The Lindblad Master Equation, Feynman-Kac Formula, and the Measurement Problem

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 \left| {{\psi _t}^{S,m,c}} \right\rangle 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 \left| {\psi _t^{S,m,c}} \right\rangle evolves for a time T that undergoes N jumps during intervals \delta t centered at times {t_1},...,{t_N} is given by:

    \[\begin{array}{l}{\left( {2\delta t{\kappa ^2}/G} \right)^N}{\rm{Tr}}\left\{ {{e^{ - i{{\tilde H}_{eff}}\left( {T - {t_N}} \right)}}} \right. \cdot \\\hat a{e^{ - i{{\hat H}_{eff}}}}\left( {{t_N} - {t_{N - 1}}} \right)\hat a...\,\hat a{e^{ - i{{\hat H}_{eff}}t}}\\ \times \left| {\psi _t^{S,m,c}} \right\rangle \left\langle {\psi _t^{S,m,c}} \right|{e^{i{{\tilde H}^\dagger }_{eff}{t_1}}}{{\hat a}^\dagger }...\,\left. {{{\hat a}^\dagger }{e^{i{{\tilde H}^\dagger }_{eff}\left( {T - {t_N}} \right)}}} \right\}\end{array}\]

So, the master equation:

    \[\begin{array}{l}{{\dot \rho }_{00}} = - i\left[ {{{\hat H}_0},{\rho _{00}}} \right] + \frac{{2{\kappa ^2}}}{G}\hat a{\rho _{00}}{{\hat a}^\dagger }\\ - \frac{{{\kappa ^2}}}{G}{{\hat a}^\dagger }\hat a{\rho _{00}} - \frac{{{\kappa ^2}}}{G}{\rho _{00}}{{\hat a}^\dagger }\hat a\end{array}\]

is valid iff the Markovian approximation is faithful and valid only on time-scales longer than 1/{\Gamma _1}, hence the jump occurs during an interval \delta t \sim 1/{\Gamma _1} centered on {t_i}. Therefore, with the Hamiltonian:

    \[{\hat H_I} = \kappa \left( {{{\hat a}^\dagger } \otimes \hat b + \hat a \otimes {{\hat b}^\dagger }} \right)\]

where \left( {\hat a,\hat b} \right);\left( {{{\hat a}^\dagger },{{\hat b}^\dagger }} \right) are the lowering/raising operators for the system and output mode respectively, it follows that the total system satisfies the master equation:

    \[\begin{array}{c}\dot \rho = - i\left[ {\hat H,\rho } \right] + {\Gamma _1}\hat b\rho {{\hat b}^\dagger } - \frac{{{\Gamma _1}}}{2}{{\hat b}^\dagger }\hat b\rho \\ - \frac{{{\Gamma _1}}}{2}\rho {{\hat b}^\dagger }\hat b + {\Gamma _2}{\sigma _z}\rho {\sigma _z} - {\Gamma _2}\rho \\ \equiv L_s^L\rho \end{array}\]

where the Pauli operator {\sigma _z} acts on the output mode and L_s^L is the Liouville superoperator. Given that it is a linear equation, it has a solution given as:

    \[\rho ({t_2}) = \exp \left\{ {L_s^L\left( {{t_2} - {t_1}} \right)} \right\}\rho ({t_1})\]

and so the evolution of the density matrix {\rho _t} is given by the Lindblad master equation:

    \[\begin{array}{l}{\partial _t}{\rho _t} = - i\left[ {{H_t},{\rho _t}} \right] + \sum\limits_{i = 1}^I {\left( {{V_i}{\rho _t}V_i^\dagger } \right.} \\\left. { - \frac{1}{2}V_i^\dagger {V_i}{\rho _t} - \frac{1}{2}{\rho _t}V_i^\dagger {V_i}} \right)\end{array}\]

where

    \[ - \left[ {{H_t},X} \right]\]

is the conservative part and {{H_t}} 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 {V_i} are the Kraus-operators, not necessarily hermitians and are typically explicitly dependent on time. The Kraus number I 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:

    \[{\partial _t}{\rho _t} = L_t^\dagger {\rho _t}\]

with L_t^\dagger 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:

    \[\left( {Y,X} \right) = {\rm{Tr}}\left( {{Y^\dagger }X} \right)\]

with {{Y^\dagger }} the hermitian conjugate of Y. We now define a pair of adjoint superoperators {L^\dagger } and L_t^\dagger as follows:

    \[\begin{array}{l}\left( {Y,L,X} \right) = {\rm{Tr}}\left( {{Y^\dagger }\left( {L,X} \right)} \right) = \\\left( {L_t^\dagger Y,X} \right) = {\rm{Tr}}\left( {{{\left( {L_t^\dagger Y} \right)}^\dagger }X} \right)\end{array}\]

Hence, we have:

    \[\begin{array}{l}{L_t}X = i\left[ {{H_t},X} \right] + \sum\limits_{i = 1}^I {\left( {V_i^\dagger } \right.} X{V_i} - \\\frac{1}{2}V_i^\dagger {V_i}X - \left. {\frac{1}{2}XV_i^\dagger {V_i}} \right)\end{array}\]

with the trace-conservation property:

    \[\left\{ {\begin{array}{*{20}{c}}{{L_t}1 = 0}\\{{L_t}\left( {{X^\dagger }} \right) = {{\left( {{L_t}X} \right)}^\dagger }}\end{array}} \right.\]

The solve quantum Master equation:

    \[{\partial _t}{\rho _t} = L_t^\dagger {\rho _t}\]

one typically introduces an evolution superoperator P_0^t defined implicitly by:

    \[{\rho _t} = {\left( {P_0^t} \right)^\dagger }{\pi _0}\]

where {\pi _0} is the initial-time-density-matrix, and the superoperator evolution is given by:

    \[P_s^t = {\exp ^ \to }\left( {\int_s^t {dvLv} } \right)\]

    \[ = \]

    \[1 + \sum\limits_{n = 1}^\infty {\int_{s \le {t_1} \le ...{t_n} \le t} {\prod\limits_{i = 1}^n {d{t_i}} } } {L_{{t_i}}}{L_{{t_2}}}...{L_{{t_n}}}\]

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

To prove:

    \[{\rho _t} = {\left( {P_0^t} \right)^\dagger }{\pi _0}\]

note that it is true at t = 0 since P_0^0 is the identity operator. Thus, from:

    \[P_s^t = {\exp ^ \to }\left( {\int_s^t {dvLv} } \right)\]

    \[ = \]

    \[1 + \sum\limits_{n = 1}^\infty {\int_{s \le {t_1} \le ...{t_n} \le t} {\prod\limits_{i = 1}^n {d{t_i}} } } {L_{{t_i}}}{L_{{t_2}}}...{L_{{t_n}}}\]

one finds that:

    \[\frac{d}{{dt}}P_s^t = P_s^t{L_t}\]

holds, and leads to:

    \[\frac{d}{{dt}}{\rho _t} = \left( {L_t^\dagger {{\left( {P_0^t} \right)}^\dagger }} \right){\pi _0} = L_t^\dagger {\rho _t}\]

entailing that it satisfies the Lindblad equation:

    \[{\partial _t}{\rho _t} = L_t^\dagger {\rho _t}\]

with initial condition {\pi _0}. Now, for the evolution operator, one writes an expression for multi-time correlations for distinct observables. For:

    \[0 \le {t_1} \le {t_2} \le ... \le {t_N} \le t\]

the time-ordered correlation is:

    \[{\left\langle {{O_1}\left( {{t_1}} \right){O_2}\left( {{t_2}} \right)...{O_N}\left( {{t_N}} \right)} \right\rangle _{{\pi _0}}}\]

    \[ = \]

    \[{\rm{Tr}}\left( {{\pi _0}P_0^{{t_1}}{O_1}P_{{t_1}}^{{t_2}}{O_2}...P_{{t_{N - 1}}}^{{t_N}}{O_N}} \right)\]

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:

    \[{\partial _t}{\rho _t} = L_t^\dagger {\rho _t}\]

our proof is complete.

Now note that in:

    \[{\left\langle {{O_1}\left( {{t_1}} \right){O_2}\left( {{t_2}} \right)...{O_N}\left( {{t_N}} \right)} \right\rangle _{{\pi _0}}}\]

    \[ = \]

    \[{\rm{Tr}}\left( {{\pi _0}P_0^{{t_1}}{O_1}P_{{t_1}}^{{t_2}}{O_2}...P_{{t_{N - 1}}}^{{t_N}}{O_N}} \right)\]

the operator {\pi _0} represents the initial density matrix of the system and the superoperator P_{{t_i}}^{{t_{i + 1}}} acts on all terms to its right.

Thus, we have the crucial …