Gaussian work extraction from random Gaussian states is nearly impossible

Quantum thermodynamics can be naturally phrased as a theory of quantum state transformation and energy exchange for small-scale quantum systems undergoing thermodynamical processes, thereby making the resource theoretical approach very well suited. A key resource in thermodynamics is the extractable work, forming the backbone of practical thermal engines. Therefore, it is of utmost importance to characterize quantum states based on their ability to be used as a source of work. From a near term perspective, quantum optical setups turn out to be ideal testbeds for quantum thermodynamics, so it is essential to assess work extraction from quantum optical states. Here, we show that Gaussian states are typically useless for Gaussian work extraction. More specifically, by exploiting the concentration of measure phenomenon, we prove that the probability that the Gaussian extractable work from an energy-bounded zero-mean multimode random Gaussian state is nonzero is exponentially small. This result can be thought of as an epsilon-no-go theorem for work extraction from Gaussian states under Gaussian unitaries, thereby revealing a fundamental limitation on the thermodynamical usefulness of Gaussian components in the quantum regime.

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