Certified randomness from quantum speed limits

TL;DR

This work shows that quantum speed limits, via time-energy constraints, can certify randomness in a semi-device-independent prepare-and-measure protocol without detailed device assumptions. By defining the quantum correlation set 𝒬_{E,Δt} and contrasting it with the classical max-average set, the authors prove that certain correlations imply intrinsic randomness and develop a dual-optimisation framework to compute lower bounds on extractable randomness. The framework accommodates open-system dynamics and provides discretisation-based guarantees, with numerical results demonstrating nonzero certifiable entropy. An explicit implementation with single-mode coherent states in a harmonic oscillator illustrates practical parameter regimes where randomness can be certified, linking time-evolution speed limits to foundational aspects of nonclassicality and semi-DI quantum information processing.

Abstract

Quantum speed limits are usually regarded as fundamental restrictions, constraining the amount of computation that can be achieved within some given time and energy. Complementary to this intuition, here we show that these limitations are also of operational value: they enable the secure generation of certified randomness. We consider a prepare-and-measure scenario with some (experimentally determined or promised) upper bound on the energy uncertainty of the average prepared quantum state, but without any further assumptions on the devices, Hilbert space or Hamiltonian. Given that we can freely choose the time at which to apply the untrusted preparation procedure, we show that this scenario admits the generation of randomness that is secure against adversaries with additional classical information. We show how to determine the amount of certified randomness given the observed correlations, discuss how interactions with the environment are taken into account, and sketch a conceivable experimental implementation. In particular, we show that single-mode coherent states admit this kind of certification of non-zero randomness in some parameter regimes, reinforcing existing demonstrations of nonclassicality in the simple harmonic oscillator. Our results extend earlier efforts to devise semi-device-independent protocols grounded in reasonable physical assumptions, and they contribute to the understanding of time-energy uncertainty relations via their operational consequences.

Figures (8)

• Figure 1: The preparation device $P$ prepares a fixed quantum state which is subsequently sent to a measurement device $M$. We treat both $P$ and $M$ as black boxes, but we assume that we have full control of the time $t_x$ at which $P$ is triggered: at $t_0$ or $t_1=t_0+\Delta t$, where $x\in\{0,1\}$ labels the input to $P$. In contrast, $M$ is implemented at a predetermined time. A classical variable $\lambda$ is allowed to influence and possibly correlate $P$ and $M$. It may be known to an adversary, but it is unknown to the user of the randomness generator.
• Figure 2: Set of quantum (blue) and classical (red) correlations for different state overlaps $\gamma \in \{0.9,0.7,0.5,0.3,0.1\}$ (from darkest to lightest).
• Figure 3: Set of quantum correlations $\mathcal{Q}_{\mathfrak{E},\Delta t}$ (blue) and classical max-average correlations $\mathcal{\bar{C}}_{\mathfrak{E},\Delta t}$ (red) for $\mathfrak{E}\Delta t=0.314$ (and thus $\gamma=0.951$ according to Eq. (\ref{['EqGamma']})).
• Figure 4: Numerical certified lower bounds on the entropy $H^\star$ under the assumption $\Delta E \leq \mathfrak{E}$ with $\mathfrak{E}\Delta t=0.5$ (in units where $\hbar=1$), obtained using the discretisation parameters $L=20$, $M=5$, and $N=S=5000$ (See Appendix \ref{['sec-algorithm']}). Points with $H^\star > 0$ correspond to correlations that allow for certified randomness extraction. As expected, $H^\star=0$ (only) on the red-rimmed classical max-average set $\mathcal{\bar{C}}_{\mathfrak{E},\Delta t}$. With this discretisation, the largest certified lower bound in the plot is $H^\star\geq 0.7284$, which occurs at $(C_0,C_1) = (\pm \sin(\mathfrak{E}\Delta t), \mp \sin(\mathfrak{E}\Delta t))$. A more exhaustive run (Appendix \ref{['sec-maxH']}) with $N=S=20000$ yields the certified lower bound $H^\star\geq 0.8113$. This value may still not be quite tight, and finer discretisation could yield marginally higher $H^\star$. See Appendix \ref{['appendix-minentropy']} for all details on the methodology we use to evaluate (\ref{['opt_prob']}).
• Figure 5: (a) Diagram of proposed experimental setup; technically similar to, but conceptually distinct from the BPSK implementation of VanHimbeeck2. The preparation device $P$ consists of a monochromatic laser emitting a coherent pulse, which is sent through a beam splitter for homodyne detection. The transmitted beam (the quantum signal, QS) is attenuated by an optical density filter (OD) and then, for the input $x=1$, delayed by some time $\Delta t$. This can be achieved using an optical wedge, whose position and thickness depends on the input $x$. By causing a time delay, the wedge introduces a corresponding phase shift. At the measurement device $M$, the signal is interfered with the reflected signal (the local oscillator, LO), which acts as a phase reference, and is measured. The quadrature $q$ is then measured, the sign of which determines the output $b$. (b) Diagram of the phase space of the coherent state, showing the relative phase $\omega\Delta t$ between the two states that may arrive at $M$. For $x=0$, the state $\ket{\alpha}=\ket{i\xi}$ is centred on the y-axis (in the rotating frame of reference of the local oscillator), evolving counter-clockwise with period $2\pi/\omega$. For $x=1$, the state is given by $\ket{\alpha(t=\Delta t)}=\ket{i\xi e^{-i\omega\Delta t}}$.

Theorems & Definitions (14)

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