Does entanglement have a weight? This might seems a very strange question. Entanglement is just correlations, what does it mean that it can have a weight? Entanglement in a quantum state can be prepared, detected, used, and this about it. Not only the theory but overwhelming experimental evidence seem to indicate that weight and entanglement do not have much to do with each other.
In the past few years, advances in quantum thermodynamics have shown that work can be stored in the quantum correlations of a state. A thermal state is the most entropic state one can prepare given a fixed amount of energy. However, with the same amount of energy, one can prepare a state with some entanglement, which can be used in clever protocols to perform for, i.e., to "store energy". If this is true, entanglement should have a weight, since we have known for more than a century that energy has a weight.
I have worked on answering a preliminary question, which could set a challenge and a direction of research : understanding which type of entanglement weighs, and why.
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