Minimal energy cost of entanglement extraction

Lucas Hackl1,2 and Robert H. Jonsson3

1Max Planck Institute of Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
2Munich Center for Quantum Science and Technology, Schellingstraße 4, D-80799 München, Germany
3QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

We compute the minimal energy cost for extracting entanglement from the ground state of a bosonic or fermionic quadratic system. Specifically, we find the minimal energy increase in the system resulting from replacing an entangled pair of modes, sharing entanglement entropy $\Delta S$, by a product state, and we show how to construct modes achieving this minimal energy cost. Thus, we obtain a protocol independent lower bound on the extraction of pure state entanglement from quadratic systems. Due to their generality, our results apply to a large range of physical systems, as we discuss with examples.

Entanglement is one of the most exciting phenomena of quantum physics. If different parts of a system are entangled, they are intimately (cor)-related with each other. Even when they are far separated from each other, measurements on one part of the system can reveal the state of the other part. Over the last decades physicists came up with intriguing applications of entanglement, such as quantum teleportation: Here the full state of, e.g., a qubit gets transferred from place to another while only a single classical bit (0 or 1) of information is sent from A to B.
For such applications, the two parties need entangled quantum systems ready in their labs. So how can such entangled resources be obtained or created? One solution could be to extract the entanglement from other systems. Excitingly, the ground states of many physical systems (such as ultra-cold atoms or quantum fields) are highly entangled. However, extracting this entanglement to our laboratory is not for free but incurs an energy cost. This is because in the process we inevitably change the state of the system and raise its energy.
What our research addresses is how to find the cheapest way possible to extract a given amount of entanglement from such systems. It is like searching for the cheapest gas station in the city, just that we now buy entanglement and pay with energy. What we find is that extracting larger amounts of entanglement becomes increasingly expensive--as if the first gallon of gas only cost $1, but the second gallon already cost 2 and third even 4. This means if you only need to extract a little bit of entanglement, you can land a bargain, but if you need a lot it will be very expensive.
What makes our results appealing is that they are very general, applying to a wide range of different systems including cold atoms and quantum fields. Moreover, our findings are quantitative and concrete, which means we can tell you the exact cost of entanglement and even how you need to couple your lab system to the entangled system to pay this low price. And even if an experimenter cannot exactly follow our recommended procedure, due to experimental limitations, she can at least estimate how far away she is from the optimal price. This is useful in the same way as it is good to know the cheapest gas price in the neighborhood to judge if the gas station across the street offers a good, decent or bad deal.

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► References

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Cited by

[1] Ivan Romualdo, Lucas Hackl, and Nelson Yokomizo, "Entanglement production in the dynamical Casimir effect at parametric resonance", arXiv:1908.00835.

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