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.
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.
Work is an essential concept in classical thermodynamics, and in the quantum regime, where the notion of a trajectory is not available, its definition is not trivial. For driven (but otherwise isolated) quantum systems, work can be defined as a random variable, associated with the change in the internal energy. The probability for the different values of work captures essential information describing the behaviour of the system, both in and out of thermal equilibrium. In fact, the work probability distribution is at the core of "fluctuation theorems" in quantum thermodynamics. Here we present the design and implementation of a quantum work meter operating on an ensemble of cold atoms, which are controlled by an atom chip. Our device not only directly measures work but also directly samples its probability distribution. We demonstrate the operation of this new tool and use it to verify the validity of the quantum Jarzynksi identity.
The work distribution function for a driven quantum system which evolves out of thermal equilibrium encodes some universal features and is a central character in the so-called fluctuation theorems. In this talk I will review recent efforts in developing methods to efficiently sample the work distribution function in modern quantum simulators. Not only I will review some theoretical work but also present the results of a conceptually simple (but technically challenging) experiment which was done to implement such methods and illustrate the validity of the quantum fluctuation theorems.
According to the axioms of quantum mechanics, there exists a quantum measurement corresponding to each observable.Energy is one of such observables, and the operator of the energy is called Hamiltonian.The Hamiltonian specifies the dynamics of closed quantum systems and it also characterizes the property of the equilibrium state in contact with a heat bath.Therefore the energy is an essential quantity in quantum mechanics, which characterizes behavior of a quantum system.On the other hand, since the Hamiltonian is not a local observable in general, a concrete construction of how to apply the quantum measurement of energy is not trivial.Thus various measurement models of energies have been proposed, and also discussions have been made on whether the time is a resource to be consumed to increase the precision of energy measurement in connection with the time-energy uncertainty relationship.Constructing a way to implement the energy measurement on an unknown Hamiltonian system is important, since it avoids constructing a new measurement scheme for each new quantum system.On the other hand, among measurements of energy, the projective measurement of energy has useful properties for applications.The projective measurement is a measurement which satisfies the condition so-called repeatable hypothesis that when we measure the state immediately after obtaining an outcome from the same measurement, we obtain the same outcome again.In this thesis, we consider two schemes to implement the projective measurement of energy for an unknown Hamiltonian system.One is a tomography based method, which is a construction suggested in the paper of Aharanov et.al. (2002).The other is the phase estimation based method, which utilizes the Hamiltonian dynamics of the system as a resource for the quantum algorithm, and realizes the projective measurement of energy on the system without identifying all parameters of the Hamiltonian.In this thesis, we assume there is a finite dimensional quantum system (quantum computer) which is able to apply any interaction between the target system and the finite dimensional system although the interaction should not depend on the target system Hamiltonian.To evaluate the performance of the measurements implemented by the two schemes, we formulate two evaluation functions called fluctuation of measurement value and non-repeatability, which can evaluate how much a measurement is different from the ideal projective measurement of an observable.We calculate a sufficient time to guarantee the fluctuation of measurement value below some small value ε, for each of the two measurement schemes.We find that the tomography based method takes the time proportional to O(d 4 ∆ max /ε), while the phase estimation based method takes O(∆ 3 max /ε 2 ), where d is the dimension of the target system and ∆ max is the difference between the largest energy eigenvalue and the smallest energy eigenvalue.We show that the phase estimation based method performs better for large particle systems.
Various approaches of defining and determining work performed on a quantum system are compared. Any operational definition of work, however, must allow for two facts: first, that work characterizes a process rather than an instantaneous state of a system and, second, that quantum systems are sensitive to the interactions with a measurement apparatus. We compare different measurement scenarios on the basis of the resulting postmeasurement states and the according probabilities for finding a particular work value. In particular, we analyze a recently proposed work meter for the case of a Gaussian pointer state and compare it with the results obtained by two projective and, alternatively, two Gaussian measurements. In the limit of a strong effective measurement strength the work distribution of projective two energy measurements can be recovered. In the opposite limit the average of work becomes independent of any measurement. Yet the fluctuations about this value diverge. The performance of the work meter is illustrated by the example of a spin in a suddenly changing magnetic field.
Estimating thermal expectation values of observables is a fundamental task in quantum physics, quantum chemistry, and materials science. While recent quantum algorithms have enabled efficient quantum preparation of thermal states, observable estimation via sampling remains costly: a straightforward implementation separates successive measurements by a full mixing time in order to ensure samples are approximately independent. In this work, we show that the sampling cost can be substantially reduced by using a single Gibbs-sampling trajectory. After a single burn-in period, we interleave coherent measurements that satisfy detailed balance with respect to the target Gibbs state. The efficiency of this approach rests on the fact that, in many settings, the autocorrelation time can be significantly shorter than the mixing time. For energy estimation (and more generally for observables commuting with the Hamiltonian), we implement the required measurements using Gaussian-filtered quantum phase estimation with only logarithmic overhead. We also introduce a weighted operator Fourier transform technique to mitigate measurement-induced disturbance for general observables.
Various approaches of defining and determining work performed on a quantum system are compared. Any operational definition of work, however, must allow for two facts, first, that work characterizes a process rather than an instantaneous state of a system, and, second, that quantum systems are sensitive to the interactions with a measurement apparatus. We compare different measurement scenarios on the basis of the resulting post-measurement states and the according probabilities for finding a particular work value. In particular, we analyze a recently proposed work-meter for the case of a Gaussian pointer state and compare it with the results obtained by two projective and, alternatively, two Gaussian measurements. In the limit of a strong effective measurement strength the work distribution of projective two energy measurements can be recovered. In the opposite limit the average of work becomes independent of any measurement. Yet the fluctuations about this value diverge. The performance of the work-meter is illustrated by the example of a spin in a suddenly changing magnetic field.