Any adequate account of how micro-causality and quantum coherence can explain the emergent-property of spacetime and how the Wheeler-DeWitt problem of time can be solved must incorporate a theory of how the Lindblad master equation solves the Markov quantum fluctuation problem as well as demonstrate how the quantum Jarzynski-Hatano-Sasa relation can be homologically defined globally for both, Minkowski space and Friedmann-Robertson-Walker generalized Cartan space-times. This is a step towards those goals. Consider a wave-function and the entropic quantum entanglement relation of the total system consisting of ‘S’, ‘m’ and the quantum-time measuring clock ‘c’ subject to Heisenberg’s UP. It follows then that the probability that any given initial state  evolves for a time that undergoes  jumps during intervals  centered at times  is given by:

So, the master equation:

is valid iff the Markovian approximation is faithful and valid only on time-scales longer than , hence the jump occurs during an interval  centered on . Therefore, with the Hamiltonian:

where  are the lowering/raising operators for the system and output mode respectively, it follows that the total system satisfies the master equation:

where the Pauli operator  acts on the output mode and  is the Liouville superoperator. Given that it is a linear equation, it has a solution given as:

and so the evolution of the density matrix  is given by the Lindblad master equation:

where

is the conservative part and  is the time-dependent Hamiltonian of the system and the other terms refer to the bath of the interactive system and reflect the effect of measurements, and are the Kraus-operators, not necessarily hermitians and are typically explicitly dependent on time. The Kraus number  depends on the bath. In the case where the system is a closed one, the Kraus operators vanish identically and the Lindblad master equation reduces to the quantum version of the Liouville equation, giving us:

with  the Lindbladian superoperator acting on the density matrix and determines its dynamics. The associated space of operators is equipped with a Hilbert-Schmidt scalar product:

with the hermitian conjugate of . We now define a pair of adjoint superoperators  and  as follows:

Hence, we have:

with the trace-conservation property:

The solve quantum Master equation:

one typically introduces an evolution superoperator defined implicitly by:

where is the initial-time-density-matrix, and the superoperator evolution is given by:

And in this time-ordered exponential, time is monotonically increasing from left to right.

To prove:

note that it is true at since  is the identity operator. Thus, from:

one finds that:

entailing that it satisfies the Lindblad equation:

with initial condition . Now, for the evolution operator, one writes an expression for multi-time correlations for distinct observables. For:

the time-ordered correlation is:

and can be evaluated in the Heisenberg representation formalism by using the full Hamiltonian of the system plus its environment. Since the total density matrix factorizes at each observation time and the weak Lindblad Master equation coupling assumption holds in that formalism, the time-ordered two-time correlation function satisfies an evolution equation which is the dual to:

our proof is complete.

Now note that in:

the operator represents the initial density matrix of the system and the superoperator  acts on all terms to its right.

Thus, we have the crucial equation:

which for systems prepared in a thermal state at:

with the Boltzmann-constant, we have:

For closed systems, one has:

which holds for any Markovian process weakly coupled with a thermal bath at provided the bath satisfies the KMS condition. Here, we shall consider generally, far from equilibrium cases, where  is not given by the canonical Gibbs-Boltzmann formula.

Deriving the Jarzynski-Hatano-Sasa identity for quantum Markovian dynamics

Even though the density-matrix  does not obey the Lindblad equation, it is a solution of the deformation-evolution equation:

Now, let us define non-stationarity via the operator:

Define the modified superoperator as such:

where acts by multiplication on the left. Such a superoperator corresponds to the auxiliary dynamics:

and yields a modified evolution superoperator via:

Given:

we can derive:

with solution:

Now, for any observable , such a solution gives us:

One can derive a quantum variant of the Jarzynski-Hatano-Sasa relation by connecting the auxiliary evolution superoperator  to the initial evolution superoperator . In order to do that, we need to prove an extension of the Feynman-Kac formula: write the Dyson-Schwinger expansion of , with  a perturbation of the Lindbladian :

where acts on all the terms to its right. Now, insert the Dyson-Schwinger expansion into the r.h.s. of:

and we get:

Reformulate the trace within the scope of the integrals as a multi-time correlation via:

and we get:

and by linearity and the relation , we get a reduction to:

where the terms inside the brackets are summable as a time-ordered exponential:

This is an extension of the Feynman-Kac formula for quantum Markov semi-groups.

and by non-commutativity of the operator algebra, the Feynman-Kac exponential is replaced by a time-ordered exponential. Hence, one gets:

and is a quantum extension of the classical Jarzynski-Hatano-Sasa identity

If we set , the above identity reduces to:

given that holds, and is a quantum measurement number-raising and book-keeping formula for correlation functions.

Now, from a first order expansion of:

we can deduce a generalized fluctuation-dissipation theorem valid in the Heisenberg-vicinity of a quantum non-equilibrium steady state.

The case of a closed isolated system determined by a time-dependent Hamiltonian, the Lindbladian reduces to the Liouville operator:

with unitary evolution. For a closed system, the evolution superoperator acts on observables as follows:

with:

The image of X under the superoperator operator defines the Heisenberg operator with representing the Heisenberg operator:

Since the superoperator is multiplicative, the r.h.s. of:

for multi-time correlations can be evaluated and one gets:

Hence, for a closed system the quantum Jarzynski-Hatano-Sasa relation is:

From multiplicativity and:

we have:

and given that we have , we can derive:

Now, from , we get the Kurchan-Tasaki quantum Jarzynski relation for closed systems. Moreover, since we have the commutation relation:

we can derive:

and for the critical case where holds, the Hänggi-Talkner quantum-Jarzynski relation for closed systems reduces to:

For open systems, which are of more foundational interest, it follows from:

that the generalized fluctuation-dissipation theorem is valid in the vicinity of any quantum non-equilibrium steady state. To see that, take a perturbation of the Lindbladian  of the form:

with

time-dependent perturbations. The density matrix satisfying:

is given by:

with  satisfying:

and the non-stationary operator becomes:

By differentiating, we get:

where:

is taken with respect to the unperturbed density matrix .

Time reversal on the states  of the Hilbert space in quantum mechanics is implemented by an anti-linear anti-unitary operator  satisfying:

for spin-0 particles without a magnetic field, is the complex conjugation operator: that is, by time reversal, the Schrödinger wave-function  becomes . In the scenario where there is a magnetic field, time-inversion must be augmented by requiring that the reversed system evolves with vector potential . Time reversal of Hilbert space observables is hence implemented by a superoperator  that acts on an operator  as such:

Hence, as promised, is multiplicative, anti-unitary, and satisfies:

We are finally in a position to define time-reversal for a quantum Markov process. Take a constant Lindbladian lying in a steady state with density-matrix . Note that the superoperator  that determines the reversed process is given by:

and the micro-reversibility condition is:

yielding the finite-time formula:

which, given two arbitrary observables , is equivalent to:

Hence, we can see that the stationary density matrix associated with the time-reversed dynamics is given by:

given that:

We have therefore the following crucial Lindbladian:

From this Lindbladian equation and , we obtain:

and:

thus connecting the Lindbladian distribution of the time-reversed system with that of the classical system.

Now, applying:

to the time-reversed system, we find that the evolution superoperator of the time-reversed system is given by:

with multi-time correlations:

Continuing with our proof, let  be a scalar such that  and define two -deformed superoperators, that act on an observable X as follows:

Def.:

and:

Now, the superoperators  interpolate between  and  when  varies from  to . Likewise,  is an interpolation from  to . The corresponding -deformed evolution superoperators are given by:

and:

Crucially, they satisfy the following duality relation that lies at the heart of the quantum fluctuation theorem:

hence, we can derive the following for the unitary operator:

and satisfies:

So we can write the operator as:

Using the duality above to any pair of observables and , and the multiplicative property and anti-unitary of , our proof is finalized by the following relation:

This is precisely the equation that axiomatically captures the essence of the quantum fluctuation theorem that explains quantum decoherence and undergirds the solution to the measurement problem, up to a Yukawa-Higgs S-matrix coupling constant. To see this, express it in terms of quantum density-matrix stochasticity via the expectation-operator:

with the asymmetric reversed operator given by:

and is closed with Hamiltonian

and the evolution operator satisfies:

by the Boltzmann law, the above equation in the Heisenberg representation is:

whose set of solutions is the set of solutions to the quantum decoherence equation describing a wave-function collapse.

Bonus: