vacuum Gaussian fields on doubled Fock spaces and analyzes when Bogoliubov transformations are unitarily implementable, emphasizing that thermal representations can be inequivalent to vacuum ones.

2. Non-vacuum State Representations

  • Thermal fields admit an Araki–Woods representation on a double Fock space whose joint vacuum reproduces thermal expectations.The construction uses two commuting representations on the same doubled space.Source: §2 · p. 9
  • The same doubled-space framework extends to squeezed states satisfying |m|2 < n(n + 1).The squeezed representation is obtained by choosing the joint Fock vacuum in the doubled space.Source: §2 · p. 9

3. Bogoliubov Transformations

  • For infinite-dimensional systems, unitary implementation of Bogoliubov transformations is not automatic and is governed by Hilbert–Schmidt conditions.The transformation is implementable exactly when both C2 − I and C−2 − I are Hilbert–Schmidt.Source: §3 · p. 10
  • The thermal transformation b(t) = √(n + 1)a1(t) + √n a2(t)∗ is not unitarily implementable, so the thermal representation is not unitarily equivalent to vacuum.This inequivalence is central to why vacuum connection rules cannot be applied naively.Source: §3 · p. 10

III. DYNAMICS DRIVEN BY QUANTUM INPUT PROCESSES

The section develops reduced dynamics and Wick-ordering tools for quantum input processes, then formulates vacuum and thermal noise models within quantum stochastic calculus.

A. Reduced Dynamics

  • Quantum input dynamics are reduced by conditional expectations, whose ordered Gaussian field averages require Wick’s Theorem.The resulting expressions involve system operators L, −L∗, and H multiplied by chronological n-point functions.Source: A. Reduced Dynamics · p. 10

B. The Vacuum Wick’s Theorem for Quantum Noise

  • Vacuum Wick ordering yields reduced Heisenberg dynamics and extends to multiple fields and scattering terms, although scattering is excluded from the subsequent dynamics.The stated limitation concerns the inclusion of scattering terms in the dynamics.Source: B. The Vacuum Wick’s Theorem for… · p. 10

C. A Thermal Wick’s Theorem for Quantum Noise

  • Thermal fields can be represented using two independent vacuum inputs, with coupling operators √(n+1)L and √nL∗, enabling thermal expectations to be computed in a double-vacuum representation.The thermal correlations are ⟨b(t)∗b(s)⟩=nδ(t−s) and ⟨b(t)b(s)∗⟩=(n+1)δ(t−s).Source: C. A Thermal Wick’s Theorem for… · p. 11

IV. QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS (QSDES)

A. Quantum Ito Calculus: Vacuum Noise

  • For scalar static beam-splitter matrices, the Heisenberg–Langevin equations do not involve scattering processes.This follows from unitarity and scalar entries of the scattering matrix.Source: A. Quantum Ito Calculus: Vacuum Noise · p. 13

B. The Series Product - Vacuum Inputs

The vacuum series product combines connected SLH components through transformed scattering, coupling, and Hamiltonian parameters, while Gaussian representations require Wick-ordered treatment rather than naive vacuum composition.

B. The Series Product - Vacuum Inputs

C. Quantum Ito Calculus: Gaussian Noise

  • For Gaussian inputs, direct use of vacuum series rules is invalid because double-Fock representations can involve additional Bogoliubov fields and are not generally unitarily equivalent.The paper identifies non-unitary equivalence of representations as the underlying mathematical problem.Source: C. Quantum Ito Calculus: Gaussian Noise · p. 13
  • The Gaussian theory instead uses Wick-ordered equations: these generalize Wick’s Theorem for Gaussian stochastic diffusion and identify the ordering compatible with the double-Fock Hudson–Parthasarathy QSDE.The paper distinguishes the correct Wick-ordered form from the expression obtained by direct substitution.Source: C. Quantum Ito Calculus: Gaussian Noise · p. 14

D. Comments

The paper distinguishes state-dependent QSDE representations from state-independent dynamics, using Wick-ordered Stratonovich forms to obtain representation-free equations. It also shows that a degenerate parametric amplifier can approximate thermal white noise in a strong-coupling limit.

D. Comments

  • The same underlying unitary dynamical evolution Ut can be expressed in representations that are not generally unitarily equivalent, complicating the corresponding QSDE descriptions.Source: §V · p. 15

V. REPRESENTATION-FREE FORM

  • The representation-free construction is established by defining Stratonovich differentials and proving equivalence with the non-Fock QSDEs for both unitary and Heisenberg evolutions.The proof uses the quantum Itō table and identities connecting the Stratonovich and Itō forms.Source: §V · p. 15
  • The unitary and Heisenberg QSDEs can be rewritten in Wick-ordered Stratonovich form without dependence on Gaussian noise parameters N and M.The resulting equations are structurally analogous to vacuum QSDEs while remaining valid across representations.Source: §V · p. 15
  • The output process associated with the representation-free Heisenberg form is not a canonical quantum noise, despite arising from the same underlying unitary evolution.The output is related to the input by d ˜Aout(t) = d ˜Ain(t) + jt(L) dt.Source: §V · p. 15

VI. APPROXIMATE SIGNAL GENERATOR FOR THERMAL STATES

  • In the strong-coupling limit, a degenerate parametric amplifier driven by vacuum fields produces an effective thermal white noise with parameter n.Its output satisfies d ˜A(t) d ˜A∗(t) = (n + 1) dt and d ˜A∗(t) d ˜A(t) = n dt.Source: A. The Thermal White Noise as… · p. 16

A. The Thermal White Noise as Idealization of the Output of a Degenerate Parametric Amplifier

  • The thermal-noise approximation is obtained from finite-dimensional cavity dynamics by taking the scaling parameter k large and deriving static input-output relations.The construction uses two vacuum input processes and yields a Bogoliubov transformation.Source: A. The Thermal White Noise as… · pp. 16–17

B. Cascade Approximation

The cascade construction realizes a system driven by thermal input as the strong-coupling limit of a vacuum-driven degenerate parametric amplifier cascaded with that system. Wick reordering converts the resulting equations into the expected thermal-noise dynamics.

B. Cascade Approximation

  • The cascade is formed by feeding the DPA output aout1(t) into G while allowing aout2(t) to propagate independently through a trivial component.The series product then describes the DPA and system as a combined network.Source: B. Cascade Approximation · p. 17
  • In the strong-coupling limit, the DPA output provides the effective Gaussian input used by the cascaded system.For static scattering S ≡ eiθ, the limiting Heisenberg equation simplifies to the thermal-noise form.Source: Figure 10 · p. 17, B. Cascade Approximation · p. 18
  • Wick reordering is required because the auxiliary vacuum operators a2(t) and a2∗(t) initially occur outside Wick order.Reordering generates the commutator terms needed to identify the resulting equation with the thermal QSDE.Source: B. Cascade Approximation · p. 18
  • The cascaded DPA-system model yields the usual input-output behavior for a system G ∼ (S, L, H) driven by thermal input.The thermal input is Ain(t) = √n + 1 A1(t) + √n A2(t)∗, while the second output is left unhindered.Source: B. Cascade Approximation · pp. 17–18

VII. THE GENERAL SERIES PRODUCT

The general series product is recovered for Gaussian inputs without scattering and extended to static beam-splitter scattering through representation-free Heisenberg dynamics.

A. Without Scattering

  • Without scattering, two systems driven in series have coupling L = L_A + L_B and Hamiltonian H = H_A + H_B + Im{L_B^*L_A}, matching the intrinsic series product.The derivation does not assume that the SLH operators commute.Source: A. Without Scattering · pp. 18–19

B. Including Scattering

  • The non-Fock theory cannot directly define scattering processes Λ_jk, but static beam-splitter effects can be incorporated through Heisenberg flows and input-output relations.This avoids explicit unitary QSDE models involving the scattering processes.Source: B. Including Scattering · pp. 19–20
  • For a static beam-splitter matrix S, models (S, L, H) and (I, S^*L, H) generate the same Heisenberg dynamics, with input-output relations related by S.The scattering terms disappear from the Heisenberg QSDE, while the output is rotated by the static beam-splitter matrix.Source: B. Including Scattering · p. 19
  • The Wick-Stratonovich formulation expresses the resulting equations in a representation-free form for mean-zero Gaussian inputs.This provides the universal QSDE description used for the general Gaussian-input series product.Source: B. Including Scattering · p. 20
  • With static scattering, the series connection of (S_A, L_A, H_A) and (S_B, L_B, H_B) retains the Fock-case series-product form.The resulting output includes the combined scattering matrix S_BS_A and coupling contribution S_BL_A + L_B.Source: B. Including Scattering · p. 20

VIII. CONCLUSIONS

The paper establishes a consistent series-product theory for general Gaussian input processes and shows that its connection rules are independent of the input state.

VIII. CONCLUSIONS

  • The theory consistently connects quantum input-output models in series for general Gaussian states with flat power spectra.The Fock-space representation is a mathematical device; Wick-Stratonovich equations remove the apparent dependence on Gaussian state parameters.Source: §VIII · p. 21
  • The connection rules are independent of the input state, and static beam-splitter components can be included despite non-Fock scattering processes being undefined.The resulting series product agrees with the rule established for the Fock case and extends quantum feedback networks to general Gaussian inputs.Source: §VIII · p. 21

Appendix A: Wick Ordering

The appendices develop Wick-ordering manipulations for Gaussian input processes and extend them to equations containing scattering terms.

Appendix A: Wick Ordering

  • The Wick-ordering derivation analyzes commutators between the input fields and the unitary evolution, using adaptedness for earlier times.The calculation also accounts for the half-delta contribution at the integration boundary.Source: Appendix A · p. 21

Appendix B: Including Scattering

  • The scattering appendix applies the Wick-ordering procedure to Schrödinger and Heisenberg equations containing reservoir creation and annihilation terms.The displayed derivation introduces system operators and scattering-related coefficients before rewriting the equations in Wick-ordered form.Source: Appendix B · pp. 21–22
  • The cited derivation rewrites an expression, re-expresses it, and then Wick orders the Schrödinger equation, without establishing claims about static scattering or a Gaussian-input framework.Source: Appendix B · p. 21
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