Timed demolition measurements

Abstract

Picture an experimental scenario where a closed quantum system, evolving through a time-independent Hamiltonian, is subject to a demolition measurement at a chosen time. The Hamiltonian, the measured observables, the initial state of the physical system and even its Hilbert space dimension are unknown; we nonetheless assume a promise or constraint on the energy distribution of the state. In this context we find that, for many natural energy constraints, the set of feasible time series or datasets can be characterized efficiently. Furthermore, under the assumption of a bounded energy spectrum, we prove that there exist ``self-testing'' datasets, whose approximate realization singles out specific Hamiltonians, states and measurement operators. Investigating to what extent the extrapolation of past measurement data is possible in this framework, we identify energy-constrained physical systems for which a non-trivial prediction at time $τ$ requires a precision in the measurement data superexponential in $τ$. We also discover two extrapolation phenomena: ``aha! datasets", which drastically increase the predictability of the future statistics of an unrelated measurement; and ``fog banks": fairly simple datasets that exhibit complete unpredictability at some future time $τ$, but full predictability at a later time $τ'>τ$. Besides their relevance for quantum foundations, our results have applications in semi-device independent quantum communication, the simulation of complex quantum systems, and the design of optimal atomic clocks.

Figures (8)

• Figure 1: Datasets. A datapoint $(P(a|x,t_i))_{a,x}$ characterises the outcome probabilities when a state $\rho$ is prepared, evolved for a time $t_i$ with Hamiltonian $H$ and measured with one of the measurements $M_x$. For the different datapoints making up a dataset only the time the system evolves for differs (shown in different rows in the figure). In order to obtain an estimate $(\tilde{P}(a|x,t_i))_{a,x}$ for the datapoint $(P(a|x,t_i))_{a,x}$, one may prepare many copies of the state $\rho$, evolve each with Hamiltonian $H$ for a time $t_i$ and perform one of the different measurements $M_x$ on it.
• Figure 2: Harmonic oscillators as quantum clocks. The timeline of the clock function with $M=5$ (purple curve) is approximated by a harmonic oscillator with truncated spectrum $(E_0, E_1, \ldots, E_{N-1})$, where $E_k= k$. In blue we display the optimal timeline of the truncated harmonic oscillator of the first $N=2,5,10,20$ energy levels, from left to right, top to bottom. The horizontal axis displays time; the vertical axis, $P(0|t)$.
• Figure 3: Hydrogen atoms as quantum clocks. The timeline of the clock function with $M=5$ (purple curve) is approximated by a hydrogen atom with truncated spectrum $(E_1, E_2, \ldots, E_{N})$, where $E_k= \frac{13.6}{k^2}$. In blue we display the optimal timeline of the truncated hydrogen atom for the first $N=2,4,6,8$ energy levels, from left to right, top to bottom. The x-axis displays time, the y-axis $P(0|t)$.
• Figure 4: Datasets $\mathbf{D}_N$ and timelines of realizations \ref{['reference_realization']}. The x-axis displays the time, the y-axis the probability to observe outcome $0$, the units are chosen the same for all four plots, where $E^+=1$. From left to right, top to bottom we consider $N=2,3,4,5$.
• Figure 5: Extrapolation problem. We display the simplest case where we consider only one two-outcome measurement $M$. In this case the pairs $(\tilde{P}(a|t_i), \delta_i)$ for $i=1, \ldots, n$ (displayed by the data points and error bars in the illustration) are sufficient to represent each noisy datapoint. Our goal in this example is to bound the range of possible ${P}(a|\tau)$ (see $\mu^-,\mu^+$). The red curve $P(t)$ displays one example of a timeline that fits the dataset. The part of the illustration where extrapolation is required is coloured palely.

Theorems & Definitions (47)

• Lemma 1
• Lemma 2
• Proposition 3
• proof
• Lemma 4
• proof
• Proposition 5
• proof
• Definition 1
• Definition 2