Review of A. Peres, Nonlinear Variants of Schrödinger's Equation Violate the Second Law of Thermodynamics (1989)

Review of A. Peres, Nonlinear Variants of Schrödinger’s Equation Violate the Second Law of Thermodynamics, Phys. Rev. Lett. 63, 1114 (1989).

Peres [1] claims that nonlinear variants of Schrödinger’s equation violate the second law of thermodynamics, and that the only evolution that is commensurate with nondecreasing entropy is Schrodinger’s unitary evolution. The same demonstration was later repeated in Peres’ book on quantum theory [2, p.278]. In this review we demonstrate that Peres’ claims that state changes must be unitary are false.

Summary

The argument begins by considering a state $\rho$ which is a mixture of two pure states:

$\rho=\lambda\vert\phi><\phi\vert + (1-\lambda)\vert\psi><\psi\vert$

and with eigenvalues given by:

$w_{\pm}=1/2\pm\sqrt{1/4-\lambda(1-\lambda)(1-x)}$

where $x=<\phi\vert\psi>$ . These eigenvalues can be used to determine the von Neumann entropy $-\mbox{Tr}[\rho \log{\rho}]$ in the form:

$S(\rho)=-w_{+}\log{w_{+}}-w_{-}\log{w_{-}}.$

It can be confirmed that this has the property that $dS/dx > 0$ for all $\lambda$ which implies that the evolution in time of $x$ must be non-positive in order to not allow the entropy to decrease in time as required by the second law of thermodynamics:

$\vert<\psi(t)\vert\phi(t)>\vert^2\leq\vert<\psi(0)\vert\phi(0)>\vert^2$ (1)

For the pure state $\vert\psi>$ , it is seen that $\sum_{k}\vert<\phi_{k}\vert\psi>\vert^2=1$ , where $\vert\psi_{k}>$ is a complete orthonormal basis. Therefore, if there is some m for which:

$\vert<\phi_m\vert\psi(t)>\vert^2<\vert<\phi(0)\vert\psi(0)>\vert^2$

then there must also be some n for which:

$\vert<\phi_n\vert\psi(t)>\vert^2>\vert<\phi(0)\vert\psi(0)>\vert^2$ .

As a consequence, the entropy of a mixture of $\vert\psi_{n}><\psi_{n}\vert$ and $\vert\phi><\phi\vert$ will decrease in a closed system. To avoid this, we must require that:

$x=<\phi\vert\psi>$ = constant, (2)

for any $\vert\psi>$ and $\vert\phi>$ to conform with the second law. Given Equation (2), Peres invokes Wigner’s Theorem from which Equation (2) implies that the evolution is either unitary or anti-unitary. Requiring continuity excludes the anti-unitary case and Peres concludes that the second law restricts the evolution of quantum states to be unitary which is linear in wavefunction.

Error in Peres’s Reasoning

Peres states:

Therefore, if the pure quantum states evolve as    and  the entropy of the mixture shall not decrease …

Peres has inserted as an assumption in his proof both $\phi(0)\rightarrow \phi(t)$ and $\psi(0)\rightarrow \psi(t)$ in order to obtain his result that the only evolution consistent with nondecreasing entropy is unitary evolution. This is tantamount to the assumption that quantum state evolution is always deterministic. The fact is that measurement, whether from the probabilistic interpretation of Born or as well the first postulate of quantum mechanics formalized by von Neumann, allows states to bifurcate or more generally to multifurcate, which are more generally defined as probabilistic morphisms or probabilistic mappings. That is, it is possible when there are $n$ outcomes of the measurement that $\phi(0)\rightarrow \{\phi_{1}(t), \cdots, \phi_{n}(t)\}$ which occur with probabilities $\{p_{1},\cdots,p_{n}\}$ and similarly with $\psi(0)$ as the initial state. Peres simply ignores the possibility of probabilistic morphisms, which is actually commonplace in quantum mechanics and in other areas of mathematics such as Markov chain theory in probability theory.

If Peres evaluated the case of probabilistic morphisms it would be clear that entropy can increase. In fact, von Neumann addressed this issue in his book in 1935 [3, p. 364] in which he also noted that entropy does not change at all but stays constant under unitary evolution:

… we show that all states  have the same entropy, i.e. that the reversible transformation of the  into the  ensemble is accomplished without the absorption or liberation of heat energy.

On the other hand. when the measurement process is utilized, von Neumann states [3, p. 379]:

We can now prove the irreversibility of the measurement process as asserted in V.1. For example, if  is a state … then in the measurement of a quantity R whose operator R has the eigenfunctions , it goes over into the ensemble  and if  is not a state, then an entropy increase has occurred …, so that the process is irreversible.

Hence it had already been known for decades before Peres’s paper that entropy is constant under unitary evolution and increases under measurement when using projection measurement operators. We now also know that all measurements that satisfy a unital operator condition are non-decreasing in entropy; there do exist non-unital measurements that decrease entropy. In particular, for a quantum system with a finite N-dimensional Hilbert space, the unitality condition is not only a sufficient, but also the necessary condition for non-diminishing entropy [4]. Trace preserving, unital processes are also called bistochastic quantum operations which have Kraus representations [5]:

We identify bistochastic quantum operations, with nontrivial Kraus representation, with pure diffusion (possibly with drift), while we associate dissipation with nonunital maps, characterized by phase space contraction parameter .

Conclusions

Peres made an attempt to find a relation between allowed system quantum time evolution and the second law of thermodynamics. However, he implicitly assumed deterministic evolution which biased the outcome to be unitary via Wigner’s theorem. The necessary and sufficient unital solution for non-decreasing entropy allows a more general range of evolution characteristics.

The key issues we have identified in this review is that state evolution must in-general by either deterministic, in which case Peres succeeded in showing the evolution must be unitary, or non-deterministic in which case the measurement operators must be unital. In the case of unitary evolution, the entropy does not change whereas in the case of unital non-deterministic evolution, entropy typically increases.

Additionally, the problem addressed by Peres is related to the quantum measurement problem and it is well-known that the solution necessitates non-linear wavefunction evolution. However, it is also known that a sufficient condition for a measurement device to not cause superluminal signaling is that the characteristics of the measurement be linear in density operator. In particular, linearity in density operator does not imply that the wave-function be linear in wave-function or evolve via Schrödinger’s equation as concluded by Peres.

Remarkably, Peres appears to have actually proven the opposite of what he desired to prove. That in order for entropy to be non-decreasing, the only two solutions are deterministic unitary evolution and non-deterministic unital evolution. Furthermore, as noted, evolution (with unphysical exceptions) must be linear in density operator in order to avoid signaling. Hence one can largely (up to unphysical exceptions) conclude that the general solution to the quantum measurement problem falls in the class of linear density operations, of which unitary evolution is a subset when evolution is deterministic.

Although a huge amount of knowledge has been obtained regarding deterministic evolution, it is equally remarkable about the dearth of knowledge regarding non-deterministic evolution that generates information, imparts significance, and is arguably as or more scientifically significant than Schrödinger’s equation. Understanding and characterizing the theory of non-deterministic quantum state evolution, is required to resolve the quantum measurement problem. Here we have taken a first step: the general theory allows non-linearity in wavefunction while being in the class of linear density operators.

Other Notes

Hilbert spaces greater than 2

Peres proved his theorem for qubits. It isn’t clear that his theorem holds for general sizes greater than 2 as this was not proven in his paper.

How the inner product fails to conserve in Probabilistic Morphisms

Given an initial ensemble as previously $\rho=\lambda\vert\phi><\phi\vert + (1-\lambda)\vert\psi><\psi\vert$ an interesting question is whether or not it is possible to maintain exactly the inner product between $\phi(t)$ and $\psi(t)$ under general measurement transformations. Suppose that $\phi(0)$ is measured to $\phi_{1}$ with measure $\mu_{1}$ and to $\phi_{2}$ with measure $\mu_{2}$ where $\mu$ is an arbitrary measure on a $\sigma$ -algebra of events defined over a set $X$ . Similarly suppose $\psi(0)$ is measured to $\psi_{1}$ with measure $\mu_{3}$ and to $\psi_{2}$ with measure $\mu_{4}$ . Assuming the measurement of $\phi$ and $\psi$ are independent, we can compute an averaged inner product using these arbitrary measures as:

$\overline{<\phi(t),\psi(t)>}:= \mu_{1}\mu_{3}\phi_{1}\psi_{1}^{\dagger} +\mu_{1}\mu_{4}\phi_{1}\psi_{2}^{\dagger} +\mu_{2}\mu_{3}\phi_{2}\psi_{1}^{\dagger} +\mu_{2}\mu_{4}\phi_{2}\psi_{2}^{\dagger}$

Now, consider the case that the $\mu$ are probability measures $\mu_{i}=P_{i}$ represented by the standard quantum mechanical formula for measurement for which $\sum_{i} P_{i}=1$ . Other than a small set of trivial cases, it was already seen that $<\phi(0),\psi(0)>\ne\overline{<\phi(t),\psi(t)>}$ . One can verify that if new measures are defined using the Euclidean norm $\mu_{i,E}:=\sqrt{P_{i}}$ than interestingly $<\phi(0),\psi(0)>=\overline{<\phi(t),\psi(t)>}$ and inner products (at least on-average) are once again conserved.

Hence the lack of conservation of inner product for the case of probabilistic morphisms appears to stem from not only the existence of multiple outcomes, but also the imposition of probability measures. However, the imposition of probability measure appears to be unique that is representative of physical situations that utilize probabilistic morphisms.

References

[1] A. Peres, “Nonlinear variants of Schr\”odinger’s equation violate the second law of thermodynamics,” Phys. Rev. Lett., vol. 63, no. 10, p. 1114, Sep. 1989, doi: 10.1103/PhysRevLett.63.1114.

[2] A. Peres, Quantum Theory: Concepts and Methods. in Fundamental Theories of Physics. Kluwer Academic Publishers, 1995. [Online]. Available: https://books.google.co.cr/books?id=rMGqMyFBcL8C

[3] J. von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton, NJ: Princeton University Press, 1955.

[4] L. Zhang and J. Wu, “Von Neumann entropy-preserving quantum operations,” Phys. Lett. A, vol. 375, no. 47, pp. 4163–4165, 2011, doi: https://doi.org/10.1016/j.physleta.2011.10.008.

[5] I. García-Mata, M. Saraceno, M. E. Spina, and G. Carlo, “Phase-space contraction and quantum operations,” Phys. Rev. A (Coll Park)., vol. 72, no. 6, p. 62315, Dec. 2005, doi: 10.1103/PhysRevA.72.062315.