Witnessing environment dimension through temporal correlations

Lucas B. Vieira1,2, Simon Milz3,2,1, Giuseppe Vitagliano4, and Costantino Budroni5,2,1

1Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
2Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
3School of Physics, Trinity College Dublin, Dublin 2, Ireland
4Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria
5Department of Physics ``E. Fermi'' University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy

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We introduce a framework to compute upper bounds for temporal correlations achievable in open quantum system dynamics, obtained by repeated measurements on the system. As these correlations arise by virtue of the environment acting as a memory resource, such bounds are witnesses for the minimal dimension of an effective environment compatible with the observed statistics. These witnesses are derived from a hierarchy of semidefinite programs with guaranteed asymptotic convergence. We compute non-trivial bounds for various sequences involving a qubit system and a qubit environment, and compare the results to the best known quantum strategies producing the same outcome sequences. Our results provide a numerically tractable method to determine bounds on multi-time probability distributions in open quantum system dynamics and allow for the witnessing of effective environment dimensions through probing of the system alone.

The amount of information that can be stored in a physical system is constrained by its dimension, i.e., the number of perfectly distinguishable states. As a consequence, a system's finite dimension imposes fundamental constraints in what behaviors it can display over time. In a sense, this dimension quantifies the "memory" of the system: how much of its past it can "remember" in order to influence its future.

A natural question arises: what is the minimum dimension a system must have in order for it to produce some observed behavior? This question can be answered with the concept of a "dimension witness": an inequality which, when violated, certifies this minimum dimension.

In this work, we investigate an application of this idea to the behavior of open quantum systems.

Physical systems are never completely isolated, and inevitably interact with their surrounding environment. As a result, information in the system can leak away into the environment at one moment, only to be partially recovered later. Therefore, the environment can act as an additional memory resource, resulting in complex correlations in time.

Even thought, in practice, the environment may be very large in size, only a small portion of it may effectively act as a memory. By establishing upper bounds on the temporal correlations achievable by repeated preparations and measurements on a small "probe" quantum system interacting with an environment of fixed size, we can construct a dimension witness for the minimum size of its effective environment.

This work provides a practical technique to obtain such bounds on temporal correlations. Our results show that there is a wealth of information contained in temporal correlations, highlighting their potential in new techniques for characterizing large complex systems by means of a small probe alone.

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