What logical structure for branches?

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

Taylor & McCulloch have a tantalizing paper about which I’ll have much to say in the future. However, for now I want to discuss the idea of the “compatibility” of branch decompositions, which is raised in their appendix. In particular, the differences between their approach and mine prompted me to think more about how we could narrow down on what sorts of logicalThis is “logic” in the same sense of identifying sets of propositions in consistent histories that comport with the axioms of a classical probability space, before discussing any questions of physics.a   axioms for branches could be identified even before we pin down a physical definition. Indeed, as I will discuss below, the desire for compatibility raises the hope that some natural axioms for branches might enable the construction of a preferred decomposition of the Hilbert space into branching subspaces, and that this might be done independently of the particular overall wavefunction. However, the axioms that I write down prove to be insufficient for this task.

Logical branch axioms

Suppose we have a binary relation “\perp\hspace{-1.1 em}\triangle” on the vectors in a (finite-dimensional) Hilbert space that indicates that two vectors (states), when superposed, should be considered to live on distinct branches. I will adopt the convention that “z = v\perp\hspace{-1.1 em}\triangle w” is interpreted to assert that z=v+w and that the branch relation v\perp\hspace{-1.1 em}\triangle w holds.This doesn’t constrain us because if we just want to assert the binary relation without asserting equality of the sum to a third vector, we write v\perp\hspace{-1.1 em}\triangle w without setting it equal to anything, and if we just want addition without asserting the relation, we write v+w=z. See a later footnote for further motivation for this convention.b   Putting aside physical properties of branches and considering just what’s necessary to take a Copenhagen interpretation given the rules of classical logic, we might demand this relation to have these properties:Thrice-recorded branches satisfy these axioms, as long as you pick a length scale require records in a sphere of that radius. Taylor-McCulloch branches do not because they won’t always be compatible. I don’t know whether Weingarten branches are compatible except in the special case of branching in spatially disjoint uncorrelated systems, in which case they are.c  

  1. Zero identity: \forall v, \,\, v\perp\hspace{-1.1 em}\triangle 0
  2. Reflexivity: v\perp\hspace{-1.1 em}\triangle w \Leftrightarrow  w\perp\hspace{-1.1 em}\triangle v
  3. Associativity: (v\perp\hspace{-1.1 em}\triangle w)\perp\hspace{-1.1 em}\triangle x \Leftrightarrow  v\perp\hspace{-1.1 em}\triangle (w\perp\hspace{-1.1 em}\triangle x)
  4. Orthogonality: v\perp\hspace{-1.1 em}\triangle w \Rightarrow  v \perp w
  5. Homogeneity: v\perp\hspace{-1.1 em}\triangle w \Rightarrow  \alpha v\perp\hspace{-1.1 em}\triangle \beta w for \alpha,\beta\in\mathbb{C}
  6. SuperpositionNote that this is not branch coarse-graining, since it need not be true that v\perp\hspace{-1.1 em}\triangle v' or even that v\perp v'. For instance, v and v' could refer to states that differ only by a single particle having a different position-space wavefunction, where as w could be a radically different distribution of matter.d  : v\perp\hspace{-1.1 em}\triangle w, v'\perp\hspace{-1.1 em}\triangle w \Rightarrow ( v+v') \perp\hspace{-1.1 em}\triangle w
  7. Binary compatibility: v\perp\hspace{-1.1 em}\triangle w =  x\perp\hspace{-1.1 em}\triangle y \Rightarrow \exists a,b,c,d such that v = a \perp\hspace{-1.1 em}\triangle b, w = c\perp\hspace{-1.1 em}\triangle d, x = a\perp\hspace{-1.1 em}\triangle c, y = b\perp\hspace{-1.1 em}\triangle d with a\perp\hspace{-1.1 em}\triangle d, b\perp\hspace{-1.1 em}\triangle c (so v\perp\hspace{-1.1 em}\triangle w =  x \perp\hspace{-1.1 em}\triangle y = a \perp\hspace{-1.1 em}\triangle b \perp\hspace{-1.1 em}\triangle c \perp\hspace{-1.1 em}\triangle d)

Some explanatory comments: Here, “v\perp w” means normal orthogonality, i.e., \langle v | w \rangle =0. Note that v \perp\hspace{-1.1 em}\triangle w means that v and w are on different branches, but that they might contain further branching within themselves (e.g., if v=x \perp\hspace{-1.1 em}\triangle x'). The zero identity isn’t physically motivated — it’s just a convention I found useful. I don’t think this convention leads any pathologies, but I definitely might have missed something.e  

From the reflexivity, associativity, homogeneity, and superposition axioms we can derive

  • Distributivity: \alpha(v \perp\hspace{-1.1 em}\triangle w) \Leftrightarrow \alpha v \perp\hspace{-1.1 em}\triangle w \Leftrightarrow v \perp\hspace{-1.1 em}\triangle \alpha w for \alpha\in\mathbb{C}
  • Fine-graining and coarse-graining:I think you could invert things so that some version of fine-graining and coarse-graining was axiomatic, with associativity then a derived property. This would arguably be more elegant since the former is more closely connected to the rules of logic.f   v \perp\hspace{-1.1 em}\triangle w, v' \perp\hspace{-1.1 em}\triangle w, v \perp\hspace{-1.1 em}\triangle v' \Leftrightarrow (v\perp\hspace{-1.1 em}\triangle v') \perp\hspace{-1.1 em}\triangle w

I think these axioms are reasonably motivated by our intuitive notion of branches, but let me emphasize: I do not at all think these logical axioms are sufficient to define branches.Indeed, the axioms here are unitarily invariant, so if one branch relation “\perp\hspace{-1.1 em}\triangle” satisfied them, there would exist another “\perp\hspace{-1.1 em}\tilde{\triangle}” such that v \perp\hspace{-1.1 em}\tilde{\triangle} w \Leftrightarrow (U v) \perp\hspace{-1.1 em}\triangle (U w)..g    I am merely trying on the hypothesis that they are necessary, and then looking at what we could conclude from that. A complete definition of branches would be augmented with physical axioms, like complexity thresholds or redundant records.

Multi-branching compatibility and the maximally fine-grained branch decomposition

The last axiom looks complicated, but the idea is simple: if you have two different ways of decomposing the wavefunction into two branches, naturally corresponding to two classically distinguishable possibilities, then there is some joint decomposition into (at most) four branches corresponding to all four logical possibilities. (Our convention that all vectors are branches with respect to the zero vector allows for the fact that some of those logical possibilities could have zero probability.) From this axiom one can quickly derive that compatibility extends to arbitrary sums:Here I have defined the big indexed sum notation “{\perp\hspace{-1.1 em}\triangle}_k b_k := b_1 {\perp\hspace{-1.1 em}\triangle} b_2 {\perp\hspace{-1.1 em}\triangle}\cdots” to act like a normal indexed sum \sum_k b_k:=b_1+b_2+\cdots along with the assertion of pairwise branching, i.e., b_k \perp\hspace{-1.1 em}\triangle b_{k'} for all k\neq k'.h  

  • Multi-way compatibility: {\perp\hspace{-1.1 em}\triangle}_i v_i =  {\perp\hspace{-1.1 em}\triangle}_j w_j \Rightarrow \exists \{a_k\} such that v_i = {\perp\hspace{-1.1 em}\triangle}_{k\in V_i} a_k and w_j = {\perp\hspace{-1.1 em}\triangle}_{k\in W_j} a_k where \{V_i\} and \{W_j\} are partitions of the indices \{k\} and where a_k \perp\hspace{-1.1 em}\triangle a_{k'} for all k\neq k' (so {\perp\hspace{-1.1 em}\triangle}_i v_i =  {\perp\hspace{-1.1 em}\triangle}_j w_j = {\perp\hspace{-1.1 em}\triangle}_{k} a_k)

In other words, given any two branch decompositions, there is a joint decomposition that is compatible with both in the sense that all branches in the original two decompositions are composed of disjoint sums of branches from the joint decomposition. From this one can quickly see that each vector in Hilbert space has a unique maximally fine-grained branch decompositionRemember, we’re just working in finite dimensions here. (“Hilbert” is just a reminder that our vectors are states in quantum mechanics.)i   defined as the joint decomposition (not containing the zero vector, by convention) for which all other branch decompositions are coarse-grainings. This furthermore implies there is a set of “unbranchable” vectors for which the maximally fine-grained branch decomposition is trivial (just the vector itself).

A universal preferred basis?

Drawing on our intuition about branches, and especially on the existence of a maximally fine-grained decomposition, one might naturally wonder whether a binary branching relation satisfying these axioms picks out a preferred decomposition \mathcal{V} = \bigoplus_i \mathcal{V}_i of the entire Hilbert space, where the maximally fine-grained branch decomposition of any vector v is given by v = \sum_i v_i, where v_i = P_i v \in \mathcal{V}_i, and where P_i projects onto the ith subspace \mathcal{V}_i. This would be very striking because it would mean that there is a “preferred basis” — really, preferred subspaces — for branches that is independent of the overall state \psi at a particular time. Different definitions of branches (e.g., based on redundant records vs. quantum state complexity) would generally pick out different preferred bases, but each would be universal for that branching relation rather than depending on the details of the (closed) system under study. Perhaps this is too much to hope for, but it would be profound if true.

Unfortunately, the above axioms are not enough to obtain this, which we can see from the following counter example. Consider a three-dimensional Hilbert space with \{a,b,c\} giving an orthonormal basis. Assert a {\perp\hspace{-1.1 em}\triangle} b, which must extend to \alpha a {\perp\hspace{-1.1 em}\triangle} \beta b = \beta b {\perp\hspace{-1.1 em}\triangle} \alpha a for any \alpha,\beta \in \mathbb{C} per the homogeneity and reflexivity axiom, and define these to be the only non-trivial branchings (i.e, v {\perp\hspace{-1.1 em}\triangle} w\Rightarrow v\propto a, w\propto b or w\propto a, v\propto b). It’s quick to check by looking at \psi = \alpha a + \beta b +\gamma c for the separate cases \gamma = 0 and \gamma\neq 0 that there is no preferred basis that always gives the maximally fine-grained branch decomposition. The issue, intuitively, is that a and b are distinct enough to be considered separate branches, but c is insufficiently distinct both from a and from b.

Thus, if we want a notion of logical branches that induces a preferred basis in this sense, we need to change/supplement the logical branch axioms above. Edit: As I show in a follow-up post, a universal basis cannot exist if we want decoherence with respect to that basis to leave feasibly measured observables invariant (assuming, as is reasonable, that those observables generate all observables under products and linear combinations).

Other assorted comments

Degrees of branchiness

The above counterexample, along with some physical arguments one could imagine, suggests that maybe we should not be working with a binary relation for branches but rather a degree of “branchiness”Taylor & McCulloch use this term within their approach.j  : Maybe the vectors v and w are very distinct different branches, but x is only sort of distinct from either of them. One could imagine a measure of branchiness \mathcal{B} between vectors which would be expected to satisfy the axioms of a metric:

  1. \mathcal{B}(v,v)=0
  2. \mathcal{B}(v,w) > 0 (or \mathcal{B}(v,w) \ge 0 for a pseudometric)
  3. \mathcal{B}(v,w)=\mathcal{B}(w,v)
  4. \mathcal{B}(v,w)\le \mathcal{B}(v,x) + \mathcal{B}(x,w)
Other approaches

This post is based on the guess that whether two vectors should be considered to be on separate branches is something that can be determined by looking at those vectors alone, and in particular does not depend on the overall state they might be a part of and/or on how things dynamically evolve in the future. I think this guess is interesting but in a later post (discussed below), we’ll find it has a serious blocker.

Direct sum of algebras and generalizing branches

The astute reader will notice that this branch relation “\perp\hspace{-1.1 em}\triangle” has a lot in common with the direct sum “\oplus” of subspaces. One difference is that the direct sum of subspaces is formally understood to be the direct sum of vector spaces that are isomorphic to the subspaces of the encompassing vector spaceThis is essentially the same as the distinction between (a) the direct sum \mathcal{M}_N \oplus \mathcal{M}_N of the complete set \mathcal{M}_N of N\times N matrices with the complete set \mathcal{M}_M of M\times M to get an algebra of block-diagonal (N+M)\times(N+M) matrices and (b) the abuse-of-terminology “direct sum” \mathcal{M}_{N;M,0}\oplus \mathcal{M}_{N,0;M} of the set \mathcal{M}_{N;M,0} of (N+M)\times(N+M) matrices that are zero outside the N\times B block with the set \mathcal{M}_{N,0;M} of (N+M)\times(N+M) matrices that are zero outside the M\times M block. Now the former, \mathcal{M}_N \oplus \mathcal{M}_N, is the mathematically careful construction. Because \mathcal{M}_{N;M,0} is isomorphic to \mathcal{M}_{N} and \mathcal{M}_{N,0;M} is isomorphic to \mathcal{M}_{M}, the latter, \mathcal{M}_{N;M,0}\oplus \mathcal{M}_{N,0;M}, is an acceptable abuse of terminology with the understanding that the “\oplus” in “\mathcal{M}_{N;M,0} \oplus \mathcal{M}_{N,0;M}” is intended (i) to assert that \mathcal{M}_{N;M,0} and \mathcal{M}_{N,0;M} are orthogonal subspace and (ii) to “sum” these subspaces by taking all possible binary sums of their members. That is essentially why we have defined “\perp\hspace{-1.1 em}\triangle” to be both a binary relation and a binary sum operation. If you hate this overloaded physics-y math, well, haters gonna hate.k  , whereas here we are trying to define a binary relation directly on the vectors in the full vector space.

Along these lines, and as I will discuss in a future post, it will probably be desirable to develop a generalized notion of branching where some branches are not distinguishable with feasible measurements so that branches are closed under some kinds of superpositions (forming a different kind of preferred subspaces). If one used a notion of branches generalized in this way, one would naturally need to modify the logical axiom, especially the notion of compatability. In particular, there would not be unique maximal branch decompositions, but rather some sort of equivalence class of decompositions.

Footnotes

(↵ returns to text)

  1. This is “logic” in the same sense of identifying sets of propositions in consistent histories that comport with the axioms of a classical probability space, before discussing any questions of physics.
  2. This doesn’t constrain us because if we just want to assert the binary relation without asserting equality of the sum to a third vector, we write v\perp\hspace{-1.1 em}\triangle w without setting it equal to anything, and if we just want addition without asserting the relation, we write v+w=z. See a later footnote for further motivation for this convention.
  3. Thrice-recorded branches satisfy these axioms, as long as you pick a length scale require records in a sphere of that radius. Taylor-McCulloch branches do not because they won’t always be compatible. I don’t know whether Weingarten branches are compatible except in the special case of branching in spatially disjoint uncorrelated systems, in which case they are.
  4. Note that this is not branch coarse-graining, since it need not be true that v\perp\hspace{-1.1 em}\triangle v' or even that v\perp v'. For instance, v and v' could refer to states that differ only by a single particle having a different position-space wavefunction, where as w could be a radically different distribution of matter.
  5. I don’t think this convention leads any pathologies, but I definitely might have missed something.
  6. I think you could invert things so that some version of fine-graining and coarse-graining was axiomatic, with associativity then a derived property. This would arguably be more elegant since the former is more closely connected to the rules of logic.
  7. Indeed, the axioms here are unitarily invariant, so if one branch relation “\perp\hspace{-1.1 em}\triangle” satisfied them, there would exist another “\perp\hspace{-1.1 em}\tilde{\triangle}” such that v \perp\hspace{-1.1 em}\tilde{\triangle} w \Leftrightarrow (U v) \perp\hspace{-1.1 em}\triangle (U w)..
  8. Here I have defined the big indexed sum notation “{\perp\hspace{-1.1 em}\triangle}_k b_k := b_1 {\perp\hspace{-1.1 em}\triangle} b_2 {\perp\hspace{-1.1 em}\triangle}\cdots” to act like a normal indexed sum \sum_k b_k:=b_1+b_2+\cdots along with the assertion of pairwise branching, i.e., b_k \perp\hspace{-1.1 em}\triangle b_{k'} for all k\neq k'.
  9. Remember, we’re just working in finite dimensions here. (“Hilbert” is just a reminder that our vectors are states in quantum mechanics.)
  10. Taylor & McCulloch use this term within their approach.
  11. This is essentially the same as the distinction between (a) the direct sum \mathcal{M}_N \oplus \mathcal{M}_N of the complete set \mathcal{M}_N of N\times N matrices with the complete set \mathcal{M}_M of M\times M to get an algebra of block-diagonal (N+M)\times(N+M) matrices and (b) the abuse-of-terminology “direct sum” \mathcal{M}_{N;M,0}\oplus \mathcal{M}_{N,0;M} of the set \mathcal{M}_{N;M,0} of (N+M)\times(N+M) matrices that are zero outside the N\times B block with the set \mathcal{M}_{N,0;M} of (N+M)\times(N+M) matrices that are zero outside the M\times M block. Now the former, \mathcal{M}_N \oplus \mathcal{M}_N, is the mathematically careful construction. Because \mathcal{M}_{N;M,0} is isomorphic to \mathcal{M}_{N} and \mathcal{M}_{N,0;M} is isomorphic to \mathcal{M}_{M}, the latter, \mathcal{M}_{N;M,0}\oplus \mathcal{M}_{N,0;M}, is an acceptable abuse of terminology with the understanding that the “\oplus” in “\mathcal{M}_{N;M,0} \oplus \mathcal{M}_{N,0;M}” is intended (i) to assert that \mathcal{M}_{N;M,0} and \mathcal{M}_{N,0;M} are orthogonal subspace and (ii) to “sum” these subspaces by taking all possible binary sums of their members. That is essentially why we have defined “\perp\hspace{-1.1 em}\triangle” to be both a binary relation and a binary sum operation. If you hate this overloaded physics-y math, well, haters gonna hate.
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