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2019 | OriginalPaper | Chapter

Conditional Erasure and the Landauer Limit

Author : Neal G. Anderson

Published in: Energy Limits in Computation

Publisher: Springer International Publishing

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Abstract

Quantum-dynamical proofs of dissipation bounds for Landauer erasure are presented, with emphasis on the crucial connection between conditioning in erasure protocols and fundamental limits on erasure costs. Bounds on erasure costs for conditional and unconditional erasure protocols are shown to follow from a very general and ecumenical physical description of Landauer erasure, a straightforward accounting of its energetic cost, refined definitions of what it means for physical system states to bear known and unknown data, and a transparent application of quantum dynamics and entropic inequalities. These results generalize and support the results of Landauer and Bennett for unconditional and conditional erasure, respectively, and do so using a theoretical methodology that sidesteps or otherwise withstands methodological objections that have been leveled against thermodynamic proofs and other theoretical arguments in the literature. The dissipation bounds obtained here coincide with those obtained elsewhere from an even more general approach that is based on a thoroughly physical conception of information and that clearly distinguishes information from entropy. This connection may help to clarify central issues in the debate over Landauer’s Principle, since the more general approach bounds dissipative costs of irreversible information loss in a range of scenarios that are both broader and less idealized than those typically considered in explorations of the Landauer limit.

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Appendix
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Footnotes
1
See the Appendix, where established properties of von Neumann entropy, unitary transformations, and trace operations used in this work are cataloged for convenience.
 
2
An agent or automaton without an existing record of the pre-erasure system state can create one through measurement as part of a more complex conditional erasure protocol, since the N pre-erasure states are mutually orthogonal and thus distinguishable from one another. Formal treatment of any such protocol would, however, have to include an additional subsystem that registers measurement outcomes and account for physical costs associated with the measurement process (e.g. creation of system/apparatus correlations and/or erasure or overwriting of the measurement outcomes on each use). This is discussed further in Sect. 4.
 
3
We adopt Orlov’s [29] labeling of erasure protocols that do and do not utilize records of pre-erasure system states as erase with copy and erase without copy protocols, respectively. Here, with and without will generally mean “conditioned upon” and “not conditioned upon,” so with copy means both “in the presence of” and “with active use of” a copy or record. Note the physical state serving as an external record need not be an identical copy of the pre-erasure state of \(\mathcal {S}\); it need only provide an unambiguous physical identifier of which encoding state \(\hat {\rho }_i^{\mathcal {S}}\) the system \(\mathcal {S}\) is in prior to erasure.
 
4
That Landauer himself regarded Bennett’s claim as a “friendly amendment” to his erasure principle is evident in [34].
 
5
The “erasing agent”, which we will also call the “operator”, is the entity (organism or automaton) tasked with executing a state-reset protocols. This entity is the “knower” of a data-bearing system’s preparation in the case of known data.
 
6
We will refer to Bennett’s Szilard-engine examples solely to reveal similarities and differences in various notions used by Bennett and by us, and should not be taken to imply that any results of this paper depend in any way upon results obtained for Szilard engines, the applicability of classical thermodynamics to Szilard engines, or any assertion that microscopic realization of a Szilard engine could operate in the face of thermal fluctuations.
 
7
Conditioning on the sidedness does not require erasing agent knowledge of the sidedness, provided that the operation—applied “blindly” by the agent—acts on both the system and the copy in a manner that conditions the operation performed on the system.
 
8
Note that reversible erasure of unknown data with a copy requires operations, applied blindly by the erasing agent, that access the copy without destroying it. See Sect. 4 and [36].
 
9
One would be similarly mistaken to see a violation of the Landauer limit in the nanomechanical OR gate recently reported by Lopez-Suarez, Neri, and Gammaitoni [37]. They report that, in their experiment, two distinct configurations of electrode charges (corresponding to two different binary input combinations “01” and “10”) similarly displace the position state of a nearby nanopillar tip (same logical output) with energy dissipation less than that Landauer limit. If this does indeed correspond to implementation of a logically irreversible (sub-)function, as the authors claim, there is no violation of the Landauer limit. Regarding the nanopillar as the “the system” in this scenario, not including the electrode tips that must remain charged to hold the nanopillar tip in its “merged position”, the “state merging” protocol is conditional and the Landauer-Bennett limit (not the Landauer limit) would apply. No Landauer cost would be expected. That a different nanopillar tip position results for electrode charges corresponding to input “11”—which should yield the same physical output as inputs “01” and “10” in a faithful physical implementation of an OR gate—is a separate worry.
 
10
While Norton has separate objections to the Szilard engine as an idealization, he accepts that “it is taken to capture the essential thermodynamic features of a more realistic one-bit memory device in a heat bath” [16] and states his thermodynamic objections in this spirit.
 
11
Phase-volume arguments are themselves artifacts of the idealization of an absolutely impenetrable partition. An arbitrarily small pinhole in the partition—a pinhole so small that only one molecule could fit through it and would almost never do so—is enough to give the molecule access to the full phase volume even when it is (temporarily) trapped on one side or the other by the partition. The presence of such a pinhole would be sufficient to undermine phase volume arguments even when the operation over finite time scales is unaffected, and any quasi-static treatment that would require the molecule to always be in equilibrium with its surroundings. With the pinhole present, a particle trapped on one side of the partition would end up on either side with equal probability when it has fully equilibrated. The same is true for a quantum particle in a symmetric double potential well with a high but finite potential barrier in the center and interacting with a heat bath. Any particle state initially localized in one well or the other—necessarily a non-equilibrium state—will ultimately equilibrate to a thermal state that does not favor occupation of either side of the chamber by the molecule.
 
12
This terminology is due to Ladyman [38].
 
13
For example, in discussing an inequality presented as an inviolable fundamental bound with no claim of achievability, the authors of [25] state that “It should be emphasized that the greater-than-or-equal-to sign—rather than a greater than sign—is very important because the equality must represent a physical possibility, at least at the conceptual level.”
 
14
Norton questions whether it is possible to express limits to computation in a “simple, sharp, principled expression(s) [16].
 
15
Recall from Sect. 3.3 that, with the way we have defined unknown data in this work, unconditional erasure of both known and unknown data carry the Landauer cost.
 
16
This inequality appears in [32] as Δ(Sb − βUb) ≤ 0, where ΔSb, ΔUb, and β are denoted here as \(\varDelta S^{\mathcal {E}}\), \(\varDelta \langle E^{\mathcal {E}}\rangle \) and (kB T)−1 but carry the same meanings. The factor of \(\ln (2)\) accounts for the differences in the base of the logarithm used to define von Neumann entropy by Partovi and ourselves; Partovi’s inequality is based on the thermodynamic definition \(S(\hat {\rho })=-Tr[\hat {\rho }\ln \hat {\rho }]\), which we have reexpressed here in terms of the information-theoretic definition \(S(\hat {\rho })=-Tr[\hat {\rho }\log _2 \hat {\rho }]\).
 
17
See, for example, Theorem 11.8 of Ref. [51].
 
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Metadata
Title
Conditional Erasure and the Landauer Limit
Author
Neal G. Anderson
Copyright Year
2019
DOI
https://doi.org/10.1007/978-3-319-93458-7_2