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Toward Physical Realizations of Thermodynamic Resource Theories

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Information and Interaction

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Abstract

“This is your arch-nemesis.” The thank-you slide of my resentation remained onscreen, and the question-and-answer session had begun.

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Notes

  1. 1.

    Conventional thermodynamics, developed during the 1800s, is arguably the first, most famous resource theory. But thermodynamics was cast in explicitly resource-theoretic terms only recently.

  2. 2.

    More precisely, the agent could extract the capacity to perform work. I omit the extra words for brevity.

  3. 3.

    Many TRT arguments rely on quasiclassicality as a simplifying assumption. After proving properties of quasiclassical systems, thermodynamic resource theorists attempt generalizations to coherent states.

  4. 4.

    \(D_0\) and \(D_\infty \) are called \(D_\mathrm{min}\) and \(D_\mathrm{max}\) in [32]. I follow the naming convention in [30]: if P denotes a d-element probability distribution and u denotes the uniform distribution \(( \frac{1}{d}, \ldots , \frac{1}{d} )\), then \(D_\infty (P ||u) = \log (d) - H_\infty (P )\), and \(D_0 (P ||u) = \log (d) - H_0 (P )\).

  5. 5.

    The authors discuss catalysis, the use of an ancilla to facilitate a transformation. Let \(R = (\rho , H_R)\) denote a state that cannot transform into \(S = (\sigma , H_S)\) by thermal operations: \(R \not \mapsto S\). Some catalyst \(C = (\xi , H_C)\) might satisfy \((\rho \otimes \xi , H_R + H_C) \mapsto (\sigma \otimes \xi , H_S + H_C)\). Catalysts act like engines used to extract work from a pair of heat baths. Engines degrade, so a realistic transformation might yield \(\sigma \otimes \tilde{\xi }\), wherein \(\tilde{\xi }\) resembles \(\xi \). For certain definitions of “resembles,” the agent can extract arbitrary amounts of work by negligibly degrading C. Brandão et al. quantify this extraction in terms of the dimension \(\mathrm{dim}( \mathscr {H}_C )\) of the Hilbert space \(\mathscr {H}_C\) on which \(\xi \) is defined. The more particles the catalyst contains, the greater the \(\mathrm{dim}( \mathscr {H}_C )\). Such one-shot results depend on the number of particles in the system represented by “one copy” of C.

  6. 6.

    The functional form of \(W_\mathrm{cost}^{\varepsilon }(R)\) is derived in [32, Suppl. Note 4]. The proof relies on Theorem 2. The proof of Theorem 2 specifies how a TRT agent can perform an arbitrary free unitary. Hence the thermal operation that generates a \(\tilde{R}\) is described indirectly.

  7. 7.

    Other goals include the mathematical isolation, quantification, and characterization of single physical quantities. Want to learn how entanglement empowers you to create more states and to perform more operations than accessible with only separable states? Use a resource theory for entanglement. Want to learn how accessing information empowers you? Use the resource theory for information [30, 34].

  8. 8.

    Such experimentalists value coherence similarly to TRT agents: in experiments, coherent entangled states offer access to never-before-probed physics. In quantum computers, entanglement speeds up calculations. TRT agents value coherence because they can catalyze transformations with coherent states [2]. Agents can also “unlock” work from coherence [43].

  9. 9.

    One might worry that \(\mathscr {M}\) might not occupy a quasiclassical state after coupling to \(\mathscr {S}\). But \(\mathscr {M}\) has a totally degenerate Hamiltonian \(\mathbbm {1}_d\). If \(\rho _M\) has coherences relative to the energy eigenbasis, free unitaries can eliminate them [30].

  10. 10.

    One might object that the measurement could project the memory’s quantum state onto a pure state. Transforming any pure state into \(\left| 0 \right\rangle \) costs no work: the transforming unitary commutes with the Hamiltonian \(\mathbbm {1}_d\) and so is free [30]. But the agent could be fined the work that one would need to erase \(\mathscr {M}\) if one refrained from measuring \(\mathscr {M}\). Imagine that a “measurement bank” implements measurements: the agent hands \(\mathscr {M}\) to a teller. Depending on the memory’s state, the teller and agent agree on a fee, which the agent pays with a charged battery. The teller measures \(\mathscr {M}\); announces the outcome; resets \(\mathscr {M}\); stores the battery’s work contents in a vault; and returns the reset memory \(( |0 \rangle \!\langle 0 |, \mathbbm {1}_d)\) and the empty battery \((|0 \rangle \!\langle 0 |, W |W \rangle \!\langle W |)\) to the agent.

  11. 11.

    Which \(\mathscr {M}^\gamma \) achieves the maximum follows from the forms of R and \(\tilde{R}\). The experimentalist chooses the form of R. The form of the \(\tilde{R}\) produced by the thermal operation is described in [32, Suppl. Note 4]. (I have assumed that the \(\tilde{R}\) produced by the thermal operation is the state produced in the experiment. But no experiment realizes a theoretical model exactly. This discrepancy should be incorporated into calculations of errors.).

  12. 12.

    I have ignored limitations on the experimentalist. For example, I have imagined that \(\mathscr {E}\) can be implemented infinitely many times and that infinitely many copies of \(\rho \) can be prepared. They cannot. These limitations should be incorporated into the error associated with \(\varDelta \).

  13. 13.

    Granted, one-shot information theory describes the simultaneous processing of finite numbers of copies of a probability distribution or quantum state. The finite numbers referred to above are numbers of sequential trials. The TRT manifestation of \(\varepsilon \) does not contradict one-shot information theory. Yet the former contradicts the spirit of the latter.

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Acknowledgements

I am grateful to Fernando Brandão, Lídia del Rio, Ian Durham, Manuel Endres, Tobias Fritz, Alexey Gorshkov, Christopher Jarzynski, David Jennings, Matteo Lostaglio, Evgeny Mozgunov, Varun Narasimhachar, Nelly Ng, John Preskill, Renato Renner, Dean Rickles, Jim Slinkman, Stephanie Wehner, and Mischa Woods for conversations and feedback. This research was supported by an IQIM Fellowship, NSF grant PHY-0803371, and a Virginia Gilloon Fellowship. The Institute for Quantum Information and Matter (IQIM) is an NSF Physics Frontiers Center supported by the Gordon and Betty Moore Foundation. Stephanie Wehner and QuTech offered hospitality at TU Delft during the preparation of this manuscript. I am grateful to Ian Durham and Dean Rickles for soliciting this paper. Finally, I thank that seminar participant for galvanizing this exploration.

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  1. Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, 91125, USA

    Nicole Yunger Halpern

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  1. Physics Department, Saint Anselm College, Manchester, New Hampshire, USA

    Ian T. Durham

  2. Unit for History and Philosophy of Science, University of Sydney, Carslaw, Australia

    Dean Rickles

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Yunger Halpern, N. (2017). Toward Physical Realizations of Thermodynamic Resource Theories. In: Durham, I., Rickles, D. (eds) Information and Interaction. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-43760-6_8

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