Category: Physics

2 Comments

Lindblad operator trace is 1st-order contribution to Hamiltonian part of reduced dynamics

  By

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]

One Comment

Weingarten’s branches from quantum complexity

  By

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.
… [continue reading]
No comments

How long-range coherence is encoded in the Weyl quasicharacteristic function

  By

In this post, I derive an identity showing the sense in which information about coherence over long distances in phase space for a quantum state \rho is encoded in its quasicharacteristic function \mathcal{F}_{\rho}, the (symplectic) Fourier transform of its Wigner function. In particular I show

(1)   \begin{align*} \int \mathrm{d}^2 \gamma |\langle \alpha|\rho|\beta\rangle|^2 = (|\mathcal{F}_\rho|^2 \ast |G|^2)(\beta-\alpha) \end{align*}

where |\alpha\rangle and |\beta\rangle are coherent states, \gamma:=(\alpha+\beta)/2 is the mean phase space position of the two states, “\ast” denotes the convolution, and G(\alpha) = e^{-|\alpha|^2} is the (Gaussian) quasicharacteristic function of the ground state of the Harmonic oscillator.

Definitions

The quasicharacteristic function for a quantum state \rho of a single degree of freedom is defined as

    \[\mathcal{F}_\rho(\xi) := \mathrm{Tr}[\rho D_\xi] = \langle\rho,D_\xi\rangle_{\mathrm{HS}},\]

where D_\xi = e^{i \xi \wedge R} is the Weyl phase-space displacement operator, \xi = (\xi_{\mathrm{x}},\xi_{\mathrm{p}}) \in \mathbf{R}^2 are coordinates on “reciprocal” (i.e., Fourier transformed) phase space, R:=(X,P) is the phase-space location operator, X and P are the position and momentum operators, “\langle \cdot,\cdot\rangle_{\mathrm{HS}}” denotes the Hilbert-Schmidt inner product on operators, \langle A,B\rangle_{\mathrm{HS}}:=\mathrm{Tr}[A^\dagger B], and “\wedge” denotes the symplectic form, \alpha\wedge\beta := \alpha_{\mathrm{x}}\beta_{\mathrm{p}} - \alpha_{\mathrm{p}}\beta_{\mathrm{x}}. (Throughout this post I use the notation established in Sec. 2 of my recent paper with Felipe Hernández.) It has variously been called the quantum characteristic function, the chord function, the Wigner characteristic function, the Weyl function, and the moment-generating function. It is the quantum analog of the classical characteristic function.

Importantly, the quasicharacteristic function obeys |\mathcal{F}_{\rho}(\xi)|\le 1 and \mathcal{F}_{\rho}(0)=1, just like the classical characteristic function, and provides a definition of the Wigner function where the linear symplectic symmetry of phase space is manifest:

(2)   \begin{align*} \mathcal{W}_{\rho}(\alpha) &:= (2\pi)^{-1}\int\! \mathrm{d}^2\xi \, e^{-i\alpha\wedge\xi}\mathcal{F}_{\rho}(\xi)\\ &= (2\pi)^{-1}\int\! \mathrm{d}\Delta x \, e^{ip\Delta x} \rho(x-\Delta x/2,x+\Delta x/2) \end{align*}

where \alpha = (\alpha_{\mathrm{x}},\alpha_{\mathrm{p}}) = (x,p) \in \mathbf{R}^2 is the phase-space coordinate and \rho(x,x')=\langle x | \rho|x'\rangle is the position-space representation of the quantum state. This first line says that \mathcal{W}_{\rho} and \mathcal{F}_{\rho} are related by the symplectic Fourier transform. (This just means the inner product “\cdot” in the regular Fourier transform is replaced with the symplectic form, and has the simple effect of exchanging the reciprocal variables, (\xi_{\mathrm{x}},\xi_{\mathrm{p}})\to (-\xi_{\mathrm{p}},\xi_{\mathrm{x}}), simplifying many expressions.)… [continue reading]

No comments

Gravitational transmission of quantum information by Carney et al.

  By

Carney, Müller, and Taylor have a tantalizing paper on how the quantum nature of gravity might be confirmed even though we are quite far from being able to directly create and measure superpositions of gravitationally appreciable amounts of matter (hereafter: “massive superpositions”), and of course very far from being able to probe the Planck scale where quantum gravity effects dominate. More precisely, the idea is to demonstrate (assuming assumptions) that the gravitational field can be used to transmit quantum information from one system to another in the sense that the effective quantum channel is not entanglement breaking.

We suggest a test of a central prediction of perturbatively quantized general relativity: the coherent communication of quantum information between massive objects through gravity. To do this, we introduce the concept of interactive quantum information sensing, a protocol tailored to the verification of dynamical entanglement generation between a pair of systems. Concretely, we propose to monitor the periodic wavefunction collapse and revival in an atomic interferometer which is gravitationally coupled to a mechanical oscillator. We prove a theorem which shows that, under the assumption of time-translation invariance, this collapse and revival is possible if and only if the gravitational interaction forms an entangling channel. Remarkably, as this approach improves at moderate temperatures and relies primarily upon atomic coherence, our numerical estimates indicate feasibility with current devices.
[Edit: See also the November 2021 errata.]

Although I’m not sure they would phrase it this way, the key idea for me was that merely protecting massive superpositions from decoherence is actually not that hard; sufficient isolation can be achieved in lots of systems.… [continue reading]

No comments

Comments on Baldijao et al.’s GPT-generalized quantum Darwinism

  By

This nice recent paper considers the “general probabilistic theory” operational framework, of which classical and quantum theories are special cases, and asks what sorts of theories admit quantum Darwinism-like dynamics. It is closely related to my interest in finding a satisfying theory of classical measurement.

Quantum Darwinism and the spreading of classical information in non-classical theories
Roberto D. Baldijão, Marius Krumm, Andrew J. P. Garner, and Markus P. Müller
Quantum Darwinism posits that the emergence of a classical reality relies on the spreading of classical information from a quantum system to many parts of its environment. But what are the essential physical principles of quantum theory that make this mechanism possible? We address this question by formulating the simplest instance of Darwinism – CNOT-like fan-out interactions – in a class of probabilistic theories that contain classical and quantum theory as special cases. We determine necessary and sufficient conditions for any theory to admit such interactions. We find that every non-classical theory that admits this spreading of classical information must have both entangled states and entangled measurements. Furthermore, we show that Spekkens’ toy theory admits this form of Darwinism, and so do all probabilistic theories that satisfy principles like strong symmetry, or contain a certain type of decoherence processes. Our result suggests the counterintuitive general principle that in the presence of local non-classicality, a classical world can only emerge if this non-classicality can be “amplified” to a form of entanglement.

After the intro, the authors give self-contained background information on the two key prerequisites: quantum Darwinism and generalized probabilistic theories (GPTs). The former is an admirable brief summary of what are, to me, the core and extremely simple features of quantum Darwinism.… [continue reading]

2 Comments

Moyal bracket with manifest (affine) symplectic covariance

  By

Moyal’s equation for a Wigner function W of a quantum system with (Wigner-transformed) Hamiltonian H is \partial_t W = \{ H,W \}_\hbar where the Moyal bracket is a binary operator on the space of functions over phase space. Unfortunately, it is often written down mysteriously as

(1)   \begin{align*} \{ A,B \}_\hbar = \frac{2}{\hbar} A \sin \left( \frac{\hbar}{2} \left(\overleftarrow{\partial}_x\overrightarrow{\partial}_p - \overleftarrow{\partial}_p\overrightarrow{\partial}_x \right) \right) B, \end{align*}

where the arrows over partial derivatives tell you which way they act, i.e., C (\overleftarrow{\partial}_x \overrightarrow{\partial}_p ) D = (\partial_x C)(\partial_p D). This only becomes slightly less weird when you use the equivalent formula \{ A,B \}_\hbar = (A \star B - B\star A)/(i\hbar), where “\star” is the Moyal star product given by

(2)   \begin{align*} A \star B = A \exp \left( \frac{i\hbar}{2} \left(\overleftarrow{\partial}_x\overrightarrow{\partial}_p - \overleftarrow{\partial}_p\overrightarrow{\partial}_x \right) \right) B. \end{align*}

The star product has the crucial feature that \widehat{A \star B} = \widehat{A}\widehat{B}, where we use a hat to denote the Weyl transform (i.e., the inverse of the Wigner transform taking density matrices to Wigner functions), which takes a scalar function over phase-space to an operator over our Hilbert space. The star product also has some nice integral representations, which can be found in books like Curtright, Fairlie, & ZachosThe complete 88-page PDF is here.a  , but none of them help me understand the Moyal equation.

A key problem is that both of these expressions are neglecting the (affine) symplectic symmetry of phase space and the dynamical equations. Although I wouldn’t call it beautiful, we can re-write the star product as

(3)   \begin{align*} A \star B = A \exp \left( \frac{i\hbar}{2} \overleftarrow{\partial}_a\overrightarrow{\partial}^a \right) B. \end{align*}

where a=x,p is a symplectic index using the Einstein summation convention, and where symplectic indices are raised and lowered using the symplectic form just as for Weyl spinors: v_a = \epsilon_{ab}v^b and w^a = \epsilon^{ab}w_b, where \epsilon is the antisymmetric symplectic form with \epsilon^{xp} = +1 = \epsilon_{px}, and where upper (lower) indices denote symplectic vectors (co-vectors).

With this, we can expand the Moyal equation as

    \begin{align*} \{ H,W \}_\hbar &= \frac{2}{\hbar} H \sin \left( \frac{\hbar}{2} \overleftarrow{\partial}_a\overrightarrow{\partial}^a \right) W \\ &= \sum_{n=0}^\infty \frac{(-\hbar^2/4)^n}{(2n+1)!} \left(\partial_{a_1}\cdots \partial_{a_{2n+1}} H\right)\left(\partial^{a_1}\cdots \partial^{a_{2n+1}} W\right) \\ &= (\partial_a H)(\partial^a W) - \frac{\hbar^2}{24}(\partial_a\partial_b\partial_c H)(\partial^a \partial^b \partial^c W) \\ &\qquad \qquad + \frac{\hbar^4}{80640}(\partial_a\partial_b\partial_c \partial_d \partial_e H)(\partial^a \partial^b \partial^c \partial^d \partial^e W) - \cdots \end{align*}

where we can see in hideous explicitness that it’s a series in the even powers of \hbar and the odd derivates of the Hamiltonian H and the Wigner function W.… [continue reading]

12 Comments

Living bibliography for the problem of defining wavefunction branches

  By
[Last updated: Jan 7, 2026.]

This post is (a seed of) a bibliography covering the primordial research area that goes by some of the following names:

Although the way this problem tends to be formalized varies with context, I don’t think we have confidence in any of the formalizations. The different versions are very tightly related, so that a solution in one context is likely give, or at least strongly point toward, solutions for the others.

As a time-saving device, I will mostly just quote a few paragraphs from existing papers that review the literature, along with the relevant part of their list of references. Currently I am drawing on eight papers: Carroll & Singh [arXiv:2005.12938]; Riedel, Zurek, & Zwolak [arXiv:1312.0331]; Weingarten [arXiv:2105.04545]; Kent [arXiv:1311.0249]; Zampeli, Pavlou, & Wallden [arXiv:2205.15893]; Ollivier [arXiv:2202.06832]; Strasberg, Reinhard, & Schindler [arXiv:2304.10258]; and Soulas, Franzmann, & Biagio [arXiv:2512.07468].

I hope to update this from time to time, and perhaps turn it into a proper review article of its own one day. If you have a recommendation for this bibliography (either a single citation, or a paper I should quote), please do let me know.

… [continue reading]
6 Comments

How to think about Quantum Mechanics—Part 8: The quantum-classical limit as music

  By ,

[Other parts in this series: 1,2,3,4,5,6,7,8.]

On microscopic scales, sound is air pressure f(t) fluctuating in time t. Taking the Fourier transform of f(t) gives the frequency distribution \hat{f}(\omega), but in an eternal way, applying to the entire time interval for t\in [-\infty,\infty].

Yet on macroscopic scales, sound is described as having a frequency distribution as a function of time, i.e., a note has both a pitch and a duration. There are many formalisms for describing this (e.g., wavelets), but a well-known limitation is that the frequency \omega of a note is only well-defined up to an uncertainty that is inversely proportional to its duration \Delta t.

At the mathematical level, a given wavefunction \psi(x) is almost exactly analogous: macroscopically a particle seems to have a well-defined position and momentum, but microscopically there is only the wavefunction \psi. The mapping of the analogyI am of course not the first to emphasize this analogy. For instance, while writing this post I found “Uncertainty principles in Fourier analysis” by de Bruijn (via Folland’s book), who calls the Wigner function of an audio signal f(t) the “musical score” of f.a   is \{t,\omega,f\} \to \{x,p,\psi\}. Wavefunctions can of course be complex, but we can restrict ourself to a real-valued wavefunction without any trouble; we are not worrying about the dynamics of wavefunctions, so you can pretend the Hamiltonian vanishes if you like.

In order to get the acoustic analog of Planck’s constant \hbar, it helps to imagine going back to a time when the pitch of a note was measured with a unit that did not have a known connection to absolute frequency, i.e.,… [continue reading]