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”.… [continue reading]

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.[continue reading]

Branching theories vs. collapse theories

This post explains the relationship between (objective) collapse theories and theories of wavefunction branching. The formalizations are mathematically very simple, but it’s surprisingly easily to get confused about observational consequences unless laid out explicitly.

(Below I work in the non-relativistic case.)

Branching: An augmentation

In its most general form, a branching theory is some time-dependent orthogonalSome people have discussed non-orthogonal branches, but this breaks the straightforward probabilistic interpretation of the branches using the Born rule. This can be repaired, but generally only by introducing additional structure or principles that, in my experience, usually turns the theory into something more like a collapse theory, which is what I’m trying to constrast with here.a   decomposition of the wavefunction: \psi= \sum_{\phi\in B(t)} \phi where B(t) is some time-dependent set of orthogonal vectors. I’ve expressed this in the Heisenberg picture, but the Schrödinger picture wavefunction and branches are obtained in the usual (non-branch-dependent) way by evolution with the overall unitary: \psi(t)=U_t \psi and \phi(t)=U_t \phi.

We generally expect the branches to fine-grain in time. That is, for any two times t and t'>t, it must be possible to partition the branches B(t') at the later time into subsets B(t',\phi) of child branches, each labeled by a parent branch \phi at the earlier time, so that each subset of children sums up to its corresponding earlier-time parent: \phi = \sum_{\phi' \in B(t',\phi)} \phi' for all \phi\in B(t) where B(t') = \bigcup_{\phi\in B(t)} B(t',\phi) and B(t',\phi)\cap B(t',\tilde\phi) for \phi\neq\tilde\phi. By the orthogonality, a child \phi' will be a member of the subset B(t',\phi) corresponding to a parent \phi if and only if the overlap of the child and the parent is non-zero. In other words, a branching theory fine-grains in time if the elements of B(t) and B(t') are formed by taking partitions P(t) and P(t') of the same set of orthogonal vectors, where P(t') is a refinement of P(t), and vector-summing each subset of the respective partition.… [continue reading]

Comments on Ollivier’s “Emergence of Objectivity for Quantum Many-Body Systems”

Harold Ollivier has put out a nice paper generalizing my best result:

We examine the emergence of objectivity for quantum many-body systems in a setting without an environment to decohere the system’s state, but where observers can only access small fragments of the whole system. We extend the result of Reidel (2017) to the case where the system is in a mixed state, measurements are performed through POVMs, and imprints of the outcomes are imperfect. We introduce a new condition on states and measurements to recover full classicality for any number of observers. We further show that evolutions of quantum many-body systems can be expected to yield states that satisfy this condition whenever the corresponding measurement outcomes are redundant.

Ollivier does a good job of summarizing why there is an urgent need to find a way to identify objectively classical variables in a many-body system without leaning on a preferred system-environment tensor decomposition. He also concisely describes the main results of my paper in somewhat different language, so some of you may find his version nicer to read.A minor quibble: Although this is of course a matter of taste, I disagree that the Shor code example was the “core of the main result” of my paper. In my opinion, the key idea was that there was a sensible way of defining redundancy at all in a way that allowed for proving statements about compatibility without recourse to a preferred non-microscopic tensor structure. The Shor-code example is more important for showing the limits of what redundancy can tell you (which is saturated in a weak sense).[continue reading]

Unital dynamics are mixedness increasing

After years of not having an intuitive interpretation of the unital condition on CP maps, I recently learned a beautiful one: unitality means the dynamics never decreases the state’s mixedness, in the sense of the majorization partial order.

Consider the Lindblad dynamics generated by a set of Lindblad operators L_k, corresponding to the Lindbladian

(1)   \begin{align*} \mathcal{L}[\rho] = \sum_k\left(L_k\rho L_k^\dagger - \{L_k^\dagger L_k,\rho\}/2\right) \end{align*}

and the resulting quantum dynamical semigroup \Phi_t[\rho] = e^{t\mathcal{L}}[\rho]. Let

(2)   \begin{align*} S_\alpha[\rho] = \frac{\ln\left(\mathrm{Tr}[\rho^\alpha]\right)}{1-\alpha}, \qquad \alpha\ge 0 \end{align*}

be the Renyi entropies, with S_{\mathrm{vN}}[\rho]:=\lim_{\alpha\to 1} S_\alpha[\rho] = -\mathrm{Tr}[\rho\ln\rho] the von Neumann entropy. Finally, let \prec denote the majorization partial order on density matrices: \rho\prec\rho' exactly when \mathrm{spec}[\rho]\prec\mathrm{spec}[\rho'] exactly when \sum_{i=1}^r \lambda_i \le \sum_{i=1}^r \lambda_i^\prime for all r, where \lambda_i and \lambda_i^\prime are the respective eigenvalues in decreasing order. (In words: \rho\prec\rho' means \rho is more mixed than \rho'.) Then the following conditions are equivalent:None of this depends on the dynamics being Lindbladian. If you drop the first condition and drop the “t” subscript, so that \Phi is just some arbitrary (potentially non-divisible) CP map, the remaining conditions are all equivalent.a  

  • \mathcal{L}[I]=0
  • \Phi_t[I]=I: “\Phi_t is a unital map (for all t)”
  • \frac{\mathrm{d}}{\mathrm{d}t}S_\alpha[\Phi_t[\rho]] \ge 0 for all \rho, t, and \alpha: “All Renyi entropies are non-decreasing”
  • \Phi_t[\rho]\prec\rho for all t: “\Phi_t is mixedness non-decreasing”
  • \Phi_t[\rho] = \sum_j p_j U^{(t)}_j\rho U^{(t)\dagger}_j for all t and some unitaries U^{(t)}_j and probabilities p_j.

The non-trivial equivalences above are proved in Sec. 8.3 of Wolf, “Quantum Channels and Operations Guided Tour“.See also “On the universal constraints for relaxation rates for quantum dynamical semigroup” by Chruscinski et al [2011.10159] for further interesting discussion.b  

Note that having all Hermitian Lindblad operators (L_k = L_k^\dagger) implies, but is not implied by, the above conditions. Indeed, the condition of Lindblad operator Hermiticity (or, more generally, normality) is not preserved under the unitary gauge freedom L_k\to L_k^\prime = \sum_j u_{kj} L_j (which leaves the Lindbladian \mathcal{L} invariant for unitary u.)… [continue reading]

Table of proposed macroscopic superpositions

Here is a table of proposals for creating enormous superpositions of matter. Importantly, all of them describe superpositions whose spatial extent is comparable to or larger than the size of the object itself. Many are quite speculative. I’d like to keep this table updated, so send me references if you think they should be included.

experimentref.object
composition
object
radius (nm)
nucleon
count
superposition
size (nm)
lifetime
(ms)
repetition
rate (Hz)
KDTL[1-3]OligoporphyrinTo achieve their highest masses, the KDTL interferometer has superposed molecules of functionalized oligoporphyrin, a family of organic molecules composed of C, H, F, N, S, and Zn with molecular weights ranging from ~19,000 Da to ~29,000 Da. (The units here are Daltons, also known as atomic mass units (amu), i.e., the number of protons and neutrons.) The distribution is peaked around 27,000 Da.a  ,00∼1.02.7 × 104100,266100,001.2410,000.00
OTIMA[4-6]Gold (Au),0005.06.0 × 106100,079100,094.0010,600.00
Bateman et al.[7]Silicon (Si),0005.51.1 × 106100,150100,140.0010,000.50
Geraci et al.[8]Silica (SiO2),0006.51.6 × 106100,250100,250.0010,000.50
Wan et al.[9]Diamond (C),0095.07.5 × 109100,100100,000.0510,001.00
MAQRO[10-13]Silica (SiO2),0120.01.0 × 101000,100100,000.0010,000.01
Pino et al.[14]Niobium (Nb)1,000.02.2 × 101300,290100,450.0010,000.10
Stickler et al.
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Lindblad operator trace is 1st-order contribution to Hamiltonian part of reduced dynamics

In many derivations of the Lindblad equation, the authors say something like “There is a gauge freedomA gauge freedom of the Lindblad equation means a transformation we can to both the Lindblad operators and (possibly) the system’s self-Hamiltonian, without changing the reduced dynamics.a   in our choice of Lindblad (“jump”) operators that we can use to make those operators traceless for convenience”. However, the nature of this freedom and convenience is often obscure to non-experts.

While reading Hayden & Sorce’s nice recent paper [arXiv:2108.08316] motivating the choice of traceless Lindblad operators, I noticed for the first time that the trace-ful parts of Lindblad operators are just the contributions to Hamiltonian part of the reduced dynamics that arise at first order in the system-environment interaction. In contrast, the so-called “Lamb shift” Hamiltonian is second order.

Consider a system-environment decomposition \mathcal{S}\otimes \mathcal{E} of Hilbert space with a global Hamiltonian H = H_S + H_{I} + H_E, where H_S = H_S \otimes I_\mathcal{E}, H_E = I_\mathcal{S}\otimes H_E, and H_I = \epsilon \sum_\alpha A_\alpha \otimes B_\alpha are the system’s self Hamiltonian, the environment’s self-Hamiltonian, and the interaction, respectively. Here, we have (without loss of generality) decomposed the interaction Hamiltonian into a tensor product of Hilbert-Schmidt-orthogonal sets of operators \{A_\alpha\} and \{B_\alpha\}, with \epsilon a real parameter that control the strength of the interaction.

This Hamiltonian decomposition is not unique in the sense that we can alwaysThere is also a similar freedom with the environment in the sense that we can send H_E \to H_E + \Delta H_E and \epsilon H_I \to \epsilon H_I - \Delta H_E.b   send H_S \to H_S + \Delta H_S and H_I \to H_I - \Delta H_S, where \Delta H_S = \Delta H_S \otimes I_\mathcal{E} is any Hermitian operator acting only on the system. When reading popular derivations of the Lindblad equation

(1)   \begin{align*} \partial_t \rho_{\mathcal{S}} = -i[\tilde{H}_{\mathcal{S}}, \rho_{\mathcal{S}}] + \sum_i\left[L_i \rho_{\mathcal{S}} L_i^\dagger - (L_i^\dagger L_i \rho_{\mathcal{S}} + \rho_{\mathcal{S}} L_i^\dagger L_i)/2\right] \end{align*}

like in the textbook by Breuer & Petruccione, one could be forgivenSpecifically, I have forgiven myself for doing this…c   for thinking that this freedom is eliminated by the necessity of satisfying the assumption that \mathrm{Tr}_\mathcal{E}[H_I(t),\rho(0)]=0, which is crucially deployed in the “microscopic” derivation of the Lindblad equation operators \tilde{H}_{\mathcal{S}} and \{L_i\} from the global dynamics generated by H.… [continue reading]

Weingarten’s branches from quantum complexity

Don Weingarten’s newI previously blogged about earlier work by Weingarten on a related topic. This new paper directly addresses my previous concerns.a   attack [2105.04545] on the problem of defining wavefunction branches is the most important paper on this topic in several years — and hence, by my strange twisted lights, one of the most important recent papers in physics. Ultimately I think there are significant barriers to the success of this approach, but these may be surmountable. Regardless, the paper makes tons of progress in understanding the advantages and drawbacks of a definition of branches based on quantum complexity.

Here’s the abstract:

Beginning with the Everett-DeWitt many-worlds interpretation of quantum mechanics, there have been a series of proposals for how the state vector of a quantum system might split at any instant into orthogonal branches, each of which exhibits approximately classical behavior. Here we propose a decomposition of a state vector into branches by finding the minimum of a measure of the mean squared quantum complexity of the branches in the branch decomposition. In a non-relativistic formulation of this proposal, branching occurs repeatedly over time, with each branch splitting successively into further sub-branches among which the branch followed by the real world is chosen randomly according to the Born rule. In a Lorentz covariant version, the real world is a single random draw from the set of branches at asymptotically late time, restored to finite time by sequentially retracing the set of branching events implied by the late time choice. The complexity measure depends on a parameter b with units of volume which sets the boundary between quantum and classical behavior.
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