That’s why Landauer was wrong
(and the physics of computing should be reconsidered)
Rolf Landauer (1927–1999) in 1961 published a paper that deeply influenced the computing community and put Shannon information theory in a new perspective (R. Landauer, Dissipation and heat generation in the computing process. IBM J. Res. Dev. 5, 183–191, 1961).
In this paper, Landauer introduced the notion of logical irreversibility, as applied to digital computing devices, by using the notion of information. Landauer writes:
“We shall call a device logically irreversible if the output of a device does not uniquely define the inputs.”
According to this definition, a computation device is ideally represented as an input–output device with some logic values entering the device and some logic values leaving the device.
The input and the output terminals being able to assume logic values can be characterized for the amount of information that they carry (see Figure 1). Typically, they carry one bit per terminal. As an example, in an ordinary OR gate we have two input terminals and one output terminal, thus we have two bits in and one bit out. In a situation like this, a necessary condition for assuring the logical reversibility is to have a number of inputs not larger than the number of outputs. An OR gate is a typical irreversible logic gate, as are AND, NAND and NOR. On the other hand, the NOT gate (one input and one output) is a logically reversible gate, because from the logic state of the output terminal is always possible to reconstruct the logic state of the input terminal.
In the same passage, Landauer writes:
“We believe that devices exhibiting logical irreversibility are essential to computing.”
In a sense they are, as far as we keep using traditional logic gates. However, as Landauer himself and, soon after, Charles Bennett (at that time at IBM) have shown, all the computation can also be made by perfectly reversible logic gates. Examples of these gates have been proposed by Edward Fredkin and Tommaso Toffoli in the early 80s of last century.
Finally, Landauer adds:
“Logical irreversibility, we believe, in turn implies physical irreversibility, and the latter is accompanied by dissipative effects.”
Undoubtedly, this is the most far-reaching result that Landauer is claiming. According to this statement, a logically irreversible device cannot be physically operated without spending energy, because dissipative effects are unavoidable. Landauer’s belief originates from the idea that the decrease of information between input and output, an operation that is sometime addressed as “information erasure”, is necessarily connected to a decrease of physical entropy in the physical system that embodies such a device.
If this statement is true, it might have relevant consequences for the development of future computers because it establishes a non-zero lower bound to how much energy needs to be dissipated during computation and thus constitutes a potential bottleneck for making more and more efficient computers.
The connection between logical irreversibility and physical irreversibility has been at the center of a very long and controversial debate, not free from misunderstandings and oversights. It is our opinion that, in this respect, Landauer was wrong: logical irreversibility does not imply physical irreversibility, and to make this point clear, we made an experiment, whose results have been published in 2016 (M.López Suárez, I.Neri, L.Gammaitoni, Sub-kBTmicro-electromechanical irreversible logic gate. Nat. Commun. 7, 12068, 2016). The experiment shows that a logically irreversible gate, made with mechanical cantilever, can be operated at room temperature with negligible dissipation and way below the expected Landauer’s limit.
Furthermore, we observe that the conceptual approach adopted by Landauer, although very popular in the information theory framework, has no direct connection with a typical physical model of the computing device, made according to the thermodynamics.
In fact, a computation is the result of a change of state of a given physical system, under the action of an external force (see Figure 2). According to Landauer, the degree of freedom bearing information is represented by the status Si of the system and by the input forces, while in a thermodynamic picture the change in entropy (that is a state variable) is uniquely related to the system itself and the forces are not accounting for any entropic contribution.
Although Shannon information and Gibbs entropy have the same formal aspect, they are not the same. The first is a mathematical quantity associated with the probability of a message, while the second is related to the probability of a microstate in a physical system. While the first has no connection with the physical nature of the message, the second is a direct consequence of the nature of the physical system itself. Thus, it seems natural to conclude that while Gibbs entropy is associated with the way the energy is exchanged according to the second principle of thermodynamics, Shannon information is not.
For further detail, please check: L. Gammaitoni, The Physics of Computing, Springer, 2021. ISBN: 978–3–030–87108–6
