Purely quantum memory in closed systems observed via imperfect measurements

TL;DR

The paper investigates memory (non-Markovianity) in closed quantum systems observed via imperfect measurements, contrasting von Neumann lumping (full collapse) with Lüders lumping (coherence-preserving) and introducing mesostates to study coarse-grained dynamics. It shows that classical lumping can be Markovian under a compatibility condition on mesostate transitions, while quantum lumping requires the absence of detectable coherence, encapsulated in the term $Q_n$, to be Markovian; otherwise, purely quantum memory emerges. An explicit $N$-site lattice example demonstrates that von Neumann-type measurements yield Markovian lumped statistics, whereas Lüders-type measurements produce non-Markovian memory except at special times $\tau_S$ and $\tau_M$ when the quantum and classical statistics coincide. The findings connect measurement-induced coherence to non-Markovian behavior, with implications for quantum thermodynamics, decoherence, and quantum information processing, and point toward extensions with dissipative (Lindblad) dynamics and more realistic measurement models, where memory can be tuned by measurement design.

Abstract

The detection and quantification of non-Markovianity, a.k.a. memory, in quantum systems is a central problem in the theory of open quantum systems. There memory is as a result of the interaction between the system and its environment. Little is known, however, about memory effects induced by imperfect measurements on closed systems, where an entanglement with the environment is not possible. We investigate the emergence and characteristics of memory in closed systems observed via imperfect stroboscopic quantum measurements yielding coarse-grained outcomes. We consider ideal and two kinds of imperfect measurements: von Neumann measurements--the analogue of classical lumping--which destroy any coherence in the system, and genuinely quantum-lumping Lüders measurements preserving certain quantum correlations. Whereas the conditions for Markov dynamics under von Neumann lumping are the same as for classical dynamics, quantum-lumping requires stronger conditions, i.e. the absence of any detectable coherence. We introduce the concept of purely quantum memory having no classical counterpart. We illustrate our results with a quantum walk on a lattice and discuss their implications for dissipative dynamics and decoherence effects induced by more realistic measurements.

Figures (3)

• Figure 1: a) Sketch of an ideal quantum measurements: The measurement apparatus distinguishes between all individual quantum states. b) von Neumann measurement: A quantum state is measured but the noise makes the outcome indistinguishable amongst groups of certain states. The latter become lumped into distinguishable sets called mesostates$A = \{z,y\}$, $B = \{x\}$..., which are clearly differentiated by the measurement apparatus. c) Lüders measurement of a single mesostate: The measurement apparatus is intrinsically unable to distinguish sets of states, such that the output are directly lumped mesostates. d) Stroboscopic measurement of a quantum system. The system shown is a quantum lattice with 4 sites and two mesostates (blue-orange).
• Figure 2: a) $\overline{T}_A^B$ as a function of $\tau$ for $N = 2,4,6$. b) Sum in Eq. \ref{['eq:suma']} for $N = 6$ and $y-z = 2,4$. The crossings with the horizontal line indicate the roots $\tau_{\rm M}, \tau_{\rm S}$. c) $\Delta D_2$ for several values of $\tau$ and different initial preparations $\rho_0, \sigma_0$ parametrized by $\theta$: $\rho_0 = \ket{\psi_\theta}\bra{\psi_\theta}$ with $\sqrt{2}\ket{\psi_\theta} = \ket{0} + \cos\theta\ket{2} + \sin\theta\ket{4}$ and $\sigma_0 = \ket{0}\bra{0}/2 + \ket{2}\bra{2}/4+\ket{4}\bra{4}/4$. The quantum lumped process is Markovian iff $\Delta D_2 \leq 0$.
• Figure 3: Schematic of the qubit chain setup: a) schematic of the interactions between the qubits and the external control. Measurements are performed on $q_2$ exclusively. b) Possible transitions between the two qubits states created by arbitrary couplings $\lambda_1$ and $\lambda_2$. Colors denote the respective indistinguishable states under measurements of $q_2$, analogous to those the studied in the quantum lattice with $N = 4$.