Weak continuous measurements require more work than strong ones

TL;DR

The paper investigates the energy cost of quantum measurements and the objectification process using a dynamical SBS-based model inspired by quantum Darwinism. It introduces a system-ancilla protocol with unitary coupling, pure dephasing across many environments to form SBS, and coarse-grained readout and resets, enabling analysis across weak to strong measurements. Three information-theoretic figures of merit (strength, efficiency, and η_{X:r}) are defined and integrated with a lower bound on measurement work, revealing a surprising result: sequences of weak measurements generally require more work than a single strong measurement achieving similar information extraction. The findings are illustrated in a qubit-cavity framework and are supported by general scaling arguments, indicating a fundamental energy-information trade-off in quantum measurement strategies with broad implications for quantum technologies.

Abstract

Understanding the energy cost of quantum measurement process and its connection to the measurement performance faces the challenge of modeling the objectification process. The latter, turns the measurement result into an objective fact, available to independent observers, and is responsible for the measurement irreversibility. To address this issue, we propose and analyze a dynamical model of quantum measurement, able to capture nonideal (weak and inefficient) measurements. In this model, the objectification is induced by a contact with a macroscopic reservoir at equilibrium which is responsible for the redundant broadcast of the measurement outcome (producing a Spectrum Broadcast Structure (SBS) state) while inducing decoherence in the pointer basis, in the line of the theory of quantum Darwinism. We analyze the performance of the obtained measurement process by introducing figures of merit to quantify the strength of the measurement and its efficiency. We also derive and a lower bound on the measurement work cost that we can relate to the measurement quality. We take as an illustration the readout of a qubit via its coupling to a harmonic oscillator. We investigate the long sequences of extremely short and weak measurements (a.k.a continuous measurements), to find under which conditions they converge to an ideal (projective) measurement and analyze their work cost. Surprisingly, we find that a sequence converging to projective measurement has a much larger work cost than an equivalent strong measurement obtained from a single intense interaction with the apparatus. We extend this result to a large class of models owing to scaling arguments. Our analysis offers new insights into the trade-offs between measurement strength, energy consumption, and information extraction in quantum measurement protocols.

Figures (17)

• Figure 1: Measurement protocol: (0) ancilla state preparation, (1) ancilla-system interaction, (2) ancilla dephasing, (3) ancilla readout, (4) ancilla reset and (5) memory reset. The color change in both the system and ancilla indicates a change of state. Blurred disks represent quantum states, while filled disks denote classical states.
• Figure 2: The ancilla undergoes dephasing through multiple independent environments. This occurs via distinct channels, each characterized by a dephasing coefficient $\gamma_i$. At the end of the process, the joint state of the ancilla and the complete system is in a Spectrum Broadcast Structure (SBS) state.
• Figure 3: Hierarchy of information flows involved in the figures of merit. The inequality from Eq. \ref{['eq:hierarchy']} suggests the following interpretation: From the total amount of information $\log d$ missing to characterize the value of $\hat{X}$ in the state ${\left|{\text{ref}}\right\rangle}$, a fraction $S[\hat{\rho}_S(t_f)]$ is transferred to the ancilla and environment (mustard arrow). A smaller amount $I_q$ is available in the ancilla (pink arrow), and an even smaller amount $\chi$ in the practically accessible degrees of freedom of the ancilla (grey arrow). Finally, only an amount $I({j};{r})$ concerns the target observable (blue arrow).
• Figure 4: Strength, $\xi$, as a function of $\epsilon$ for the qubit measurement model. The strength depends only on $\epsilon$ and converges to 1. The qubit was initialized in the reference state ${\left|{\text{ref}}\right\rangle}=\frac{1}{\sqrt{2}}\left({\left|{g}\right\rangle}+{\left|{e}\right\rangle}\right)$.
• Figure 5: Measure of efficiency $\eta$ in terms of $\bar{\alpha}$ and $\epsilon$ in the coarse-grained qubit measurement scenario where the two measurement results correspond to the total absence of excitation in the field and the presence of one or more excitations, respectively. The qubit was initialized in the reference state ${\left|{\text{ref}}\right\rangle}=\frac{1}{\sqrt{2}}\left({\left|{g}\right\rangle}+{\left|{e}\right\rangle}\right)$.