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Emergence of Jumps in Quantum Trajectories via Homogenization

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Abstract

In the strong noise regime, we study the homogenization of quantum trajectories i.e. stochastic processes appearing in the context of quantum measurement. When the generator of the average semigroup can be separated into three distinct time scales, we start by describing a homogenized limiting semigroup. This result is of independent interest and is formulated outside of the scope of quantum trajectories. Going back to the quantum context, we show that, in the Meyer–Zheng topology, the time-continuous quantum trajectories converge weakly to the discontinuous trajectories of a pure jump Markov process. Notably, this convergence cannot hold in the usual Skorokhod topology.

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Notes

  1. Completely positive maps \(\Phi :M_d({{\mathbb {C}}})\rightarrow M_d({{\mathbb {C}}})\) are linear maps such that for any \(n\in {\mathbb {N}}\), \(\Phi \otimes {{\,\mathrm{Id}\,}}_{M_n({{\mathbb {C}}})}:M_d({{\mathbb {C}}})\otimes M_n({{\mathbb {C}}})\rightarrow M_d({{\mathbb {C}}})\otimes M_n({{\mathbb {C}}})\) is positive.

  2. The Hamiltonian dynamics can also be intrinsic, independent of any additional drive due to the experimentalist.

  3. The acronym GKSL stands for Gorini–Kossakowski–Sudarshan–Lindblad.

  4. By analogy with discrete quantum channels.

  5. In other words a \(C^*\) commutative subalgebra of \(M_d ({{\mathbb {C}}})\).

  6. To be more precise \({{\mathbb {L}}}^0\) is a quotient space where two functions are considered as equal if they coincide almost everywhere with respect to the Lebesgue measure.

  7. We recall that if \(\chi \) is a topological space and \((X_n)_n\) is a sequence of \(\chi \)-valued random variables, we say it converges weakly (or in law) to the \(\chi \)-valued random variable X if and only if for any bounded continuous function \(f:\chi \rightarrow {\mathbb {R}}\), \(\lim _{n \rightarrow \infty } {{\mathbb {E}}} \big [ f(X_n)\big ] ={{\mathbb {E}}} \big [ f(X)\big ]\).

  8. Observe that the transition rate is independent of the \(L^{(1)}_k\)’s and of \(H^{(0)}\). It depends of course of \(H^{(1)}, L^{(0)}_k, L_k^{(2)}\) but also of \(H^{(2)}\) through the eigenvalues \(\tau _{i,j}\) given in Eq. (2.6).

  9. Here also we denote with the same notation the corresponding operator norm on \({\mathrm{End}} (V)\).

  10. This property is equivalent to one of the following assertions:

    • the restriction to \({{\,\mathrm{Ker}\,}}\left[ \left( {{\mathcal {L}}}^{(2)} \right) ^m \right] \) is diagonalizable, with \(m \in {{\mathbb {N}}}\) being the algebraic multiplicity.

    • the algebraic multiplicity m of \(\lambda =0\) matches the geometric multiplicity \(\dim {{\,\mathrm{Ker}\,}}{{\mathcal {L}}}^{(2)}\).

  11. The operator 2-norm of a matrix \(M\in M_d ({{\mathbb {C}}})\) is the operator norm of M considered as an operator on \({{\mathbb {C}}}^d\), the latter being equipped with its standard Hilbert structure.

  12. This operator is defined on the whole matrix space but we identify it with its restriction to diagonal matrices \(\Pi {{\mathcal {L}}}^*\Pi : {{\,\mathrm{Span}\,}}_{{\mathbb {C}}}\{E_{i,i}: i=1,\dotsc ,d\}\rightarrow {{\,\mathrm{Span}\,}}_{{\mathbb {C}}}\{E_{i,i}: i=1,\dotsc ,d\}; X\mapsto \Pi {{\mathcal {L}}}^*\Pi (X)\).

  13. In this formula we adopt the convention that the semigroup \(e^{tT^*}\) acts on a diagonal matrix \({\text {diag}} (x_1,\dotsc ,x_d)\) by transforming it in the diagonal matrix \({\text {diag}} (e^{tT^*}(x_1,\dotsc ,x_d))\).

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Acknowledgements

T.B. would like to thanks Martin Fraas for enlightening discussions. The research of T.B., R.C. and C.P. has been supported by project QTraj (ANR-20-CE40-0024-01) of the French National Research Agency (ANR). The research of T.B. and C.P. has been supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. The work of C.B. and R.C. has been supported by the projects RETENU ANR-20-CE40-0005-01, LSD ANR-15-CE40-0020-01 of the French National Research Agency (ANR) and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovative programme (grant agreement No 715734). J. N. has been supported by the start up grant U100560-109 from the University of Bristol and a Focused Research Grant from the Heibronn Institute for Mathematical Research. This work is supported by the 80 prime project StronQU of MITI-CNRS: “Strong noise limit of stochastic processes and application of quantum systems out of equilibrium”.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, 31062, Toulouse Cedex 9, France

    Tristan Benoist, Reda Chhaibi & Clément Pellegrini

  2. Université Côte d’Azur, CNRS, LJAD, Parc Valrose, 06108, NICE Cedex 02, France

    Cédric Bernardin & Raphaël Chetrite

  3. Interdisciplinary Scientific Center Poncelet (CNRS IRL 2615), Moscow, Russia, 119002

    Cédric Bernardin

  4. University of Bristol, University Walk, Clifton, Bristol, UK

    Joseph Najnudel

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  1. Tristan Benoist
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  2. Cédric Bernardin
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Benoist, T., Bernardin, C., Chetrite, R. et al. Emergence of Jumps in Quantum Trajectories via Homogenization . Commun. Math. Phys. 387, 1821–1867 (2021). https://doi.org/10.1007/s00220-021-04179-8

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