Blog - Noether's theorem: quantum vs stochastic (Rev #24)

This blog article in progress, written by Ville Bergholm. See also Blog - quantum network theory (part 1) and Blog - quantum network theory (part 2). To see the discussion of the article being written, visit the Azimuth Forum. If you want to write your own article, please read

*guest post by Ville Bergholm*

In 1915 Emmy Noether discovered an important connection between the symmetries of a system and its conserved quantities. Her result has become a staple of modern physics and is known as Noether’s theorem.

The theorem and its generalizations have found particularly wide use in quantum theory. Those of you following the network series on Azimuth might recall Part 11 where John Baez and Brendan Fong proved a version of Noether’s theorem for stochastic systems. Their result is now published here:

• John Baez and Brendan Fong, A Noether theorem for stochastic mechanics, *J. Math. Phys.* 54:013301 (2013).

One goal of the network theory series here on Azimuth has been to merge ideas appearing in quantum theory with other disciplines. John and Brendan proved their stochastic version of Noether’s theorem by exploiting ‘stochastic mechanics’ which was formulated in the network theory series to mathematically resemble quantum theory. Their result, which we will outline below, was different than what would be expected in quantum theory, so it is interesting to try to figure out *why*.

Recently Jacob Biamonte, Mauro Faccin and myself have been working to try to get to the bottom of these differences. What we’ve done is prove a version of Noether’s theorem for Dirichlet operators. As you may recall from Parts 16 and 20 of the network theory series, these are the operators that generate both stochastic *and* quantum processes. In the language of the series, they lie in the intersection of stochastic and quantum mechanics. So, they are a subclass of the infinitesimal stochastic operators considered in John and Brendan’s work.

The extra structure of Dirichlet operators—compared with the wider class of infinitesimal stochastic operators—provided a handle for us to dig a little deeper into understanding the intersection of these two theories. By the end of this article, astute readers will be able to prove that Dirichlet operators generate doubly stochastic processes.

Before we get into the details of our proof, let’s recall first how conservation laws work in quantum mechanics, and then contrast this with what John and Brendan discovered for stochastic systems. (For a more detailed comparison between the stochastic and quantum versions of the theorem, see Part 13 of the network theory series.)

In standard (closed system) quantum theory, the unitary time evolution of a state $|\psi(t)\rangle$ is generated by a self-adjoint matrix $H$ which is called the Hamiltonian. So, $|\psi(t)\rangle$ satisfies Schrödinger’s equation:

$i \hbar \displaystyle{\frac{d}{dt}} |\psi(t) \rangle = H |\psi(t) \rangle.$

The state of a system starting off at time zero in the state $|\psi_0 \rangle$ and evolving for a time $t$ is then given by

$|\psi(t) \rangle = e^{-i t H}|\psi_0 \rangle.$

The observable properties of a quantum system are associated with self-adjoint operators. The expected value of a self-adjoint operator $O$ in the state $|\psi \rangle$ is

$\langle O \rangle_{\psi} = \langle \psi | O | \psi \rangle.$

$O$ is a constant of the motion if and only if it commutes with the Hamiltonian $H$:

$[O, H] = 0 \quad \iff \quad \displaystyle{\frac{d}{dt}} \langle O \rangle_{\psi(t)} = 0 \quad \forall \: |\psi_0 \rangle, \forall t.$

In stochastic mechanics, the story changes a bit. Now the Hamiltonian $H$ is an **infinitesimal stochastic** operator, i.e. a real-valued square matrix with non-negative off-diagonal entries and columns summing to zero. And now $|\psi(t)\rangle$ is a normalized probability distribution obeying the master equation:

$\displaystyle{\frac{d}{dt}} |\psi(t) \rangle = H |\psi(t) \rangle.$

$H$ generates a continuous-time Markov process $e^{t H}$, which maps the space of normalized probability distribution to itself. So, a probability distribution evolves as follows:

$|\psi(t)\rangle = e^{t H} |\psi_0 \rangle.$

In this context an **observable** $O$ is a real diagonal matrix, and its expected value is given by

$\langle O\rangle_{\psi} = \sum_{i \in X} \langle i | O |\psi\rangle = \langle \hat{O} | \psi \rangle,$

The version of Noether’s theorem for stochastic systems is stated as:

**Noether’s Theorem for Markov Processes (Baez–Fong).** Suppose $H$ is an infinitesimal stochastic operator and $O$ is an observable. Then

$[O,H] =0$

if and only if

$\displaystyle{\frac{d}{dt}} \langle O \rangle_{\psi(t)} = 0$

and

$\displaystyle{\frac{d}{dt}}\langle O^2 \rangle_{\psi(t)} = 0$

for all $t$ and all $\psi(t)$ obeying the master equation.

From this theorem, we can see immediately that every symmetry in stochastic mechanics given by $[O,H]=0$ still leads to $O$ representing a **conserved quantity**, meaning that

$\displaystyle{\frac{d}{dt}} \langle O\rangle_{\psi(t)} = 0$

for any $\psi_0$ and all $t$.

However, John and Brendan showed is that—unlike in quantum theory—you need more than just the expectation value of the observable $O$ to be constant to obtain the equation $[O,H]=0$. You really need both

$\displaystyle{\frac{d}{dt}} \langle O\rangle_{\psi(t)} = 0$

together with

$\displaystyle{\frac{d}{dt}} \langle O^2\rangle_{\psi(t)} = 0$

for all initial data $\psi_0$ to be sure that $[O,H]=0$. So it’s a bit subtle, but symmetries and conserved quantities have a rather different relationship than they do in quantum theory.

But what if the infinitesimal generator of our Markov process is also self-adjoint? In other words, what if $H$ is both an infinitesimal stochastic matrix but also its own transpose: $H = H^\top$? Then it’s called a **Dirichlet operator**… and we found that in this case, we get a stochastic version of Noether’s theorem that more closely resembles the usual quantum one:

**Noether’s Theorem for Dirichlet Operators** If $H$ is an infinitesimal stochastic operator with $H = H^\top$, and $O$ is an observable, then

$[O, H] = 0 \quad \iff \quad \displaystyle{\frac{d}{dt}} \langle O \rangle_{\psi(t)} = 0 \quad \forall \: |\psi_0 \rangle, \forall t \ge 0.$

**Proof.** Since $H$ is symmetric we may use a spectral decomposition

$H = \sum_k E_k |\upsilon_k \rangle \langle \upsilon_k |$. As in John and Brendan’s proof, the $\implies$ direction is easy to show. In the other direction, we have

$\displaystyle{\frac{d}{dt}} \langle O \rangle_{\psi(t)} = \langle \hat{O} | H e^{Ht} |\psi_0 \rangle = 0 \quad \forall \: |\psi_0 \rangle, \forall t \ge 0$

$\iff \quad \langle \hat{O}| H e^{Ht} = 0 \quad \forall t \ge 0$

$\iff \quad \sum_k \langle \hat{O} | \upsilon_k \rangle E_k e^{t E_k} \langle \upsilon_k| = 0 \quad \forall t \ge 0$

$\iff \quad \langle \hat{O} | \upsilon_k \rangle E_k e^{t E_k} = 0 \quad \forall t \ge 0$

$\iff \quad |\hat{O} \rangle \in \Span\{|\upsilon_k \rangle | E_k = 0\} = \ker \: H,$

where the third equivalence is due to $\{ |\upsilon_k \rangle \}_k$ being a linearly independent set of vectors. For any infinitesimal stochastic operator $H$ the corresponding transition graph consists of $m$ connected components iff we may reorder (permute) the states of the system such that $H$ becomes block-diagonal with $m$ blocks. Now it is easy to see that the kernel of $H$ is spanned by $m$ eigenvectors, one for each block. Since $H$ is also symmetric, the elements of each such vector can be chosen to be ones within the block and zeros outside it. Consequently

$|\hat{O} \rangle \in \ker \: H$

implies that we can choose the eigenbasis of $O$ to be $\{|\upsilon_k \rangle\}_k$, which implies

$[O, H] = 0$.

Alternatively,

$|\hat{O} \rangle \in \ker \: H$ implies $

$|\hat{O^2} \rangle \in \ker \: H \quad \iff \quad \ldots \quad \iff \quad \displaystyle{\frac{d}{dt}} \langle O^2 \rangle_{\psi(t)} = 0 \quad \forall \: |\psi_0 \rangle, \forall t \ge 0,$

where we have used the above sequence of equivalences backwards. Now, using John and Brendan’s original proof, we can obtain $[O, H] = 0$.

In summary, by restricting ourselves to the intersection of quantum and stochastic generators, we have essentially recovered the quantum version of Noether’s theorem. However, this simplification comes at a cost. We find that the only observables $O$ that remain invariant under a symmetric $H$ are of the very restricted type described above, where the observable has to have the same value in every state in a connected component.

Suppose that a graph Laplacian matrix $H$ generates a 1-parameter Markov semigroup as follows:

$U(t) = e^{t H}$

defined for all non-negative times $t$.

**Puzzle 1.** Suppose that also $H = H^\top$, so that $H$ is a Dirichlet operator and hence $i H$ generates a 1-parameter unitary group. Show the following. Let $n$ label any node of the underlying adjacency matrix corresponding to $H$. Then the indegree and outdegree of any node $n$ is equal (graphs with this property are called **balanced**).

**Puzzle 2.** Now assume further that $U(t)$ is doubly stochastic for all times $t$, e.g.

$\sum_i U_{ij} = \sum_j U_{ij} = 1$

and show that this is equivalent to the condition on the generator

$\sum_i H_{ij} = \sum_j H_{ij} = 0$

**Puzzle 3.** Prove that a doubly stochastic continuous-time Markov semigroup is generated by a balanced graph. Observe further that symmetric graphs are a strict subclass of balanced graphs.

**Puzzle 4.** Let $A$ be a possibly time-dependent stochastic observable, and write $\langle A\rangle$ for its expected value with respect to some initial state $\psi_0$ evolving as $e^{t H}\psi_0$. Show that

$\frac{d}{d t}\langle A\rangle = \langle [A, H] \rangle+ \left\langle \frac{\partial A}{\partial t}\right\rangle$

Using this (or some other method), prove a stochastic version of the Ehrenfest theorem.

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