Quantum Thermodynamic Uncertainty Relation for Continuous Measurement

We use quantum estimation theory to derive a thermodynamic uncertainty relation in Markovian open quantum systems, which bounds the fluctuation of continuous measurements. The derived quantum thermodynamic uncertainty relation holds for arbitrary continuous measurements satisfying a scaling condition. We derive two relations; the first relation bounds the fluctuation by the dynamical activity and the second one does so by the entropy production. We apply our bounds to a two-level atom driven by a laser field and a three-level quantum thermal machine with jump and diffusion measurements. Our result shows that there exists a universal bound upon the fluctuations, regardless of continuous measurements.

Figure 1

Quantum trajectories and measurements of (a) jump measurement (photon counting) and (b) diffusion measurement (homodyne detection) in a two-level atom. Upper panels are quantum trajectories of ${\rho }_{ee}\equiv ⟨{\epsilon }_e|\rho |{\epsilon }_e⟩$ and lower panels are measurement outputs.

Figure 2

Quantum Fisher information and the results of computer simulations of jump measurement. (a) The quantum Fisher information ${\mathcal{I}}_Q(0)=T(\mathrm{\Upsilon }+\mathrm{\Psi })$ (solid line), $T\mathrm{\Upsilon }$ (dashed line), and $T\mathrm{\Psi }$ (dotted line) as a function of $\kappa$, where $T=1$, $\mathrm{\Omega }=1$, and $\mathrm{\Delta }=1$. (b) $\mathrm{Var}[{\mathrm{\Theta }}_{\mathcal{N}}]/\mathbb{E}[{\mathrm{\Theta }}_{\mathcal{N}}{]}^2$ as a function of $T(\mathrm{\Upsilon }+\mathrm{\Psi })$ (circles) and $T\mathrm{\Upsilon }$ (triangles) for the jump measurement, where $\mathrm{\Delta }\in [0.1,10.0]$, $\mathrm{\Omega }\in [0.1,10.0]$, $\kappa \in [0.1,10.0]$, and $T=1000$. The dashed line corresponds to $1/[T(\mathrm{\Upsilon }+\mathrm{\Psi })]$ for the circles and $1/[T\mathrm{\Upsilon }]$ for the triangles.

Figure 3

Illustration of the model and results of computer simulation for the thermal machine. (a) Thermal machine consisting of three levels $|{\epsilon }_A⟩$, $|{\epsilon }_B⟩$, and $|{\epsilon }_g⟩$. The transitions between each of the states are coupled with heat reservoirs with the inverse temperature ${\beta }_r$ ($r=1$, 2, 3). (b) $\mathrm{Var}[{\mathrm{\Theta }}_C^{\prime }]/\mathbb{E}[{\mathrm{\Theta }}_C^{\prime }{]}^2$ (circles) as a function of $T\mathrm{\Sigma }$ for the transformed jump measurement, where ${\beta }_1\in [0.1,1.0]$, ${\beta }_2\in [0.01,0.1]$, ${\beta }_3\in [1.0,10.0]$, ${\omega }_1\in [1.0,10.0]$, ${\omega }_2\in [1.0,10.0]$, $R_{ij}^{\prime }\in [0.0,1.0]$, $|{\zeta }_{ij}|\in [0.0,0.2]$, $\gamma =0.1$, and $T=100$. $R_{ji}^{\prime }$ and ${\zeta }_{ji}$ are defined for $i\ne j$ and satisfy $R_{ji}^{\prime }=-R_{ij}^{\prime }$ and $|{\zeta }_{ji}|=|{\zeta }_{ij}|$. The dashed line corresponds to the lower bound $2/[T\mathrm{\Sigma }]$.