# A Collective Measurement To Estimate Work In Quantum Systems

What is work in thermodynamics? This basic question is still debated nowadays, especially when dealing with small quantum systems. In 1824, Sadi Carnot defined work as useful energy which can be used to e.g. lift a weight in his seminal paper *Reflections on the Motive Power of Fire*. In turn, a common way to generate work is by performing a cycle in a heat engine operating between two thermal reservoirs at different temperatures. When the size of such an engine is small (e.g. in the context of biological systems), different work outcomes can be obtained even when exactly the same cycle is performed. This is due to thermal fluctuations, which are negligible in large macroscopic systems but play a crucial role in the microscopic regime.

To give a meaningful definition of work for such small systems, it is convenient to define work as a stochastic quantity: instead of associating work with a single number, one associates a probability distribution with different possible work outcomes given a certain thermodynamic process.

When dealing with quantum systems, the notion of work is challenged again. This is essentially due to the phenomena of quantum superposition: quantum systems can behave “as if” they were in more than one state at the same time, leading to the puzzling phenomena of quantum interference. It is not difficult to imagine that this might make the notion of (stochastic) work “blurry,” as several different energies can potentially coexist simultaneously.

The natural way out of this situation is to perform a measurement: the postulates of quantum mechanics allow us to assign a probability to the possible outcomes of the magnitude being observed. However, it is challenging to find a measurement for work: this is because work is not associated to a single-time quantity (such as the position or the energy), but rather to a process, e.g., a cycle of a heat engine. Not surprisingly, there is not a unique way of measuring a “process-dependent” quantity; compare, for example, a continuously measured system with a system that is only measured at the endpoints of the process. As a consequence, different measurement schemes might lead to different quantum work distributions, which has lead to controversy and a rich scientific debate in the last years.

The most standard and widely used approach to define quantum work is the two energy measurement approach. Essentially, it consists of performing an energy measurement at the beginning and at the end of the process, and work is given by the difference of outcomes of such measurements. This approach has proven to be widely successful, in particular by extending the celebrated fluctuation theorems to the quantum regime. There are, however, scenarios where this approach might not be desirable due to the back-action that the first measurement induces, which destroys the quantum coherence that the initial state might possess.

Back-action is ubiquitous in quantum measurements: unlike classical systems, quantum systems are perturbed by the very act of observing them. To minimize such a back-action, several strategies can be pursued. One is weak measurements, which disturb the state very little at the price of also obtaining little information about it. Another conceptually different approach is global measurements; that is, measurements that simultaneously act on several copies of the system. It seems rather natural to imagine that, by processing more than one copy, more information can be obtained while reducing the back-action induced on each copy individually. This appears particularly relevant in “process-dependent” quantities, such as work, as a different measurement can be carried out on different copies at different instances of time.

In a recent article, a collective measurement has been suggested for estimating quantum work. The advantage of this new approach is that it reproduces exactly the results of the two energy measurement approach for initial states with no coherence (i.e., in which the two energy measurement approach has no back-action) while reducing the back-action in the presence of coherence.

Recently, a group from the University of Science and Technology of China has conducted the collective measurement scheme experimentally for the first time. It has been deterministically realized in an all-optical setup. The core idea of the experiment is to encode the first (second) copy into the path (polarisation) degree of freedom of a single photon. After the preparation of the two-qubit states, they are fed in the desired collective measurement. A comparative experiment is also performed, for simulating the traditional two-projective-energy-measurement scheme. Both measurement schemes are based on linear optical technology.

The experimental results show that the collective measurement scheme can reduce the measurement back-action by yielding transition probabilities that are closer to the unmeasured evolution. These results are expected to stimulate new conceptual and technological developments in quantum thermodynamics and quantum information science, where collective measurement plays an important role in numerous tasks.

These findings have been published in an article entitled Experimentally reducing the quantum measurement back action in work distributions by a collective measurement, recently published in *Science Advances*.