Abstract
In this paper we introduce a definition for conditional energy changes due to general quantum measurements, as the change in the conditional energy evaluated before, and after, the measurement process. By imposing minimal physical requirements on these conditional energies, we show that the most general expression for the conditional energy after the measurement is simply the expected value of the Hamiltonian given the post-measurement state. Conversely, the conditional energy before the measurement process is shown to be given by the real component of the weak value of the Hamiltonian. Our definition generalises well-known notions of distributions of internal energy change, such as that given by stochastic thermodynamics. By determining the conditional energy change of both system and measurement apparatus, we obtain the full conditional work statistics of quantum measurements, and show that this vanishes for all measurement outcomes if the measurement process conserves the total energy. Additionally, by incorporating the measurement process within a cyclic heat engine, we quantify the non-recoverable work due to measurements. This is shown to always be non-negative, thus satisfying the second law, and will be independent of the apparatus specifics for two classes of projective measurements.
Highlights
Measurements play an important role in thermodynamic processes
The energetic statistics obtained by the proposed definition generalises existing results in the literature, which are valid in specific circumstances: (i) if the measured observable involves an initial and final energy measurement, we regain the work distribution of the TPM protocol; (ii) if the observable measured is the Heisenberg-evolved Hamiltonian, the energy statistics is equivalent to the quasi-probability distribution over the random variable of work introduced in [13]; and (iii) if the measurement process first projects the system onto one of its pure state components, we obtain the definition for internal energy change along a quantum trajectory used in stochastic thermodynamics
In the present work we have defined the change of energy, conditional on the outcome of a general quantum measurement, as the difference in conditional energies of the system, evaluated before and after the measurement process
Summary
Measurements play an important role in thermodynamic processes. This has been established ever since the introduction of Maxwell’s demon [1] and the subsequent insights gained in the thermodynamic role of information [2,3,4,5,6,7]. The change in energy is defined as the difference in expected values of the Hamiltonian at the start and end of the trajectory in question Such an approach, implicitly assumes that we know which pure state the system initially occupies. The energetic statistics obtained by the proposed definition generalises existing results in the literature, which are valid in specific circumstances: (i) if the measured observable involves an initial and final energy measurement, we regain the work distribution of the TPM protocol; (ii) if the observable measured is the Heisenberg-evolved Hamiltonian, the energy statistics is equivalent to the quasi-probability distribution over the random variable of work introduced in [13]; and (iii) if the measurement process first projects the system onto one of its pure state components, we obtain the definition for internal energy change along a quantum trajectory used in stochastic thermodynamics. We shall refer to the conditional states of S after observing the measurement outcome x as ρ(x) := IxM(ρ)/pM ρ (x) ≡ trA[ S+A(x)], and the orthogonal states of A representing outcome x will be denoted ξ(x) := PAxξ PAx/pM ρ (x) ≡ trS[ S+A(x)], where ξ := trS[U (ρ ⊗ ξ)U †] is the state of A after premeasurement
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