2.1. The Setup

Von Neumann begins by considering N noninteracting, quantum systems, denoted by S₁, …, S_N. For the purpose of this article, we take these systems to be two-state quantum systems and only consider, say, the spin states of a spin half particle.Footnote ³ Each of these systems is placed in a box K_i, with $i=1$ , … N, whose walls shield its contained system off from its environment and thus prevent any interaction between systems. The boxes K₁, …, K_N are now all placed into another much larger box $\overline{K}$ of volume V. For simplification, von Neumann assumes that there are no force fields, in particular no gravitational fields, present in $\overline{K}$ . This in turn means that there is no gravitational interaction between the boxes K₁, …, K_N, even though they may exchange kinetic energy via collisions. The boxes can thus be taken to behave just like molecules of a gas. Von Neumann calls this ‘quantum gas’ a [S₁, …, S_N] gas, where [S₁, …, S_N] is the statistical ensemble associated with the systems. A more detailed analysis of von Neumann’s understanding of such an ensemble, or Gesamtheit, will be presented in the appendix. The large box $\overline{K}$ can be brought into thermal contact with a heat bath at temperature T, and in that case, after some equilibration time, the [S₁, …, S_N] gas itself will be at temperature T.Footnote ⁴ Finally, we add an empty box ${\overline{K}}^{\prime }$ of equal volume V to the right of $\overline{K}$ .

2.2. The Process

The [S₁, …, S_N] gas now undergoes a series of transitions. The following various stages are also illustrated in figure 1.Footnote ⁵

Figure 1.

Illustration of von Neumann’s argument. (I) Individual two-state quantum systems are each in state $|0〉=w_1|+〉+w_2|-〉$ , indicated by bicolored circles. For illustrative purposes, we assume $w_1=w_2=1/\sqrt{2}$ . (II) After the measurement in the |+〉/|−〉 basis, the system is now described by a statistical mixture of |+〉 (white) and |−〉 systems (black). (III) The |+〉 and |−〉 systems are now separated by using semipermeable membranes to ‘push’ the |−〉 systems into the right box $\overline{K}$ . (IV) The |+〉 and |−〉 systems are now completely separated. (V) The two boxes are compressed to half of their respective volumes. (VI) The |+〉 and |−〉 gases are now transformed back into their original state of superposition. (VII) The partition is removed. No heat bath is present, but we assume that it exists in the background and takes up the dissipated entropy at 4 → 5.

Von Neumann now argues as follows: all transitions between stages II and VII take place in a reversible fashion, which means that the total entropy change of gas and heat bath between II and VII must be zero. Since a total amount of $\Delta S={\mathrm{Nk}}_B\left[w_1^2\ln w_1^2+w_2^2\ln w_2^2\right]$ has been dumped into the heat bath during the compression stage, and since the (normed) entropy of the final |0〉 gas is zero by definition, the entropy of the gas must have been $S=S_{+}+S_{-}=-{\mathrm{Nk}}_B\left[w_1^2\ln w_1^2+w_2^2\ln w_2^2\right]$ before. Later in his book, von Neumann explains that the measurement process (‘Prozess 1’) is responsible for the entropy increase between stages I and II (Reference von Neumann1996, 202–6).

The above considerations are easily generalized to more dimensions. For a system described by a density matrix ρ with eigenvectors |ϕ ₁〉, …, |ϕ _n〉 and eigenvalues w ₁, …, w_n, the entropy is then given by $S_{\rho }=-\mathrm{Tr}\rho \ln \rho =-{\sum }_{i=1}^n{w}_i\ln w_i$ .

5.1. Redundancy of the Step 4 Location Measurement

I begin the discussion with some general observations, in order to provide more clarity. For this, we recall that according to Hemmo and Shenker, the only difference between their and von Neumann’s original thought experiment is that a further measurement, a location measurement (step 4), is needed to determine the molecule’s location before the compression stage. For gases at the thermodynamic limit, this measurement becomes redundant, since the amount of molecules on each side of the box becomes proportional to their respective occupying volume. Not so for single molecules, for which, before the empty side of the box can be compressed (with probability 1), a location measurement is required in order to determine which side the particle is on.

Contrary to Hemmo and Shenker’s assertions, however, the location measurement during step 4 is not needed. A spin z measurement already took place at step 2, and the outcome of this measurement will be fully correlated with the position of the particle after the separation in step 3. And so instead of introducing yet another auxiliary system that performs a location measurement on the particle, it would have been sufficient to read out the measurement result of the spin z measurement.

In the case of collapse, for example, the particle will have already collapsed into a spin eigenstate during the step 2 measurement. The correlations established during the location measurement will thereby all be classical, and reading out the spin measurement result is sufficient to predict the particle’s location after the separation. In the case of no-collapse interpretations, the system and (spin) measurement apparatus become entangled during step 2:

(6)$\begin{array}{}& |\Psi {〉}^{\left(2\right)}=\frac{1}{\sqrt{2}}\left(\left|{+}_z{〉}_P\right|+{〉}_M+\left|{-}_z{〉}_P\right|{-}_M\right)|L{〉}_P\mathrm{.}\end{array}$

During the separation in step 3, the particle’s spatial degree of freedom becomes entangled with its spin degree of freedom. This means that the state of the overall system is

(7)$\begin{array}{}& |\Psi {〉}^{\left(3\right)}=\frac{1}{\sqrt{2}}\left(\left|{+}_z{〉}_P\right|+{〉}_M|L{〉}_P+|{-}_z{〉}_P|-{〉}_M|R{〉}_P\right)\text{.}\end{array}$

Therefore, for both collapse and no-collapse cases, it is in fact sufficient to read out the measurement result of the step 2 spin measurement in order to determine the location of the particle after the separation process.

Having two instead of one measurement would not be much of a problem, if it were not the case that in Hemmo and Shenker the two measurements have different consequences for the von Neumann entropy. This is the first inconsistency in their argument: whereas Hemmo and Shenker agree that after the spin z measurement at step 2 the postspin measurement density matrix of the particle is given by

(8)$\begin{array}{}& {\rho }^{(2,\mathrm{red})}=\frac{1}{2}\left(|{+}_z〉〈{+}_z{|}_P+|{-}_z〉〈{-}_z{|}_P\right)|L〉〈L{|}_P\text{,}\end{array}$

they do not apply the same reasoning to the postlocation measurement state of the system at step 4. Instead, they use a ‘collapsed’ density matrix to calculate the von Neumann entropy:

(9)$\begin{array}{}& {\rho }^{(4,+)}=|{+}_z〉〈{+}_z{|}_P|L〉〈L{|}_P\text{ or }{\rho }^{(4,-)}=|{-}_z〉〈{-}_z{|}_P|R〉〈R{|}_P\text{.}\end{array}$

In the first case, the (spin) measurement has therefore increased the entropy, whereas in the second case the (location) measurement has effectively reduced it. What is going on?

Let me first try to assemble what the authors themselves could have had in mind. In von Neumann’s original argument, the step 2 spin measurement is nonselective,Footnote ⁸ which means that even if the system has de facto collapsed into one of its eigenstates, an external agent would not be able to determine into which state the system has collapsed and would therefore describe the system by a density matrix ρ ^(2,red).Footnote ⁹ The system would be in a so-called proper mixture, meaning that it is possible to understand ρ as representing a probability distribution over pure states.Footnote ¹⁰ The von Neumann entropy of the system at step 2 has thereby increased compared to its previous state, in agreement with what von Neumann considers as being the irreversibility of a ‘Prozess 1’.

The step 4 location measurement, however, is selective—it establishes correlations between the agent who performs the measurement and the system.Footnote ¹¹ These correlations then allow the agent to perform further operations on the system, such as the step 5 compression of the box. For Hemmo and Shenker, the von Neumann entropy of the system at step 4 has therefore decreased from step 3.

Since Hemmo and Shenker want to include the nonselective spin measurement at step 2, the step 4 location measurement is indeed a necessary requirement for the single-particle case, given that the compression is not being allowed to be conditional on the outcome of the step 2 measurement. Without the selective measurement, the work-free compression against vacuum could not take place. The limiting case of infinite particles (and in fact von Neumann’s original account) does not require this selective measurement since the amount of particles within each chamber of the box becomes equal.

The problem with including a second measurement on the system is that this second measurement also introduces a second measurement apparatus. It will be shown shortly that Hemmo and Shenker’s conclusion is based on an erroneous calculation of the von Neumann entropy when the system is correlated to this second measurement apparatus. Before elaborating on this point, however, I would like to discuss another shortcoming of their argument.

5.2. Violation of the Second Law

Hemmo and Shenker notably claim that the thermodynamic entropy change is zero throughout the whole cycle and, in particular, that at the end of the cycle “the measuring device [can be returned to its initial ready state] unitarily” (Reference Hemmo and Shenker2006, 161) and, hence, without any heat cost. To appreciate the consequences of this claim, let us assume that it is indeed possible to unitarily bring the measurement apparatus back to its original position without a compensating heat transfer into the environment, as Hemmo and Shenker claim. We may then construct a slightly amended version of their proposed cycle. For this amended version, the only thing we change is step 5, which instead of being a compression we turn into an isothermal expansion. This means that instead of compressing the empty side against the vacuum, we let the particle push against the partition in a quasi-static, isothermal fashion. Given that the position of the particle is ‘known’ as a result of the location measurement, it is possible to attach a weight to the partition, thereby extracting kTln2 units of work from the system during the expansion, while the according amount of heat is delivered from the heat reservoir. After the work extraction, the measurement apparatus is brought back to its initial state (which according to Hemmo and Shenker can be done for free). The partition is then reinserted into the original system (for free), the position of the particle is measured again (for free), and the above process is repeated, thereby extracting arbitrarily large amounts of work from this one-particle engine with the sole effect being that heat is extracted from a single reservoir. This constitutes a direct violation of the Kelvin-Planck statement of the second law (Planck Reference Planck1991).

Von Neumann himself, as the authors acknowledge, states that “in the sense of phenomenological thermodynamics, each conceivable process constitutes valid evidence, provided that it does not conflict with the two fundamental laws of thermodynamics” (Reference von Neumann1996, 192; trans. Reference von Neumann and Beyer1955, 359).Footnote ¹² And so at this point, one might already conclude that Hemmo and Shenker’s argument fails, as their suggested unitary reset does conflict with one of the fundamental laws of thermodynamics.

It should be noted, however, that there exists some controversy in the literature about whether we ought to expect the second law to hold in the single-particle case and whether thermodynamic considerations in these cases are indeed acceptable (Maxwell Reference Maxwell1878; Earman and Norton Reference Earman and Norton1999; Hemmo and Shenker Reference Hemmo and Shenker2012; Norton Reference Norton2013).Footnote ¹³ In fact Hemmo and Shenker elsewhere have expressed skepticism on whether the second law holds in such cases (Hemmo and Shenker Reference Hemmo and Shenker2012). While it seems reasonable to assume that in the article under consideration the second law is required to hold (after all, the article is about comparing the thermodynamic entropy with the von Neumann entropy, and the authors explicitly state that they “do not address this issue” of skepticism here; Hemmo and Shenker Reference Hemmo and Shenker2006, 158), it is worth emphasizing that the above violation is a reliable violation of the second law. This means that even if one is willing to sacrifice the strict second law that states that entropy should never decrease, and instead adapts either a statistical or a probabilistic version of the second law,Footnote ¹⁴ the problem with the above case is that we are confronted with a violation that allows us to extract work from the heat reservoir 100% of the time. It allows us to extract work reliably and continuously and, so, is a particularly serious version of a Maxwell’s demon.

A common strategy to recover the second law from situations like the one presented above is to invoke Landauer’s principle (Landauer Reference Landauer1961; Bennett Reference Bennett1973). It states that there is a heat cost involved with the resetting of the measurement apparatus to its initial state, and it is widely accepted in the physics community.Footnote ¹⁵ For the case presented above, the resetting of the memory cell would then lead to an increase of heat in the environment, thereby offsetting the previous entropy decrease.Footnote ¹⁶

And indeed, this is what is required in the current case, for contrary to Hemmo and Shenker’s assertion, a unitary reset of the measurement device is not possible. To see this, we consider the end of the cycle. The measurement device then is in one of the two mutually exclusive states |−〉_M or |+〉_M. As can be easily seen, there then exists no unitary operator that reliably maps the memory cell back to its initial, ready state ${\left\{\right|-〉}_M,|+{〉}_M\}\mapsto |\mathrm{ready}{〉}_M$ . The only way to reset the measurement device unitarily is if one recorded beforehand which one of the two mutually exclusive states the device is in. To do so, however, one requires a measurement on the measurement device itself, performed by a second measurement device. But then one would want to reset this second measurement device unitarily, too, for which a third device would be needed, and so on. Eventually, one would run out of resources, and a Landauer type reset, at a cost, becomes unavoidable.

Which Entropy?—Notwithstanding the above criticism, Hemmo and Shenker’s main point is that the von Neumann entropy (as opposed to the thermodynamic entropy) during the step 4 location measurement decreases and that this gives us reason to reject that their conceptual equivalence still stands, or seems to. “As a result of the location measurement, the von Neumann entropy decreases back to its original value” (Hemmo and Shenker Reference Hemmo and Shenker2006, 163).

In this section I discuss this claim, and in particular I show that

1. i) The physically relevant entity is the joint entropy of system and measurement apparatus. It remains the same.

2. ii) It is the system’s so-called conditional entropy that decreases during the measurement, not the system’s marginal von Neumann entropy.

I begin with a justification of the two claims. In phenomenological thermodynamics, the joint entropy of two systems is always the sum of their respective entropies. The von Neumann (in the classical case, the Gibbs) entropy, however, is generally subadditive and additive only in the absence of correlations between two systems:

(10)$\begin{array}{}& H\left(S,M\right)\le H\left(S\right)+H\left(M\right)\text{,}\end{array}$

where S stands for ‘system’ and M stands for ‘measurement apparatus’ or ‘memory cell’. In the concrete case of the step 4 location measurement, we can model the measurement apparatus M as a box containing a single molecule divided by a partition. It can then be in one of two mutually exclusive states, corresponding to the position of the molecule, left (l) or right (r). We assume it needs to be in a ‘ready’ state before the measurement, which we chose to be l. The entropy is zero, in this case. If we consider the case of collapse, then at the time of the measurement the system will have already collapsed into a spin eigenstate. The correlations between the location of the system and the measurement apparatus are then all essentially classical, and the von Neumann entropy before the measurement can be rewritten as

(11)$\begin{array}{}& H\left(S\right)=-k_B\sum _{s=l,r}p\left(s\right)\log p\left(s\right)\text{,}\end{array}$

where p(s) is the probability of the system being in the left or right chamber of the box.

Before the step 4 location measurement, system and measurement apparatus are not correlated, and their joint entropy is given by $H_3(S,M)=H_3\left(S\right)+H_3\left(M\right)$ , where 3 is taken to denote ‘stage 3’, or, in other words, ‘before the step 4 measurement’. During the measurement, the memory cell will align itself with the position of the particle, and the two systems become correlated. The joint entropy now cannot be expressed anymore as the sum of the individual entropies and instead becomes

(12)$\begin{array}{}& H_4(S,M)=H_4\left(S\right|M)+H_4(M)\le H_4(S)+H_4(M)\text{,}\end{array}$

with H(S|M) being the so-called conditional entropy, which quantifies how much S is correlated with M and which is given by

(13)$\begin{array}{}& H\left(S\right|M)& =-k_B\sum _{s,m}p(s,m)\ln p\left(s\right|m)\text{(14)}& & =-k_B\sum _{m}p\left(m\right)\sum _{s}p\left(s\right|m\left)\ln p\right(s\left|m\right)\text{(15)}& & =k_B\sum _{m}p\left(m\right)H\left(s\right)\text{.}\end{array}$

Here p(s) and p(m) are the probabilities that system and memory cell are found in macrostate $s=l_s$ , r_s or $m=l_m$ , r_m respectively, p(m, s) is their joint probability, and p(s|m) the conditional probability, with $H\left(s\right)=-k_B{\sum }_{s}p\left(s\right|m\left)\ln p\right(s\left|m\right)$ . The conditional entropy is nonnegative and is maximal when the system and measurement apparatus are uncorrelated, $0\le H\left(S\right|M)\le H(S)$ , in which case equation (12) reduces to equation (10). It is often considered to be the entropy relative to an agent (in this case the measurement apparatus).

Let us now go back to Hemmo and Shenker’s claim that the von Neumann entropy of the system decreases during the location measurement. Does it? The answer is no. What decreases, however, is the conditional entropy relative to the measurement apparatus:

(16)$\begin{array}{}& H_3\left(S\right|M)\ge H_4(S\left|M\right)\text{.}\end{array}$

It reduces to zero because system and measurement apparatus become perfectly correlated during the measurement. And so when Hemmo and Shenker claim that the system’s entropy has decreased, what they mean is that the system’s conditional entropy has decreased. But the conditional entropy is distinct from the marginal entropy. Rewriting the joint entropy of system and measurement apparatus demonstrates this:

(17)$\begin{array}{}& H_4(S,M)=H_4\left(S\right|M)+H_4(M)=H_4(M\left|S\right)+H_4\left(S\right)\text{.}\end{array}$

As opposed to phenomenological thermodynamics, which treats systems as black boxes, (classical and quantum) statistical mechanics is able to detect correlations between subsystems, allowing us to mathematically handle the concept of ‘measurement’ in the first place. If we associate entropy with the potential to (reliably) extract work from a system, then the conditional entropy quantifies this ability to a certain extent: a memory cell endowed with an automaton would now be able to (reliably) extract work from the system by allowing it to isothermally expand into the other half of the box, thereby raising a weight, contrary to an external agent who is not correlated to the particle location. But this is just the ordinary Maxwell’s demon scenario applied to a one-particle setting.Footnote ¹⁷

What becomes important for thermodynamic treatments in such a setting is the joint entropy of system and measurement apparatus, as the joint system (ideally) has no correlations with the outside and can thus be treated as a thermodynamic black box. And it turns out that the behavior of the thermodynamic entropy of the joint system is exactly mirrored by the behavior of the von Neumann entropy. The joint entropy of system and measurement apparatus does not change during the location measurement but remains the same:

(18)$\begin{array}{}& H_3(S,M)=H_4(S,M)\text{.}\end{array}$

And so, to summarize the above: all that changes during the location measurement is the conditional entropy, not the joint entropy of the system or the marginal entropy H(S). Furthermore, the joint entropy, just like the thermodynamic entropy of the joint system, remains the same during the location measurement.

5.3. No-Collapse Scenarios

Let us now consider the case of no-collapse scenarios. In no-collapse scenarios, following the measurement in step 4, the measurement apparatus and the location degree of freedom become entangled. The reduced density matrix after tracing out the decohering environment therefore is given by an improper mixture due to the neglect of the correlations with the environment. After the step 4 measurement, the density matrix of the combined system and measurement apparatus is given by

(19)$\begin{array}{}& {\rho }^{(4,P+M)}=\frac{1}{2}\left(|{+}_z〉〈{+}_z{|}_P|L〉〈L{|}_P|L〉〈L{|}_M+|{-}_z〉〈{-}_z{|}_P|R〉〈R{|}_P|R〉〈R{|}_M\right)\text{,}\end{array}$

where now |L〉_M and |R〉_M represent the states of the measurement apparatus. The correlations between the measurement apparatus and the system are of a classical nature, and so also in the absence of collapse, the von Neumann entropy of system and apparatus has not changed during the step 4 measurement.