Assessing Quantum Dimensionality from Observable Dynamics

Using tools from classical signal processing, we show how to determine the dimensionality of a quantum system as well as the effective size of the environment’s memory from observable dynamics in a model-independent way. We discuss the dependence on the number of conserved quantities, the relation to ergodicity and prove a converse showing that a Hilbert space of dimension $D+2$ is sufficient to describe every bounded sequence of measurements originating from any $D$-dimensional linear equations of motion. This is in sharp contrast to classical stochastic processes which are subject to more severe restrictions: a simple spectral analysis shows that the gap between the required dimensionality of a quantum and a classical description of an observed evolution can be arbitrary large.

Figure 1

(a) Eigenvalues of the evolution operator lie on the unit disc and can be obtained from the observable time dependence of any expectation value. For a classical description, they have to be located in a region depending on the number $d_c$ of degrees of freedom. For $d_c=2$, this is the real line, for $d_c=3$, 4, the dark and light grey regions become additionally accessible. (b) A simple damped oscillation leads to eigenvalues $e^{-\gamma \pm i\omega }$, implying that a quantum mechanical description can access the entire unit disc even for $d=2$.

Figure 2

Top: a finite sequence $⟨A(t)⟩$, $t\le 100$ obtained from a randomly chosen unitary dynamics of a spin-1 quantum system ($d=3$). As there are three conserved quantities (the eigenstates of the evolution), we have $\dim \mathcal{V}\le 7$. Bottom: log-plot of the 15 largest singular values $s_j$ of the corresponding $50\times 50$ matrix $V$ as a function of the standard deviation of added Gaussian noise (1%–10% of the signal). While for small noise the largest 7 singular values are clearly separated by a threshold, this washes out as the error increases—for too much noise the data could as well be explained by smaller $\dim \mathcal{V}$ as expressed quantitatively by Eq. (13).