Branches and matrix-product states

I’m happy to use this bully pulpit to advertise that the following paper has been deemed “probably not terrible”, i.e., published.

When the wave function of a large quantum system unitarily evolves away from a low-entropy initial state, there is strong circumstantial evidence it develops “branches”: a decomposition into orthogonal components that is indistinguishable from the corresponding incoherent mixture with feasible observations. Is this decomposition unique? Must the number of branches increase with time? These questions are hard to answer because there is no formal definition of branches, and most intuition is based on toy models with arbitrarily preferred degrees of freedom. Here, assuming only the tensor structure associated with spatial locality, I show that branch decompositions are highly constrained just by the requirement that they exhibit redundant local records. The set of all redundantly recorded observables induces a preferred decomposition into simultaneous eigenstates unless their records are highly extended and delicately overlapping, as exemplified by the Shor error-correcting code. A maximum length scale for records is enough to guarantee uniqueness. Speculatively, objective branch decompositions may speed up numerical simulations of nonstationary many-body states, illuminate the thermalization of closed systems, and demote measurement from fundamental primitive in the quantum formalism.

Here’s the figureThe editor tried to convince me that this figure appeared on the cover for purely aesthetic reasons and this does not mean my letter is the best thing in the issue…but I know better! a   and caption:


Spatially disjoint regions with the same coloring (e.g., the solid blue regions \mathcal{F}, \mathcal{F}', \ldots) denote different records for the same observable (e.g., \Omega_a = \{\Omega_a^{\mathcal{F}},\Omega_a^{\mathcal{F}'},\ldots\}). (a) The spatial record structure of the Shor-code family of states, which can exhibit arbitrary redundancy (in this case four-fold) for two incompatible observables. (b) The solid orange observable pair-covers the hashed blue observable because the top two orange records overlap all blue records. However, if one of the top two orange records is dropped, then neither observable pair-covers the other, and hence both are compatible, despite many overlaps of individual records. (c) Any spatially bounded set of records can be contained inside a single record of a sufficiently dilated but otherwise identical set of records for an incompatible observable; such a state is given in Eq. (9). (d) Any observable with records satisfying the hypothesis of the Corollary for some length \ell cannot pair-cover, or be pair-covered by, any other such observable.

It is my highly unusual opinion that identifying a definition for the branches in the wavefunction is the most conceptually important problem is physics. The reasoning is straightforward: (1) quantum mechanics is the most profound thing we know about the universe, (2) the measurement process is at the heart of the weirdness, and (3) the critical roadblock to analysis is a definition of what we’re talking about. (Each step is of course highly disputed, and I won’t defend the reasoning here.) In my biased opinion, the paper represents the closest yet anyone has gotten to giving a mathematically precise definition.

On the last page of the paper, I speculate on the possibility that branch finding may have practical (!) applications for speeding up numerical simulations of quantum many-body systems using matrix-product states (MPS), or tensor networks in general. The rough idea is this: Generic quantum systems are exponentially hard to simulate, but classical systems (even stochastic ones) are not. A definition of branches would identify which degrees of freedom of a quantum system could be accurately simulated classically, and when. Although classical computational transitions are understood in many certain special cases, our macroscopic observations of the real world strongly suggest that all systems we study admit classical descriptions on large enough scales.Note that whether certain degrees of freedom admit a classical effective description is a computational question. The existence of such a description is incompatible with them exhibiting quantum supremacy, but it is compatible with them encoding circumstantial evidence (at the macro scale) for quantum mechanics (at the micro scale). b   There is reason to think a general (abstract) description of the quantum-classical transition is possible, and would allow us to go beyond special cases.

In the rest this blog post I’m going to construct a simple MPS state featuring branches with records that are arbitrarily redundant. The state’s key features will be that it is translationally invariant and has finite correlation length. Translational invariance makes the state much easier to study and compare with the literature, and is a property shared with the simple inflationary model I’m studying with Elliot Nelson. Finite correlation length eliminates the trivial solution of a generalized GHZ state, guarantees that there are an infinite number of branches (if there are any at all), and is in some sense more natural.

An MPS with branches

Our strategy will be to build the state up from a classical probability distribution that already has the key features. Let \vec{b} = (b_1, \dots, b_N), be a classical state of a finite 1D lattice of N bits (b_n = \pm 1), and consider the canonical ensemble starting with a uniform distribution and adding an energetic penalty of \gamma for misaligned nearest-neighborsWe assume periodic boundary conditions: b_{N+1} \equiv b_1 c  :

(1)   \begin{align*} P(\vec{b}) &= Z^{-1}  e^{\frac{\gamma}{2}\sum_n b_n b_{n+1}} \propto \exp\left[-\gamma\sum_n \left(\frac{b_{n+1}-b_n}{2}\right)^2\right] \end{align*}

with Z = [2\, \mathrm{cosh} (\gamma/2)]^N. It’s not hard to check that the classical expectation value obeys \langle b_n \rangle = 0 and

(2)   \begin{align*} \langle b_n b_{n+m} \rangle = \left(\frac{e^\gamma -1}{e^\gamma +1}\right)^m . \end{align*}

If we define our correlation length \ell by the asymptotic behavior \vert \langle b_n b_{n+m} \rangle\vert \sim e^{-m/\ell}, then for large \vert\gamma\vert (large \ell) we have

(3)   \begin{align*} \ell = \frac{e^{\vert \gamma \vert}}{2} \end{align*}

The idea is to define our quantum state as a superposition of configurations of a spin chain weighted by this distribution: \vert \Psi \rangle \equiv \sum_{\vec{b}} P(\vec{b}) \bigotimes_n \vert \chi(b_n) \rangle_n. Contiguous regions of size \ell will be highly correlated. The coarse-grained variables that are being recorded are something like “whether this local region is mostly +1 or mostly -1”, but we don’t necessarily want each qubit to have a perfect record of this information. Rather, we choose

(4)   \begin{align*}  \vert \chi(b_n) \rangle_n &\equiv \cos(b_n\theta) \vert \uparrow \rangle_n + \sin(b_n\theta) \vert \downarrow \rangle_n \\ &= \cos(\theta) \vert \uparrow \rangle_n + b_n\sin(\theta) \vert \downarrow \rangle_n \end{align*}

for some fixed (typically small) angle \theta. For \vert\theta\vert < \pi/4 one can’t reliably infer the classical variable from a measurement on a single spin, but one can make an arbitrarily reliable inference by measuring lots of them: \vert\langle +1 \vert -1 \rangle\vert^m = (1-2 \sin^2 \theta)^m. If we define our “recording length” r by the asymptotic behavior \vert\langle +1 \vert -1 \rangle\vert^m \sim e^{-m/r}, then for small \theta we have

(5)   \begin{align*} r = \frac{1}{2\theta^2} \end{align*}

Thus our final (unnormalized) state is

(6)   \begin{align*} \mkern-18mu \vert \Psi (\ell,r)\rangle \equiv \sum_{\vec{b}} \exp\left[\frac{-\ln (2 \ell)}{2}\sum_n b_n b_{n+1}\right] \bigotimes_n \Big[\cos \sqrt{1/2r} \vert \uparrow \rangle_n + b_n\sin \sqrt{1/2r} \vert \downarrow \rangle_n\Big] \hspace{2cm}\textcolor{white}{.} \end{align*}

for arbitrary positive parameters \ell and r. The state inherits from the classical probability distribution the properties of finite correlation length and translational invariance. Any contiguous set of m \ll \ell spins is very likely to all be in the same state, and by measuring a subset of m' \gg r spins one can distinguish between the cases \vert +1 \rangle and \vert -1 \rangle with high reliability. Therefore in the limit r \ll \ell it makes sense to define the redundancy R \sim r/\ell; this is the approximate number of disjoint records that are available about each branch, where there is a (binary) branch density of \ell^{-1}.

Now lets turn this into a matrix-product state.This will not be a general algorithm for building MPSs since it exploits the special form of the state (6) we’re trying to build, which allows us to identify the b‘s with the \alpha‘s. d   That is, it should take the formAgain, periodic boundary conditions e  The tikzpicture code was obtained from Piotr Migdał. f  

(7)   \begin{align*} \vert \Psi \rangle &= \sum_{i_1,\dots,i_N}\sum_{\alpha_1,\dots,\alpha_N} A^{i_1}_{\alpha_1 \alpha_2}A^{i_2}_{\alpha_2 \alpha_3}\dots A^{i_{N-1}}_{\alpha_{N-1} \alpha_{N}} A^{i_N}_{\alpha_{N} \alpha_{1}} \vert i_1\dots i_N\rangle\\ &= \sum_{i_1,\dots,i_N} \mathrm{Tr}\left[ A^{i_1} A^{i_2}\dots A^{i_{N-1}}A^{i_N}\right] \vert i_1\dots i_N\rangle \end{align*}

Rendered by QuickLaTeX.com

where the i_n = \uparrow,\downarrow range over the local Hilbert spaces of the qubits and the \alpha_n are the “bonds” representing contraction of matrices of as-yet unspecified dimension (depicted in the figure as lines connecting the A‘s horizontally). The trick is to pretend that our classical probability distribution was achieved by starting with a quantum state of appropriately weighted bras,

(8)   \begin{align*} \langle P \vert \equiv \sum_{\vec{b}} P(\vec{b}) \left[ \bigotimes_n \, {}_n \! \langle b_n \vert \right], \end{align*}

and contracting (i.e., multiplying from the right to inner-product over the fictional bras) with a specially designed state that picks out the correct conditional states of the original Hilbert space:

(9)   \begin{align*} \vert \Omega \rangle % &\equiv \bigotimes_n \vert \omega \rangle_n\\ &\equiv \bigotimes_n \left[\sum_{b_n = \pm 1} \vert b_n \rangle_n \otimes \vert \chi(b_n) \rangle_n \right]\\ &= \bigotimes_n \Big[\vert +1 \rangle_n \left(\cos\theta \vert \uparrow \rangle_n + \sin\theta \vert \downarrow \rangle_n \right) + \vert -1 \rangle_n \left(\cos\theta \vert \uparrow \rangle_n - \sin\theta \vert \downarrow \rangle_n \right)\Big].\hspace{2cm}\textcolor{white}{.} \end{align*}

The object

(10)   \begin{align*} \langle P \vert \Omega \rangle = \sum_{i_1,\dots,i_N}\sum_{b_1,\dots,b_N} A^{i_1}_{b_1 b_2}\dots  A^{i_N}_{b_{N} b_{1}} \vert i_1\dots i_N\rangle.   \end{align*}

then takes the form of (7) and is equal to \vert \Psi(\ell,r)\rangle so long as we choose A to satisfy

(11)   \begin{align*} A^{i_n}_{b_n b_{n+1}} % &= \langle b_n \vert \Omega \rangle_n\\ &= e^{\frac{\gamma}{2} b_n b_{n+1}} \left(\cos\theta \vert \uparrow \rangle_n + b_n \sin\theta \vert \downarrow \rangle_n \right)%\\ %&= e^{\frac{\gamma}{2} b_n b_{n+1}} \big[\delta_{b_n,+1} \left(\delta_{i_n,\uparrow} \cos \theta + \delta_{i_n,\downarrow} \sin \theta\right) \\ %& \qquad\qquad\qquad\quad + \delta_{b_n,-1} \left(\delta_{i_n,\uparrow} \cos \theta - \delta_{i_n,\downarrow} \sin \theta\right)\big],\hspace{1cm}\textcolor{white}{.} \end{align*}

or, more cleanly,

(12)   \begin{align*} A^\uparrow = \cos \sqrt{\frac{1}{2r}} \left(\begin{array}{cc}\sqrt{2 \ell} &  1/\sqrt{2 \ell}\\ \phantom{-}1/\sqrt{2 \ell} &  \phantom{-}\sqrt{2 \ell} \end{array}\right), \\  A^\downarrow = \sin \sqrt{\frac{1}{2r}} \left(\begin{array}{cc}\sqrt{2 \ell} &  1/\sqrt{2 \ell}\\ -1/\sqrt{2 \ell} &  -\sqrt{2 \ell} \end{array}\right). \end{align*}

[I think Martin Ganahl and Guifre Vidal for discussion.]

Footnotes

(↵ returns to text)

  1. The editor tried to convince me that this figure appeared on the cover for purely aesthetic reasons and this does not mean my letter is the best thing in the issue…but I know better!
  2. Note that whether certain degrees of freedom admit a classical effective description is a computational question. The existence of such a description is incompatible with them exhibiting quantum supremacy, but it is compatible with them encoding circumstantial evidence (at the macro scale) for quantum mechanics (at the micro scale).
  3. We assume periodic boundary conditions: b_{N+1} \equiv b_1
  4. This will not be a general algorithm for building MPSs since it exploits the special form of the state (6) we’re trying to build, which allows us to identify the b‘s with the \alpha‘s.
  5. Again, periodic boundary conditions
  6. The tikzpicture code was obtained from Piotr Migdał.
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