Category: Physics

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One scaling framework for three Lindblad limits

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This note shows how three microscopic derivations of a Markovian Lindblad (GKSL) generator are really just variations on the same basic framework, differing in the relative strengths of the system, bath, and interaction Hamiltonians:

  • the weak-coupling/Davies limit (Born–Markov plus a secular average),
  • the singular-coupling/white-noise limit (fast bath with appropriately rescaled coupling), and
  • the fast-system limit (“super-secular”).

The first two are familiar. The third is partially degenerate, and so less often discussed, but appears, e.g., in the classic decoherence literature on chiral molecules. The three variations are often presented in ways that make them look completely distinct, which obscures their close relationship.

The scaling ansatz and the Markov limit

Fix a system–bath decomposition and consider a one-parameter family of Hamiltonians

(1)   \begin{align*} H^{(\lambda)}=\lambda^{-a}H_S+\lambda^{-c}V+\lambda^{-b}H_B, \qquad \lambda\to 0, \end{align*}

where H_S acts on the system Hilbert space, H_B on the bath Hilbert space, and V couples them. Think of a,b,c \in \mathbb{R} as encoding how quickly the system rotates (\lambda^{-a}), how quickly the bath decorrelates (\lambda^{-b}), and how strongly the coupling acts (\lambda^{-c}), all while keeping the physical time variable t fixed.

We assume the usual minimal structure used in essentially all microscopic derivations of the Lindblad equation:

  • Factorized initial state: \rho(0)=\rho_S(0)\otimes \eta_B.
  • Stationary bath state: [\eta_B,H_B]=0.
  • Centered coupling: \operatorname{Tr}_B(\eta_B V)=0, i.e., no first-order drift.

A convenient form is

(2)   \begin{align*} V=\sum_\alpha A_\alpha\otimes B_\alpha, \qquad \operatorname{Tr}_B(\eta_B B_\alpha)=0. \end{align*}

The question is: for which (a,b,c) can the reduced dynamics

    \[\rho_S^{(\lambda)}(t)=\operatorname{Tr}_B\!\Big(e^{-iH^{(\lambda)}t}\big(\rho_S(0)\otimes\eta_B\big)e^{iH^{(\lambda)}t}\Big)\]

have a nontrivial, finite \lambda\to 0 limit that is a time-homogeneous quantum dynamical semigroup, i.e., a Lindblad equation?

Why the scaling constraint 2c = b is essentially forced

Go to the interaction picture with respect to the free Hamiltonian

    \[H_0^{(\lambda)}=\lambda^{-a}H_S+\lambda^{-b}H_B.\]

Then the interaction-picture coupling is

    \[V_I^{(\lambda)}(t)=\lambda^{-c}\sum_\alpha A_\alpha^{(\lambda)}(t)\otimes B_\alpha^{(\lambda)}(t),\]

with

    \[A_\alpha^{(\lambda)}(t)=e^{i\lambda^{-a}H_S t}A_\alpha e^{-i\lambda^{-a}H_S t},\qquad B_\alpha^{(\lambda)}(t)=e^{i\lambda^{-b}H_B t}B_\alpha e^{-i\lambda^{-b}H_B t}.\]

Bath correlations in the stationary state \eta_B are

    \[C_{\alpha\beta}(t)=\operatorname{Tr}_B\!\big(\eta_B B_\alpha(t)B_\beta(0)\big),\qquad B_\alpha(t)=e^{iH_B t}B_\alpha e^{-iH_B t}.\]

Because B_\alpha^{(\lambda)}(t)=B_\alpha(t/\lambda^b), one has

(3)   \begin{align*} \operatorname{Tr}_B\!\big(\eta_B B_\alpha^{(\lambda)}(t)B_\beta^{(\lambda)}(s)\big)=C_{\alpha\beta}\!\left(\frac{t-s}{\lambda^b}\right). \end{align*}

Now recall what every Born–Markov style derivation is doing at second order: the reduced generator is built from a memory integral of the form

(4)   \begin{align*} \frac{d}{dt}\rho_S^{(\lambda)}(t)\;\approx\;-\int_0^\infty ds\;\operatorname{Tr}_B\!\Big([V_I^{(\lambda)}(t),[V_I^{(\lambda)}(t-s),\rho_S^{(\lambda)}(t)\otimes \eta_B]]\Big), \end{align*}

where “\approx” means

  1. truncate at second order in the interaction,
  2. replace \rho_S^{(\lambda)}(t-s) by \rho_S^{(\lambda)}(t) inside the integral, and
  3. extend the upper limit to \infty using the decay of bath correlations.
… [continue reading]
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Checking our forecast for quantum computing in 2024

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Scott Aaronson has a nice post discussing recent experimental progress in quantum computing from Google, Microsoft, and Quantinuum. In particular, he says

Let me end by sticking my neck out. If hardware progress continues at the rate we’ve seen for the past year or two, then I find it hard to understand why we won’t have useful fault-tolerant QCs within the next decade. (And now to retreat my neck a bit: the “if” clause in that sentence is important and non-removable!)

If you’ll permit me to indulge, I’d like to compare the reported progress to the predictions from the 2020 forecasting paper by Jaime Sevilla and me.

I understand the Google results to be that they have ~10-3 physical two-qubit error rates and they they would have ~10-6 logical error rates if they had ~1500 physical qubits, but they actually have 101 in the present device.

For forecasting purposes, Jaime and I defined a figure of merit, the “generalized logical qubit” (GLQ) count. See Eq. (1) and Fig. 1. It essentially equals the surface-code logical qubit count in the limit of low errors and many qubits (assuming we demand 10-18 logical error rate), but is smoothly defined with fractional values when error rates are larger and/or physical qubit counts are smaller. Our naive (log linear regression) forecast from data at the time was that 1 GLQ would be achieved 2026-2033 and 4,100 GLQs (the rough number necessary to be cryptographically dangerous) would be achieved in 2029-2060 (both 90% confidence intervals — not Bayesian credences!). Our forecasts for 2024 were 0.02-0.3 GLQs.

I’m pleased to see that the above Google numbers correspond to a GLQ count of about 0.03, which is consistent with our model (though this is perhaps not impressive since it’s only been 4 years and our confidence intervals were wide).… [continue reading]

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Generalizing wavefunction branches to indistinguishable subspaces

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[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]

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What logical structure for branches?

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[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]

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Branching theories vs. collapse theories

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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]

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

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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]

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Unital dynamics are mixedness increasing

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[EDIT 2025-6-25: The original version of this post incorrectly included mixed-unitary and Renyi-entropy-increasing in the equivalent conditions.]

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 \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 two conditions and drop the “t” subscript, so that \Phi is just some arbitrary (potentially non-divisible) CP map, the remaining three conditions are all equivalent.a  

  • \sum_k [L_k, L_k^\dagger]=0
  • \mathcal{L}[I]=0
  • \Phi_t[I]=I: “\Phi_t is a unital map (for all t\ge 0)”
  • \Phi_t[\rho]\prec\rho for all t\ge 0: “\Phi_t is mixedness non-decreasing”
  • \frac{\mathrm{d}}{\mathrm{d}t}\mathrm{Tr}[f(\Phi_t[\rho])] \ge 0 for all \rho, t\ge 0, and concave functions f:\mathbb{R}\to\mathbb{R}

The last condition implies that all Renyi entropies, including the Shannon entropy, are non-decreasing: \frac{\mathrm{d}}{\mathrm{d}t}S_\alpha[\Phi_t[\rho]] \ge 0 for all \rho, t\ge 0, and \alpha.

The non-trivial equivalences above are proved by Thm 8.8 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  

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Table of proposed macroscopic superpositions

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Here is a table of proposed experiments for creating enormous superpositions of matter, along with two that have actually been performed (KDTL and MUSCLE, in green). 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. [Updated: 2026-Jan-23]

experimentref.object
composition		object
radius (nm)		nucleon
count		superp.
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.
… [continue reading]