November 12, 2022. Alternating between losing strategies can (apparently paradoxically) give a winning strategy. I explain how to implement this in quantum mechanics, and why it is not a paradox.

Sometimes, alternating between two losing strategies becomes a winning strategy. This is called Parrondo’s paradox. As we’ll see, it isn’t much of a paradox, but it is interesting, particularly due to its connection to thermodynamics.

The ratchet

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To illustrate, consider a molecule sitting in a hot bath. It will bump into other particles and jump around randomly. If it’s subject to gravity and sitting on an even slope $E$, it will gradually slide down, albeit with random jumps back and forth. Instead of an even slope, we could imagine a bumpy slope $B$, which still tilts to the left overall. The molecule may get stuck in a local divot, or it may drift to the left. What it cannot do, however, is drift to the right.

Suppose that it did. Then we would be turning thermal energy—the random jumps—into gravitational potential energy. This violates the Second Law of thermodynamics, which forbids us turning thermal energy into useful work. To be clear, each time the particle jumps to the right, we seem to have violated the Second Law, but really, what matters is the average. Random fluctuations are fine, but the average trajectory must always get stuck or go left. So, both $E$ and $B$ are “losing strategies” in the sense that on average the particle does not go right.

Now suppose we can press a button which alternates instantaneously between $E$ and $B$. Provided the bumps are big enough to trap the particle, we can manufacture a violation of the Second Law as follows. Wait for it to get trapped in a divot in $B$; quickly switch to $E$ so it jumps right; switch back to $B$ so it gets trapped again. This is a type of “Brownian ratchet” which slowly raises the molecule up the hill. This converts the thermal energy of random jumps into gravitational potential energy on average, and thus violates the Second Law. Of course, in order to do this, we must be very strategic about alternation. This resolves the paradox. Briefly:

A quantum ratchet

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It turns out to be possible to convert this alternation strategy directly into a quantum-mechanical version of Parrondo’s paradox. Let us consider a system with Hamiltonian $H$, and two observables, $E$ (not energy!) and $B$. For simplicity, we’ll suppose that neither of these changes with time, just like the slopes; the time-dependence will be in the alternation only. To say that both $E$ and $B$ are “losing strategies” here means that the expectation decreases with time:

\[\partial_t\langle E\rangle \leq 0, \quad \partial_t\langle B\rangle \leq 0.\]

Suppose the loss rate is bounded below by some minimum loss rate $m(t)$, i.e.

\[|\partial_t \langle E\rangle|, |\partial_t \langle B\rangle| \geq m(t).\]

Let’s define our alternating strategy in terms of an operator which weights $E$ and $B$ in a time-dependent fashion:

\[A(t) = \alpha(t) E + (1-\alpha(t)) B.\]

To implement Parrondo’s paradox, we will simply seek an alternation strategy $\alpha(t) \in [0, 1]$ such that $\langle A(t)\rangle$ increases with time. Since expectations are linear, we have that

\[\langle A(t)\rangle = \alpha(t) \langle E\rangle + (1-\alpha(t)) \langle B\rangle,\]

and hence time derivative

\[\partial_t\langle A(t)\rangle = \dot{\alpha}(t) (\langle E\rangle - \langle B\rangle ) + \alpha(t) \partial_t\langle E\rangle + (1-\alpha(t)) \partial_t\langle B\rangle.\]

For this to be positive, we require

\[\begin{align*} \dot{\alpha}(t) (\langle E\rangle - \langle B\rangle) & > \alpha(t)|\partial_t\langle E\rangle| + (1 - \alpha(t))|\partial_t\langle B\rangle| \\ &\geq \alpha(t)m(t) + (1- \alpha(t))m(t) = m(t), \end{align*}\]

using our loss bound $m(t)$. Put differently, we will turn our losing strategies into a winning strategy provided we switch to whatever strategy is locally better quickly enough, where “quickly enough” means

\[| \dot{\alpha}(t)| > \frac{m(t)}{|\langle E\rangle - \langle B\rangle|}. \tag{1} \label{qparr}\]

Of course, we could phrase the Parrondo condition in terms of probability distributions rather than observables, but the quantum-mechanical version is to my mind more fundamental.

The Parrondo condition $(\ref{qparr})$ tells us clearly what powers we need to have to implement the paradox. We need to be able to locally assess the difference in expectations,

\[\langle E\rangle - \langle B\rangle,\]

which is analogous to knowing where the molecule is on the slope. We then needs to switch to the better slope, at a speed inversely proportional to the difference, like lowering the divot to allow the molecule to jump out, up the slope.