Exploring the extent of validity of quantum work fluctuation theorems in the presence of weak measurements

Abstract

Work fluctuation theorems have been one of the important achievements in the field of nonequilibrium Statistical Physics, both in the classical and quantum regimes. Conventionally, the work done on a quantum system is defined by means of a two-point measurement scheme, where a projective measurement of the Hamiltonian is performed both at the beginning and at the end of the process. Recently, quantum work fluctuation theorems in the context of generalized measurements have received a lot of attention. Here, we define a weak value of work, within the broad frame-work of generalized measurements and show that the deviation from the exact work fluctuation theorems are much less in this formalism as compared to previous efforts in the literature, using a two-level system as the model. We find that the original form of Jarzynski equality (valid for projective two-point measurements) does not remain exact in this framework. Nevertheless, the deviations are in general small, so that an approximate effective temperature of the thermal bath can be deduced using our results. Further, in the limit of the measurements being projective, the exact form of the work fluctuation theorems is recovered.

Appendix A: Comparison between projective and weak measurements

Let us consider a two-state system with eigenstates $|0\rangle$ and $|1\rangle$, whose quantum state is defined by the density operator

$$\begin{aligned} \rho (t) = p_0|0\rangle \langle 0| + p_1|1\rangle \langle 1|. \end{aligned}$$

(A1)

Here, $p_0$ and $p_1$ are the probabilities of obtaining the states $|0\rangle$ and $|1\rangle$, respectively, in a projective measurement. In other words, a projective measurement is the application of an operator whose action leaves the state in one of the eigenstates of the system. As can be observed, operators of the form $\Pi _0=|0\rangle \langle 0|$ and $\Pi _1=|1\rangle \langle 1|$ are projective operators. Such a measurement, when applied to a quantum observable (take the Hamiltonian as an example) will yield an eigenvalue of the operator. The state of the system just after the measurement (at time t, say) is given by

$$\begin{aligned} \rho _i(t^+) = \frac{\Pi _i\rho (t)\Pi _i}{p_i}, \end{aligned}$$

(A2)

where the denominator is required for normalization.

One artifact of the projective measurement is that it heavily disturbs the evolution of the state of the system by forcing it to jump to one of its eigenstates. This could be an undesired feature for a quantum system, since there remains no correlation between the pre-measurement and post-measurement states. Quantum projective measurement usually deals with the scenario of a direct correspondence between system and apparatus states. However, in practice, the system-apparatus correspondence may not be achieved in a general measurement scenario, leading to generalized measurements, also called the positive operator-valued measures (POVMs). Weak measurements are a special case of generalized measurements. They have a number of special features that has attracted attention in recent times [41]. Thus, for example, they have been used to amplify detector signals, enabling the sensitive estimation of unknown small evolution parameters [42].The weak value has motivated the development of new methods for the direct measurement of quantum states [43] and has provided a measurable window into nonclassical features of quantum mechanics.

One defines operators $F_i$ that replaces the projection operators $\Pi _i$, where i corresponds to the reading of a meter that shows the outcome of the measurement. However, the post-measurement state is in general not an eigenstate. Rather, it remains in a superposition of its eigenstates, given by

$$\begin{aligned} \rho _i(t^+) = \frac{F_i\rho (t)F_i^\dagger }{{{\,\mathrm{Tr}\,}}[F_i\rho (t)F_i^\dagger ]}. \end{aligned}$$

(A3)

The generalized measurement operators are positive definite and sum up to give the identity operator.

A special case of this broad class of operators consists of weak operators, that disturbs the state of the system minimally, so that they are typically of the form [34, 36]

$$\begin{aligned} F_i = \gamma _i(\hat{I}+\hat{\varepsilon _i}), \end{aligned}$$

(A4)

where ${\hat{I}}$ is the identity operator, $\gamma _i$ is a constant scalar coefficient, and the operator $\varepsilon _i$ is of the form

$$\begin{aligned} {\hat{\varepsilon }} = \begin{pmatrix} \varepsilon _{1i} &{} 0\\ 0 &{} \varepsilon _{2i} \end{pmatrix}, (\varepsilon _{1i}, \varepsilon _{2i} \ll 1). \end{aligned}$$

Obviously, these operators, being very close to identity operator, do not change the pre-measurement state substantially. However, the price that is to be paid comes in the form of a lack of an appreciable information gained about the state through such a measurement. Nevertheless, it can be shown that a sequence of weak measurements can be used to construct a generalized measurement, including the projective one [44].