Work Measurement as a Generalized Quantum Measurement

We present a new method to measure the work $w$ performed on a driven quantum system and to sample its probability distribution $P(w)$. The method is based on a simple fact that remained unnoticed until now: Work on a quantum system can be measured by performing a generalized quantum measurement at a single time. Such measurement, which technically speaking is denoted as a positive operator valued measure reduces to an ordinary projective measurement on an enlarged system. This observation not only demystifies work measurement but also suggests a new quantum algorithm to efficiently sample the distribution $P(w)$. This can be used, in combination with fluctuation theorems, to estimate free energies of quantum states on a quantum computer.

Figure 1

Quantum circuit for the estimation of work probability distribution. The initial state of the auxiliary $M$-qubit system is a pure state ${\rho }_{\mathcal{A}}=|{\psi }_{\mathcal{A}}⟩⟨{\psi }_{\mathcal{A}}|$ with $|{\psi }_{\mathcal{A}}⟩=|x=0⟩$. When the initial state of the system is pure given by $|\varphi ⟩=\sum _{n=1}^{D_{\mathcal{S}}}{c}_n|{\varphi }_n⟩$ then the state of $\mathcal{S}$ and $\mathcal{A}$ just before the measurement is $|\mathrm{\Psi }{⟩}_{\mathcal{S},\mathcal{A}}=\frac{1}{D}\sum _{n,m=1}^{D_{\mathcal{S}}}{c}_n⟨{\tilde{\varphi }}_m|U_E|{\varphi }_n⟩\sum _{t,x}{e}^{i(2\pi /D)t(x-(E_{m,n}/4E_M)D)}|w⟩\otimes |{\tilde{\varphi }}_m⟩$.

Figure 2

Top: Comparison of the coarse-grained version of the exact probability distribution [given by Eq. (1)] with the probability distribution generated by the algorithm [Eq. (7)]. For this example a system $\mathcal{S}$ of 10 qubits was used (therefore giving $2^{20}$ different possible values of work), while the ancilla $\mathcal{A}$ was composed of only 5 qubits. Bottom: Free energy estimation using Jarzynski’s equality and work values sampled from the exact distribution, $P(w)$, and the distribution resulting from the algorithm, $P_D(w)$. The exact value of the free energy difference is also shown, calculated as the ratio of the partition functions.