Generalizing wavefunction branches to indistinguishable subspaces

[This post describes ideas generated in discussion with Markus Hauru, Curt von Keyserlingk, and Daniel Ranard.]

An original dream of defining branches based on redundant records (aka redundant classical information, aka GHZ-like correlations) was that it would be possible to decompose the wavefunction of an evolving non-integrable quantum system at each point in time into macroscopically distinguishable branches that individually had bounded amounts of long-range entanglement (i.e., could be efficiently expressed as a matrix product state) even though the amount of long-range entanglement for the overall state diverges in time. If one could numerically perform such a decomposition, and if the branches only “fine-grain in time”, then one could classically sample from the branches to accurately estimate local observables even if the number of branches increases exponentially in time (which we expect them to do).

However, we now think that only a fairly small fraction of all long range entanglement can be attributed to redundantly recorded branches. Thus, even if we found and efficiently handled all such classical information using a decomposition into a number of branches that was increasing exponentially in time (polynomial branch entropy), most branches would nevertheless still have an entanglement entropy across any spatial partition that grew ~linearly in time (i.e., exponentially increasing bond dimension in the MPS representation) until saturating.

In this post I’ll first write down a simple model that suggests the need to generalize the idea of branches in order to account for most long-range entanglement. Then I will give some related reasons to think that this generalized structure will take the form not of a preferred basis, but rather preferred subspaces and subsystems, and together these will combine into a preferred “branch algebra”. Finally, I’ll argue against the possibility that these subspaces can be state-independent. That is, you can’t expect to branch a state by projecting onto fixed subspaces, independent of the state. This will lead, in a future post, to a speculative connection with quantum error correction.

Toy model of the problem

Assume for simplicity that the world consists of quantum systems on a lattice that is divided into $R$ spatial regions, where the Hilbert space of each region can be divided into system and environment. (For instance, the environment might represent photons or phonons present in each region.) Thus, the entire Hilbert space looks like

$\mathcal{H} = \bigotimes_{r=1}^R \left( \mathcal{S}_r \otimes \mathcal{E}_r\right).$

This perfect system-environment split isn’t true for the physically compelling models we care about, but I think the same basic argument will extend to, e.g., the generalized subsystem structure given by the matrix-algebra structure theorem (Artin-Wedderburn).

Let $D_s = \mathrm{dim}(\mathcal{S}_r)$ and $D_e = \mathrm{dim}(\mathcal{E}_r)$ , so that $D_H \equiv \mathrm{dim}(\mathcal{H}) = D_s^R D_e^R$ . If we want a branch decomposition of a system variable to be redundantly recorded on all $R$ regions, the maximum number of classically distinguishable states (i.e., maximum number of branches) is $D_s$ . Let us also assume that the environment decoheres the system and thermalizes, e.g., a low-energy subspace of scattering magnons. This means that the state of the entire environment (not just one region), conditional on a particular branch labeled by $s$ , is effectively in some state $|e_s\rangle_{E}$ that locally looks thermal and which has an MPS bond dimension^a that grows exponentially in time. The total state will be of the form

$\Psi = \sum_{s=1}^{B} \left[\bigotimes_{r=1}^R |s\rangle_{S_r} \right]\otimes |e_s\rangle_{E}.$

In order for the environment to decohere all the branches, we of course need to have $B \le D_e^R$ .

Even if branch finding goes perfectly and all $B$ branches are correctly identified, the bond dimension at the middle of the lattice for a single branch will grow like it does for a non-integrable system with dimension $D_e^{R} \ge B$ (which of course is infeasible to simulate for large environments and large times). More specifically we expect the bond dimension of each branch to grow exponentially until it reaches $D_e^{R/2} \ge \sqrt{B}$ because the state $|e_s\rangle_{E}$ is thermalizing. Crucially, it seems unlikely that in general we can lower bound the rate at which redundantly recorded branches form, relative to the rate at which the environmental state on a branch thermalizes, since copying info over a large scale is expected to happen on a slower, macroscopic timescale (e.g., the Lyapunov exponent).^b By finding and sampling the branches we would effectively only be lowering the bond dimension from $D_e^{R/2} B$ to $D_e^{R/2}\ge \sqrt{B}$ ,

This would be an “exponential speedup” in the weak sense that it could reduce the computational costs of simulating the system for time $T$ by a $\mathrm{poly}(B)$ factor that is exponential in $T$ and (also the system size $L$ , since the branch production rate should be proportional to space). But the remaining computational costs are still growing exponentially with $T$ (and $L$ ), so that the maximum simulation time achievable with any fixed amount of compute would increase by only a constant factor.

Therefore, if we want to be able to classically sample branches to keep bond dimension under control and enable long-time evolution, it looks like we will need to find a more generalized notion of branches that can be used to decompose thermal (and thermalizing) states.

What do redundantly recorded branches look like in the future?

Suppose some records form at an early time, and then we just unitarily evolve the corresponding branches to late times when everything is thermalized. We intuitively expect them to remain just as good for sampling for two main reasons:

The apparent effectiveness of assuming wavefunction collapse a la the Copenhagen interpretation despite the presumed continued unitary evolution of the global wavefunction, i.e., we don’t have to worry about the other worlds in many worlds, even when those worlds correspond to previously macroscopic fluctuations that have since locally thermalized.
In the limit of infinite time, each time-evolved branch should uniformly explore the microcanonical ensemble^c, the set of states with the same values of the conserved quantities (energy, and perhaps others). Detecting interference between these states is, for the overwhelming dominant fraction of them, completely infeasible.

We can formalize this by first defining the non-interference condition: for some fixed set $\mathcal{F}$ of observables representing “feasible” measurements, an orthogonal decomposition of the wavefunction $\psi = \sum_i \psi_i$ is non-interfering if

$\langle \psi_i | F | \psi_j \rangle \approx 0$

for all $F\in \mathcal{F}$ and for all $i \neq j$ . Non-interference is the condition that guarantees^d that sampling is effective:

$\langle \psi | F | \psi \rangle = \sum_i \langle \psi_i | F | \psi_i \rangle + \sum_i \sum_{j\neq i} \langle \psi_i | F | \psi_j \rangle \approx \sum_i \langle \psi_i | F | \psi_i \rangle.$

Depending on the choice of $\mathcal{F}$ , non-interference at a given time is guaranteed by the existence of redundant records at that time, e.g., one can check that $R$ redundant records implies the non-interference of branches when $\mathcal{F}$ is taken to be the set of all $N$ -single-site observables (i.e, $N$ -point correlators) when $N \le R-1$ . Furthermore, the overwhelming majority of pairs of orthogonal states from the microcanonical ensemble have negligible interference, so this condition should hold in the distant future.

So the non-interference condition and holds at early times if records are present, it almost certainly holds in the distant future, and the effectiveness of wavefunction collapse suggests it should hold during the times in between. But there is a big difference between recorded branches and the effectively indistinguishable quasi-branches from the microcanonical ensemble they eventually evolve to: the branches of a state with records are very distinct and pick out a preferred basis, whereas you can decompose the state into a superposition of any set of orthogonal states in the subspace of the microcanonical ensemble because any pair of orthogonal states in the subspace are non-interfering. So rather than have preferred branches, it seems a generalized notion will only provide us with a preferred subspace.

(Note: Even for normal branches, it’s not immediately clear whether the preferred basis should depend on the dynamics and/or the overall state, and the same applies to the preferred subspaces of generalized branches. Our particular example of the subspace of the microcanonical ensemble depends on the dynamics and not the state, but, e.g., it could be a subset of a state-dependent subspace independent of the dynamics. Here I am agnostic about such dependancies, but I discuss state dependence near the end of this post.^e)

Importantly, how do we characterize the entanglement as it evolves from the GHZ-like structure of redundant records to the hyper-delocalized correlations of the thermalized states in the microcanonical ensemble? However we characterize it, isn’t it plausible such entanglement could be generated without passing through a preliminary phase of having records? (After all, all states in the microcanonical ensemble are essentially on equal footing.) And couldn’t this be responsible for much or even all of the remaining exponential growth in bond dimension? If so, it seems imperative that we be able to detect it in that un-recorded form since we’ve already seen it won’t be enough to detect records when they are produced and then track the corresponding states as they evolve into the future.

Toward a branch algebra

The limiting case of thermalized states suggests we should consider generalizing our notion of branches (a preferred basis) to some sort of preferred subspaces where any orthogonal decomposition of the state in that subspace is an acceptable way to break it up into branches, at least for the purposes of sampling. For instance, suppose $\psi = \alpha v+\beta w$ where $v$ and $w$ are orthogonal but otherwise Haar-random vectors conditional on the same conserved values. (Here, $\alpha,\beta\in \mathbb{C}$ .) Then $v$ and $w$ will be non-interfering, and hence useful for sampling, for any reasonable choice of feasible measurements $\mathcal{F}: \langle v | F | w \rangle = 0$ for $F\in\mathcal{F}$ . But this is also true for the decomposition $\psi = \alpha' x+\beta' y$ if $x$ and $y$ are orthonormal vectors in the 2D subspace spanned by $v$ and $w$ (so $x=\nu_x v + \mu_x w$ and $y=\nu_y v + \mu_y w$ ) since $x$ and $y$ will also be Haar-random (conditional on orthogonality).

On the other hand, if a state $\psi = \alpha v+\beta w$ can be decomposed into branches $v$ and $w$ that are redundantly recorded (or perhaps are distinguishable according to some weaker^f criterion that nevertheless still implies non-interference), then alternative orthogonal decompositions into states in the spanning space are not on equal footing. But if we consider a different state $\psi' = \alpha v'+\beta w$ that is the same as $\psi$ except that one branch $v$ is replaced by a state $v'$ in the same subspace corresponding to that record outcome, then the corresponding decomposition for $\psi' = \alpha v'+\beta w$ will also be redundantly recorded. Likewise, if we instead just consider modifying the coefficients in $\psi = \alpha v+\beta w$ to get $\psi'' = \alpha'' v+\beta'' w$ , we again expect to get branches based on $v$ and $w$ ; see my discussion of logical properties of branches. Thus there appear to be whole subspaces (in this case, containing $v$ and $v'$ ) that are macroscopically distinguishable from other subspaces (in this case, containing $w$ ).

Taken together, these cases weakly suggest a preferred algebraic structure

(1) $\begin{align*} \mathcal{B} = \bigoplus_j \mathcal{M}_{n_j} \oplus \bigoplus_k \mathcal{I}_{\ell_k} \end{align*}$

that could be called the “branch algebra”. (Note that, for now, the binary operations above are all direct sums, not tensor products.) Here, $n_j$ and $\ell_k$ are positive integers, $\mathcal{M}_{n}$ is the complete matrix algebra of $n\times n$ matrices, and $\mathcal{I}_{\ell}$ is the trivial algebra of multiples of the $\ell\times\ell$ unit matrix. This would induce the branch decompositions

(2) $\begin{align*} \psi = \sum_j \phi_j + \sum_k \chi_{k} \end{align*}$

where the $\phi_j$ and $\chi_{k}$ are distinguishable and indistinguishable branches in the subspaces associated with $\mathcal{M}_{n_j}$ and $\mathcal{I}_{\ell_k}$ , respectively. The indistinguishable branches can be further decomposed as $\chi_{k} = \sum_r\chi_{k,r}$ using any basis $\{\chi_{k,r}\}_r$ of subspace associated with $\mathcal{I}_{\ell_k}$ .

More specifically, the interpretation would be that

The $\mathcal{M}_{n_j}$ represent the unbranched subspaces of Hilbert space, i.e., subspaces in which full coherence is preserved. Intuitively, this would for instance include the subspace generated by changing the wavefunction of a single particle in a distinguishable branch.
Regular (non-generalized) branching happens between these subspaces, so that regular branch decompositions look like $\psi = \sum_j \psi_j$ where $\psi_j$ lives in the $n_j$ -dimensional subspace that $\mathcal{M}_{n_j}$ acts on.
Generalized branches correspond to the identity algebras $\mathcal{I}_{\ell_k}$ . Within the subspace associated with such an algebra, you not only lose coherence, you also have no preferred basis because all states in the subspace are effectively indistinguishable.

In other words, we might represent the branch structure not with a preferred orthogonal basis or preferred set of orthogonal subspaces, but rather a preferred algebra, which would be something like the algebra of the information accessible to feasible measurements.

Equivalence between indistinguishability of branches and the subspace of non-interference

Based on quantum state complexity, Taylor & McCulloch define notions of distinguishability and interferability that are duals for complementary bases: their “interference complexity proxy” for two orthogonal states $a$ and $b$ is equal to the “distinguishability complexity proxy” for $c=(a+e^{i\theta}b)/\sqrt{2}$ and $d=(a-e^{i\theta}b)/\sqrt{2}$ maximized over $\theta$ . This motivates us to make the above connection between distinguishability and a generalized notion of branching subspaces (without making any reference to complexity) more precise.

Suppose a state has a branch decompositions $\psi = \alpha v+\beta w$ where $v$ and $w$ are orthonormal. By virtue of being branches, they should satisfy the non-interference condition:

(3) $\begin{align*} \langle v | F | w\rangle = 0 \end{align*}$

for all $F\in\mathcal{F}$ in some set of feasible observables $\mathcal{F}$ . This guarantees^g that the pure state $\rho = |\psi \rangle\langle\psi |$ cannot be distinguished from the mixture $\tilde{\rho} = |\alpha|^2 |v \rangle\langle v|+|\beta|^2 |w \rangle\langle w|$ by any measurement $F\in\mathcal{F}$ .

Let’s further consider the indistinguishability condition

(4) $\begin{align*} \langle v | F | v\rangle = \langle w | F | w\rangle \end{align*}$

for all $F\in\mathcal{F}$ . If $v$ and $w$ are both non-interfering and indistinguishable (in addition to orthonormal), then we can check that in fact any orthonormal basis of the subspace spanned by $v$ and $w$ is non-interfering and indistinguishable. The reason is just that (4) and (3) fix the 4 matrix elements of $F$ in that subspace such that $F$ (which is Hermitian) is proportional to the identity there.^h Thus, if $\psi$ has a decomposition into non-interfering and indistinguishable states $v$ and $w$ , then it has such a decomposition into any orthonormal basis of the subspace that they span.

On the other hand, suppose that $\psi = \alpha v+\beta w$ where $v$ and $w$ are orthonormal and non-interfering, but not indistinguishable, i.e., there is a feasible observable $F\in\mathcal{F}$ such that $\langle v | F | v\rangle \neq \langle w | F | w\rangle$ . This means that $F$ is not proportional to the identity in the subspace, and hence not diagonal (i.e., non-interfering) in other bases.ⁱ ^j Thus, non-interfering but distinguishable decompositions are unique in this subspace.

For a given set of feasible measurement observables $\mathcal{F}$ , it is natural to classify branches a pair of non-interfering branches as either distinguishable or indistinguishable with respect to $\mathcal{F}$ .

Fully general branch algebra

By Artin-Wedderburn, the most general possible finite-dimensional algebra is (isomorphic to)^k

(5) $\begin{align*} \mathcal{B} = \bigoplus_i \left(\mathcal{M}_{n_i}\otimes\mathcal{I}_{\ell_i}\right). \end{align*}$

We can motivate using this fully general structure for the branch algebra, rather than just the special case (1), by simply considering how we should handle tensoring together two systems.

Suppose we have a large quantum system in a superposition of being fully thermalized at one of two different temperatures, $T_1$ and $T_2$ . Per the previous section, we’d schematically express this as $\mathcal{B} = \mathcal{I}^{(T_1)}\oplus \mathcal{I}^{(T_2)}$ where $\mathcal{I}^{(T_i)}$ is the algebra of all operators proportional to the projector $\Pi_i$ that projects onto to subspace of the microcanonical ensemble associated with $T_i$ . Now suppose we tensor on a qudit system that can be manipulated fully coherently, so that the branch algebra would be trivial: the complete matrix algebra $\mathcal{M}_d$ . Then branch algebra of the combined system would presumably be

(6) $\begin{align*} \tilde{\mathcal{B}} &= \mathcal{M}_d \otimes \mathcal{B} \\ &= \left( \mathcal{M}_d \otimes \mathcal{I}^{(T_1)} \right)\oplus \left( \mathcal{M}_d \otimes \mathcal{I}^{(T_2)} \right) \end{align*}$

Or if the qudit would be well-branched three ways at $T_1$ , so that it individually had branch algebra $\mathcal{M}_{d/3}\oplus\mathcal{M}_{d/3}\oplus\mathcal{M}_{d/3}$ at that temperature but no branches at $T_2$ , we’d expect

(7) $\begin{align*} \tilde{\mathcal{B}} &= \left[ \left( \mathcal{M}_{d/3}\oplus\mathcal{M}_{d/3}\oplus\mathcal{M}_{d/3} \right)\otimes \mathcal{I}^{(T_1)} \right]\oplus \left( \mathcal{M}_d \otimes \mathcal{I}^{(T_2)} \right) \\ &= \left(\mathcal{M}_{d/3}\otimes \mathcal{I}^{(T_1)}\right) \oplus \left(\mathcal{M}_{d/3}\otimes \mathcal{I}^{(T_1)}\right) \oplus \left(\mathcal{M}_{d/3}\otimes \mathcal{I}^{(T_1)}\right) \oplus \left( \mathcal{M}_d \otimes \mathcal{I}^{(T_2)} \right) \end{align*}$

This is of course an example of the more general algebra (5), and the natural generalization of (2) is

(8) $\begin{align*} \psi = \sum_i \psi_i = \sum_i \sum_s \phi_{i,s} \otimes \chi_{i,s} \end{align*}$

where $\phi_{i,s}\in\mathcal{V}_i$ and $\chi_{i,s}\in\mathcal{W}_i$ for the subspaces $\mathcal{V}_{i}$ and $\mathcal{W}_{i}$ associated with $\mathcal{M}_{n_i}$ and $\mathcal{I}_{\ell_i}$ . The choices $\phi_{i,s}$ and $\chi_{i,s}$ are fully determined by $\psi$ by way of the Schmidt decomposition $\psi_i=\sum_s \phi_{i,s} \otimes \chi_{i,s}$ (except for degeneracies), although we could just as well decompose $\psi_i$ in some other way that gave the same reduced state on $\mathcal{V}_i$ .^l Another view on this is that the pure state $\rho = |\psi\rangle\langle\psi |$ will be indistinguishable, through the measurement of any $B\in \mathcal{B}$ , from the state

(9) $\begin{align*} \rho = \sum_i \sigma_i \otimes \eta_i \end{align*}$

where $\sigma_i = \sum_s |\phi_{i,s}\rangle\langle\phi_{i,s} |$ and $\eta_i = \sum_{s} |\chi_{i,s}\rangle\langle\chi_{i,s}|$ .

Against a state-independent branch algebra

The above is an argument that branches should be associated with an algebra $\mathcal{B} = \bigoplus_i \left(\mathcal{M}_{n_i}\otimes\mathcal{I}_{\ell_i}\right)$ that applies not just to a particular state $\psi$ but also to states that are “nearby” in some sense. Indeed, the matrix algebras $\mathcal{M}_{n_i}$ were supposed to correspond to coherent quantum information that was feasibly accessible in the sense that $|\psi\rangle$ and $|\psi'\rangle = A|\psi\rangle$ could be feasibly distinguished for $A\in\mathcal{M}_{n_i}$ , e.g., $\psi$ describes a superposition of macroscopically distinct outcomes and $A$ merely excites a single atom on one branch. So we would get the branches of both $\psi$ and $\psi'$ by projecting onto the same subspaces labeled by $i$ .

It’s then natural to wonder whether branch structure should be described by a single algebra that is independent of the overall state (even if the algebra might depend on the Hamiltonian). Here’s a reason to think that, even if we fix the dynamics, the answer is no.

The commutant $\mathcal{B}'$ of $\mathcal{B}$ represents the information that branches guarantee is inaccessible in the sense that a hypothetical external actor measuring these observables on a pure state $\rho=|\psi\rangle\langle\psi|$ — effectively “decohering the branches” — shouldn’t change the expectation value of anything we could feasibly measure. In other words, (2) and (9) should be infeasible to distinguish.

So suppose that, for all state, the expectation value of a feasibly measurable observable $F$ is unchanged by the measurement of any (often infeasible-to-measure) observable in $\mathcal{B}'$ . This implies $F$ is in $\mathcal{B}$ . To see this, considering the CP map $\Phi$ of a maximally fine-grained projective measurement of an observable in $\mathcal{B}'$ . To leave the expectation value of $F$ unchanged, regardless of the states, $F$ must be a fixed point of the adjoint^m map $\Phi^\dagger$ , which is in fact equal to the map itself for projective measurements: $\Phi^\dagger = \Phi$ . The only observable that is a fixed point of all observables in an algebra (and so is diagonal in all bases) is the one proportional to the identity on that algebra. So, $F$ commutes with $\mathcal{B}'$ , i.e, $F\in \mathcal{B}$ .

Therefore, if decohering the branches should not be feasibly detectable, then the feasible measurement must be a subset of $\mathcal{B}$ . But, as previously mentioned, the feasible measurements plausible generate the entire algebra of all operators, so the branch decomposition would always be trivial, consisting of just the single branch $\psi$ .

One can summarize our conclusion like this: Let’s say that an algebra $\mathcal{B}$ “induces” a branch decomposition on a state $\psi$ as described earlier, and a branch decomposition is “allowed” if decohering a branch decomposition preserves all feasibly measured observables. If the feasible measurements generate the maximal algebra $\mathcal{M}$ of all operators on the entire Hilbert space, then the only algebra that induces an allowed branch decomposition for all $\psi$ is $\mathcal{M}$ , i.e., there is never non-trivial branching.

Here is another characterization: any non-trivial measurement (including POVMs) must disrupt (the expectation value) of some single-site observables for some state. So you can’t hope that there is always a set of subspaces that can be decohered for any wavefunction without affecting feasibly measured observables. If there is generally a branch algebra that induces wavefunction branches, it must have some dependence on the wavefunction.

And indeed, I think that’s reasonable. Just as traditional quantum error correction codes are only protected in some subspace, I suspect it doesn’t make sense to talk about a branch algebra that would apply to all wavefunctions (even for a particular choice of dynamics), only a subspace. It may (and I think will) make sense to talk about something like a branch algebra, but it won’t be universal for all states. I will discuss the possible connection to error correction in my next post.

Footnotes

(↵ returns to text)

For those unfamiliar with tensor networks, the MPS bond dimension of a state at a location on a 1D lattice is just the number of non-zero eigenvalues in the density matrix of the state after tracing out the portion of the lattice to one side of that location. (Strictly speaking this will often be maximum/infinite constant because the state will have a huge number of exponentially tiny eigenvalues, but if we truncate the eigenvalues below some fixed tiny threshold the number of surviving eigenvalues will generally grow exponentially in time.) For lattices in higher dimensions, I’m basically using “bond dimension” to be the number of non-zero eigenvalues when tracing out the half the lattice.↵
I find the toy model quite illuminating, but this sentence is basically the heart of it.↵
There is some subspace $\mathcal{Y}$ of states with fixed values of the conserved quantities, and I think the “microcanonical ensemble” is the normalized projector onto this subspace (a density matrix). The normalized restriction of the Haar measure to states in this subspace (a probability distribution) is, strictly speaking, different, but it can also be called the “microcanonical ensemble” when there is no chance of confusion.↵
It’s logically possible that pairs of redundantly recorded branches interfere before local thermalization, and that the apparent effectiveness of the Copenhagen interpretation arises from some average cancelation between interference terms, but showing this would be a major discovery and I’m not aware of any account along these lines.↵
The best description of branches could depend on the dynamics and/or on the overall state. First, consider dependence on dynamics. Some authors like Adrian Kent want to ensure branching is perfect (no recombining) by “looking into the future” and only declaring things to have branched when we are assured they don’t recombine; thus their definition of branches depend on the dynamics. On the other hands, a records-based definition, the Weingarten definition, and the Taylor-McCulloch definition are all independent of the dynamics, i.e., they only depend on the state at a fixed time. It then becomes a dynamical question whether they avoid recombining. (With such a definition there is always a choice of dynamics that would screw you, but the hope is that some second-law behavior ensures that such branches don’t recombine for all the dynamics in our universe.) Now consider dependence on the overall state. A records-based definition and the Weingarten definition depend on the overall state, while the Taylor-McCulloch definition does not, but as a consequence it’s unclear if it defines a basis at all. Even at a fixed time (putting aside the issues of dynamics and recombination), Taylor-McCulloch only defines a basis insofar as something like the “compatibility” conjecture in their appendix is true. (Strictly speaking that conjecture is false as written, and will require more understanding of generalized branches before it can be fixed).)↵
Redundant records is in some sense much stronger than orthogonality since it allows many fewer such branches in total than the dimension of the Hilbert space. Non-interference is weaker in the logical sense, since it is implied by redundancy, but is also weaker in the sense one can easily construct full bases that span the whole Hilbert space with vectors that are orthogonal and non-interfering.↵
Although strictly speaking the indistinguishability of $\rho$ and $\tilde{\rho}$ requires only the weaker condition $\alpha^* \beta \langle v | F | w\rangle + \beta^* \alpha \langle w | F | v\rangle = 0$ , we intuitively expect that if $\psi = \alpha v+\beta w$ is a branch for some state $\psi$ then $\psi' = \alpha' v+\beta' w$ should be a branch for another state $\psi'$ with any $\alpha',\beta'\in\mathbb{C}$ . In order that we always have $\rho'$ and $\tilde{\rho}'$ indistinguishable, the stronger non-interference condition $\langle v | F | w\rangle = 0$ must hold. See my post on logical properties for branches.↵
Or explicitly: if $x=\nu_x v + \mu_x w$ and $y=\nu_y v + \mu_y w$ are orthonormal, so $1 = \langle x | x\rangle = |\nu_x|^2 + |\mu_x|^2$ , $1 = \langle y | y\rangle = |\nu_y|^2 + |\mu_y|^2$ , $0 = \langle x | y\rangle = \nu_x^*\nu_y + \mu_x^*\mu_y$ ,
then the non-interference (3) and indistinguishability (4) of $v$ and $w$ imply $\langle x | F | y\rangle &= \nu_x^*\nu_y \langle v | F | v\rangle + \mu_x^*\nu_y \langle w | F | v\rangle + \nu_x^*\mu_y \langle v | F | w\rangle + \mu_x^*\mu_y \langle w | F | w\rangle = (\nu_x^*\nu_y + \mu_x^*\mu_y) \langle v | F | v\rangle = 0$ and $\langle x | F | x\rangle - \langle y | F | y\rangle &= |\nu_x|^2 \langle v | F | v\rangle + \mu_x^*\nu_x \langle w | F | v\rangle + \nu_x^*\mu_x \langle v | F | w\rangle + |\mu_x|^2 \langle w | F | w\rangle - |\nu_y|^2 \langle v | F | v\rangle - \mu_y^*\nu_y \langle w | F | v\rangle - \nu_y^*\mu_y \langle v | F | w\rangle - |\mu_y|^2 \langle w | F | w\rangle = (|\nu_x|^2 + |\mu_x|^2 - |\nu_y|^2 + |\mu_y|^2) \langle v | F | v\rangle = 0$ for all $F\in\mathcal{F}$ , i.e., $x$ and $y$ are also non-interfering and indistinguishable.↵
Or explicitly: For any other orthonormal vectors $x$ and $y$ in the subspace spanned by $v$ and $w$ we have $\langle x | F | y\rangle &= \nu_x^*\nu_y \langle v | F | v\rangle + \mu_x^*\nu_y \langle w | F | v\rangle + \nu_x^*\mu_y \langle v | F | w\rangle + \mu_x^*\mu_y \langle w | F | w\rangle = \mu_x^*\mu_y (\langle w | F | w\rangle- \langle v | F | v\rangle)\neq 0$ ,
i.e., $x$ and $y$ are not non-interfering (so long as $mu_x,\mu_y\neq 0$ , i.e., the $x$ and $y$ aren’t just the same as $v$ and $w$ up to a phase).↵
The symmetry between non-interference and indistinguishability for different bases also manifests in Taylor & McCulloch and Aaronson et al., but we can see here that it is not specific to quantum complexity.↵
Strictly speaking the most general algebra is $\mathcal{B} = \mathcal{Z}_z \oplus \bigoplus_i \left(\mathcal{M}_{n_i}\otimes\mathcal{I}_{\ell_i}\right)$ where $\mathcal{Z}_z$ consisting solely of the zero operators on dimension $z$ (so that $\mathcal{B}$ annihilates the associated $z$ -dimensional subspace). But I don’t really know how to make sense of this unless $\mathcal{Z}_z$ is orthogonal to $\psi$ , in which case it doesn’t affect the decomposition anyway. If you want, you can think of this as another reason (beyond what’s discussed in the final section of this post) to expect the proper branch algebra to depend on the state.↵
I haven’t thought yet much about the freedom to decompose the $\chi_{i,s}$ further within $\mathcal{W}_{i}$ , but that freedom is there and important.↵
The adjoint of a CP map $\Phi[\rho] = \sum_i K_i\rho K_i^\dagger$ with Krauss operators $K_i$ is the CP map $\Phi^\dagger[\rho] = \sum_i K_i\dagger \rho K_i$ .↵

Generalizing wavefunction branches to indistinguishable subspaces

Toy model of the problem

What do redundantly recorded branches look like in the future?

Toward a branch algebra

Equivalence between indistinguishability of branches and the subspace of non-interference

Fully general branch algebra

Against a state-independent branch algebra

Footnotes

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