Physicists often define a Lindbladian superoperator as one whose action on an operator can be written as
for some operator with positive anti-Hermitian part, , and some set of operators . But how does one efficiently check if a given superoperator is Lindbladian? In this post I give an “elementary” proof of a less well-known characterization of Lindbladians:
A superoperator generates completely positive dynamics , and hence is Lindbladian, if and only if , i.e.,
for all . Here “” denotes a partial transpose, is the “superprojector” that removes an operator’s trace, is the identity superoperator, and is the dimension of the space upon which the operators act.
Thus, we can efficiently check if an arbitrary superoperator is Lindbladian by diagonalizing and seeing if all the eigenvalues are positive.
A quick note on terminology
The terms superoperator, completely positive (CP), trace preserving (TP), and Lindbladian are defined below in Appendix A in case you aren’t already familiar with them.
Confusingly, the standard practice is to say a superoperator is “positive” when it is positivity preserving: . This condition is logically independent from the property of a superoperator being “positive” in the traditional sense of being a positive operator, i.e., for all operators (matrices) , where
is the Hilbert-Schmidt inner product on the space of matrices. We will refer frequently to this latter condition, so for clarity we call it op-positivity, and denote it with the traditional notation .
It is reasonably well known by physicists that Lindbladian superoperators, Eq. (1), generate CP time evolution of density matrices, i.e., is completely positive when and satisfies Eq.… [continue reading]
Wigner’s quasi-probability distribution function in phase-space is a special (Weyl–Wigner) representation of the density matrix. It has been useful in describing transport in quantum optics, nuclear physics, quantum computing, decoherence, and chaos. It is also of importance in signal processing, and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space or path integral formulations. In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory: The variables (observables) in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, but are composed together in novel algebraic ways.
Here are some quotes. First, the phase-space formulation should be placed on equal footing with the Hilbert-space and path-integral formulations:
When Feynman first unlocked the secrets of the path integral formalism and presented them to the world, he was publicly rebuked: “It was obvious”, Bohr said, “that such trajectories violated the uncertainty principle”.
However, in this case, Bohr was wrong. Today path integrals are universally recognized and widely used as an alternative framework to describe quantum behavior, equivalent to although conceptually distinct from the usual Hilbert space framework, and therefore completely in accord with Heisenberg’s uncertainty principle…
Similarly, many physicists hold the conviction that classical-valued position and momentum variables should not be simultaneously employed in any meaningful formula expressing quantum behavior, simply because this would also seem to violate the uncertainty principle…However, they too are wrong.
We report quantum ground state cooling of a levitated nanoparticle in a room temperature environment. Using coherent scattering into an optical cavity we cool the center of mass motion of a nm diameter silica particle by more than orders of magnitude to phonons along the cavity axis, corresponding to a temperature of μK. We infer a heating rate of kHz, which results in a coherence time of μs – or coherent oscillations – while the particle is optically trapped at a pressure of mbar. The inferred optomechanical coupling rate of kHz places the system well into the regime of strong cooperativity (). We expect that a combination of ultra-high vacuum with free-fall dynamics will allow to further expand the spatio-temporal coherence of such nanoparticles by several orders of magnitude, thereby opening up new opportunities for macroscopic quantum experiments.
Ground-state cooling of nanoparticles in laser traps is a very important milestone on the way to producing large spatial superpositions of matter, and I have a long-standing obsession with the possibility of using such superpositions to probe for the existence of new particles and forces like dark matter. In this post, I put this milestone in a bit of context and then and then toss up a speculative plot for the estimated dark-matter sensitivity of a follow-up to Delić et al.’s device.
One way to organize the quantum states of a single continuous degree of freedom, like the center-of-mass position of a nanoparticle, is by their sensitivity to displacements in phase space.… [continue reading]
Our paper discussed in the previous blog post might prompt this question: Is there still a way to use Landauer’s principle to convert the free energy of a system to its bit erasure capacity? The answer is “yes”, which we can demonstrate with a simple argument.
Summary: The correct measure of bit-erasure capacity N for an isolated system is the negentropy, the difference between the system’s current entropy and the entropy it would have if allowed to thermalize with its current internal energy. The correct measure of erasure capacity for a constant-volume system with free access to a bath at constant temperature is the Helmholtz free energy (divided by , per Landauer’s principle), provided that the additive constant of the free energy is set such that the free energy vanishes when the system thermalizes to temperature . That is,
where and are the internal energy and entropy of the system if it were at temperature . The system’s negentropy lower bounds this capacity, and this bound is saturated when .
Traditionally, the Helmholtz free energy of a system is defined as , where and are the internal energy and entropy of the system and is the constant temperature of an external infinite bath with which the system can exchange energy.Here, there is a factor of Boltzmann’s constant in front of because I am measuring the (absolute) entropy in dimensionless bits rather than in units of energy per temperature. That way we can write things like .a (I will suppress the “Helmholtz” modifier henceforth; when the system’s pressure rather than volume is constant, my conclusion below holds for the Gibbs free energy if the obvious modifications are made.)… [continue reading]
People often say to me “Jess, all this work you do on the foundations of quantum mechanics is fine as far as it goes, but it’s so conventional and safe. When are you finally going to do something unusual and take some career risks?” I’m now pleased to say I have a topic to bring up in such situations: the thermodynamic incentives of powerful civilizations in the far future who seek to perform massive computations. Anders Sandberg, Stuart Armstrong, and Milan M. Ćirković previously argued for a surprising connection between Landauer’s principle and the Fermi paradox, which Charles Bennett, Robin Hanson, and I have now critiqued. Our comment appeared today in the new issue of Foundations of Physics:
In their article [arXiv:1705.03394], 'That is not dead which can eternal lie: the aestivation hypothesis for resolving Fermi's paradox', Sandberg et al. try to explain the Fermi paradox (we see no aliens) by claiming that Landauer's principle implies that a civilization can in principle perform far more (~1030 times more) irreversible logical operations (e.g., error-correcting bit erasures) if it conserves its resources until the distant future when the cosmic background temperature is very low. So perhaps aliens are out there, but quietly waiting. Sandberg et al. implicitly assume, however, that computer-generated entropy can only be disposed of by transferring it to the cosmological background. In fact, while this assumption may apply in the distant future, our universe today contains vast reservoirs and other physical systems in non-maximal entropy states, and computer-generated entropy can be transferred to them at the adiabatic conversion rate of one bit of negentropy to erase one bit of error.
I am briefly stirring from my blog-hibernationThis blog will resume at full force sometime in the future, but not just yet.a to present a collection of frequently asked questions about experiments seeking to investigate quantum Darwinism (QD). Most of the questions were asked by (or evolved from questions asked by) Phillip Ball while we corresponded regarding his recent article “Quantum Darwinism, an Idea to Explain Objective Reality, Passes First Tests” for Quanta magazine, which I recommend you check out.
Who is trying see quantum Darwinism in experiments?
I am aware of two papers out of a group from Arizona State in 2010 (here and here) and three papers from separate groups last year (arXiv: 1803.01913, 1808.07388, 1809.10456). I haven’t looked at them all carefully so I can’t vouch for them, but I think the more recent papers would be the closest thing to a “test” of QD.
What are the experiments doing to put QD the test?
These teams construct a kind of “synthetic environment” from just a few qubits, and then interrogate them to discover the information that they contain about the quantum system to which they are coupled.
What do you think of experimental tests of QD in general?
Considered as a strictly mathematical phenomenon, QD is the dynamical creation of certain kinds of correlations between certain systems and their environments under certain conditions. These experiments directly confirm that, if such conditions are created, the expected correlations are obtained.
The experiments are, unfortunately, not likely to offer many insight or opportunities for surprise; the result can be predicted with very high confidence long in advance.… [continue reading]
In particular, he sketched the essential equivalence between matrix product states (MPS) and restricted Boltzmann machinesThis is discussed in detail by Chen et al. See also good intuition and a helpful physicist-statistician dictionary from Lin and Tegmark.b (RBM) before showing how he and collaborators could train an efficient RBM representations of the states of the transverse-field Ising and XXZ models with a small number of local measurements from the true state.
[This is akin to a living review, which will hopefully improve from time to time. Last edited 2020-4-8.]
This post will collect some models of decoherence and branching. We don’t have a rigorous definition of branches yet but I crudely define models of branching to be models of decoherenceI take decoherence to mean a model with dynamics taking the form for some tensor decomposition , where is an (approximately) stable orthonormal basis independent of initial state, and where for times and , where is the initial state of and is some characteristic time scale.a which additionally feature some combination of amplification, irreversibility, redundant records, and/or outcomes with an intuitive macroscopic interpretation.
(Note in particular that I am not just listing models for which you can mathematically take a classical limit ( or ) and recover the classical equations of motion; Yaffe has a pleasingly general approach to that task , but I’ve previously sketched why that’s an incomplete explanation for classicality.)
I have the following desiderata for models, which tend to be in tension with computational tractability: