all possible paths - lessons from machine learning

[scruffy]

In the quantum world, probabilities represent "all possible paths". But what exactly is "possible"?

In physics, what's possible is determined by a model, in this case the Schrodinger equation and it's geometric embeddings.

But what if you don't know the model ahead of time?

In that case, you have to build it. The classical way of building models is Bayesian inference, which works off "hypotheses".

Linear hypotheses are asymptotic, which means the more data you get, the better you can predict. However they fail in many cases, resulting in things like Simpson's paradox. The problem is especially acute when the model involves causality - which in physics is related to irreversibility.

The general problem is we begin with unknown dimensionality. We have to look at a dataset and "estimate" how many parameters it takes to describe it. We do that by extracting correlations between data points and groups of points, and we partition the dataset in various ways while we're doing that (using fancy terms like "cluster analysis").

However there is a procedure in machine learning that successively carves out subspaces of the correlations, it's a version of "pruning" paths. In Judea Pearl's do-calculus it's called "interventions". It means removing some of the possible paths, and then rebuilding the information matrix.

This is relevant for open systems. Which are those where the bath has an unknown composition with unknown effects. The basic idea is formulated by Sumio Watanabe, and it's more or less the same approach Renyi takes towards entropy.

Sumio Watanabe - Wikipedia

en.wikipedia.org

The basic idea is to identify "singular" instances of the Fisher information matrix. It turns out, neural networks are especially good at this. They can generate singular subspaces and apply them to the datasets in near real time.

Here is some background

Singular Learning Theory - LessWrong

Singular Learning Theory (SLT) is a novel mathematical framework that expands and improves upon traditional Statistical Learning theory using techniques from algebraic geometry, bayesian statistics, and statistical physics. It has great promise for the mathematical foundations of modern machine...

www.lesswrong.com

And further background

Singular Learning Theory for Alignment

Lightning unenlightened

metauni.org

Here is a direct application to quantum physics

[2310.06301] Dynamical versus Bayesian Phase Transitions in a Toy Model of Superposition

 

[Nobody911]

In the quantum world, probabilities represent "all possible paths". But what exactly is "possible"?

In physics, what's possible is determined by a model, in this case the Schrodinger equation and it's geometric embeddings.

But what if you don't know the model ahead of time?

In that case, you have to build it. The classical way of building models is Bayesian inference, which works off "hypotheses".

Linear hypotheses are asymptotic, which means the more data you get, the better you can predict. However they fail in many cases, resulting in things like Simpson's paradox. The problem is especially acute when the model involves causality - which in physics is related to irreversibility.

The general problem is we begin with unknown dimensionality. We have to look at a dataset and "estimate" how many parameters it takes to describe it. We do that by extracting correlations between data points and groups of points, and we partition the dataset in various ways while we're doing that (using fancy terms like "cluster analysis").

However there is a procedure in machine learning that successively carves out subspaces of the correlations, it's a version of "pruning" paths. In Judea Pearl's do-calculus it's called "interventions". It means removing some of the possible paths, and then rebuilding the information matrix.

This is relevant for open systems. Which are those where the bath has an unknown composition with unknown effects. The basic idea is formulated by Sumio Watanabe, and it's more or less the same approach Renyi takes towards entropy.

Sumio Watanabe - Wikipedia

en.wikipedia.org

The basic idea is to identify "singular" instances of the Fisher information matrix. It turns out, neural networks are especially good at this. They can generate singular subspaces and apply them to the datasets in near real time.

Here is some background

Singular Learning Theory - LessWrong

Singular Learning Theory (SLT) is a novel mathematical framework that expands and improves upon traditional Statistical Learning theory using techniques from algebraic geometry, bayesian statistics, and statistical physics. It has great promise for the mathematical foundations of modern machine...

www.lesswrong.com

And further background

Singular Learning Theory for Alignment

Lightning unenlightened

metauni.org

Here is a direct application to quantum physics

[2310.06301] Dynamical versus Bayesian Phase Transitions in a Toy Model of Superposition

In the quantum world, probabilities represent the likelihood of different outcomes for a particle or system. These probabilities arise from the wave nature of particles, as described by the principles of quantum mechanics.

When we say that different paths are possible, we mean that at a quantum level, particles can exhibit behaviors that seem counterintuitive based on classical physics.

In quantum mechanics, the concept of superposition allows particles to exist in multiple states simultaneously until a measurement is made, collapsing the wave function to a specific state. This idea of all possible paths reflects the uncertainty inherent in quantum systems and the idea that particles can exhibit wave-like behavior, such as interference patterns.

The notion of what is possible in the quantum world is defined by the laws of quantum mechanics, which describe the behavior of particles at microscopic scales. These laws govern the behavior of particles in terms of probabilities and allow for phenomena such as superposition, entanglement, and quantum tunneling.

Ultimately, the possibilities in the quantum world are determined by the mathematical framework of quantum mechanics and the observed behavior of particles in experiments.

Quantum mechanics provides a powerful mathematical framework to describe the behavior of particles at the quantum level. One example is the Schrödinger equation, which is foundational in quantum mechanics and governs the evolution of a quantum system over time. By solving this equation, physicists can predict the probabilities of different outcomes of a quantum experiment.

Another important mathematical tool in quantum mechanics is linear algebra, which is used to describe the state of a quantum system using vectors and operators. This formalism allows physicists to perform calculations and make predictions about the behavior of particles in complex quantum systems.

Furthermore, quantum field theory utilizes sophisticated mathematical frameworks such as functional analysis and group theory to describe the behavior of quantum fields and their interactions. These mathematical tools are essential for understanding phenomena such as particle interactions and the behavior of quantum fields in different spacetime regions.

Overall, the mathematical frameworks of quantum mechanics provide a rigorous and precise way to describe the behavior of particles at the quantum level, enabling physicists to make predictions and understand the fundamental aspects of the quantum world.

 

[scruffy]

In the quantum world, probabilities represent the likelihood of different outcomes for a particle or system. These probabilities arise from the wave nature of particles, as described by the principles of quantum mechanics.

When we say that different paths are possible, we mean that at a quantum level, particles can exhibit behaviors that seem counterintuitive based on classical physics.

In quantum mechanics, the concept of superposition allows particles to exist in multiple states simultaneously until a measurement is made, collapsing the wave function to a specific state. This idea of all possible paths reflects the uncertainty inherent in quantum systems and the idea that particles can exhibit wave-like behavior, such as interference patterns.

The notion of what is possible in the quantum world is defined by the laws of quantum mechanics, which describe the behavior of particles at microscopic scales. These laws govern the behavior of particles in terms of probabilities and allow for phenomena such as superposition, entanglement, and quantum tunneling.

Ultimately, the possibilities in the quantum world are determined by the mathematical framework of quantum mechanics and the observed behavior of particles in experiments.

Quantum mechanics provides a powerful mathematical framework to describe the behavior of particles at the quantum level. One example is the Schrödinger equation, which is foundational in quantum mechanics and governs the evolution of a quantum system over time. By solving this equation, physicists can predict the probabilities of different outcomes of a quantum experiment.

Another important mathematical tool in quantum mechanics is linear algebra, which is used to describe the state of a quantum system using vectors and operators. This formalism allows physicists to perform calculations and make predictions about the behavior of particles in complex quantum systems.

Furthermore, quantum field theory utilizes sophisticated mathematical frameworks such as functional analysis and group theory to describe the behavior of quantum fields and their interactions. These mathematical tools are essential for understanding phenomena such as particle interactions and the behavior of quantum fields in different spacetime regions.

Overall, the mathematical frameworks of quantum mechanics provide a rigorous and precise way to describe the behavior of particles at the quantum level, enabling physicists to make predictions and understand the fundamental aspects of the quantum world.

Except it fails!

The reason is because the Ricci calculus is only valid in local neighborhoods.

This is why we have to study learning methods to arrive at a better understanding. We want OPEN systems, not closed systems.

Here is an example that can be used to illustrate a critical point, which is a phase transition.

You see the two distinct shapes, and the question is, what happens as they get closer and closer together?

From the physics point of view, you begin with two distinct Hamiltonians, and at some point they merge and become one. (cf entanglement)

One question is, how close can they get, before singular behavior predominates? We are describing a PHASE TRANSITION, which would apply to such things as Bose-Einstein condensates, but also to the stable states of nonlinear non-equilibrium thermodynamics.

The reason machine learning is important in this view is because it quantifies this interaction in ways the quantum theory can't, because quantum theory is fundamentally local.

Here is a simple but thorough summary of the math from the machine learning point of view, based on Watanabe's information geometry.

DSLT 0. Distilling Singular Learning Theory — LessWrong

TLDR; In this sequence I distill Sumio Watanabe's Singular Learning Theory (SLT) by explaining the essence of its main theorem - Watanabe's Free Ener…

www.lesswrong.com

https://www.lesswrong.com/posts/4eZtmwaqhAgdJQDEg/dslt-1-the-rlct-measures-the-effective-dimension-of-neural

https://www.lesswrong.com/posts/CZHwwDd7t9aYra5HN/dslt-2-why-neural-networks-obey-occam-s-razor

https://www.lesswrong.com/posts/tZwaGp5wMQqKh3krz/dslt-3-neural-networks-are-singular

https://www.lesswrong.com/posts/aKBAYN5LpaQMrPqMj/dslt-4-phase-transitions-in-neural-networks

The key is the free energy formula, which translates directly into the information space.

 

[scruffy]

Here's the deal - we want to describe entanglement in physical terms.

Right now, the quantum theory waves its hands over the issue - it says, "oh well, the wave functions merge".

We can do better. We want to know exactly how and why they merge. And it turns out, this behavior has a lot to do with the information space.

The merging of wave functions is an example of a PHASE TRANSITION. And phase transitions are strictly described (only) by information geometry.

We can draw a very close analogy and example, from the phase transitions in neural networks. BECAUSE, there, we have a formalism that describes how the posterior behaves after learning (which is to say, after information transfer). The updating of the posterior is analogous to the updating of the probability matrix in an open (quantum or thermodynamic) system.

The beauty and value of the Watanabe approach is it extends the description beyond the linear case. Which we already know is NECESSARY in the case of open systems.

The Watanabe approach quantifies the critical point at the moment of phase transition. It does so in terms of the changes to the information matrix.

The real prize in this approach is that it can handle changes to the dimensionality. So for example if you're talking about adding or subtracting a path, this may equate NOT ONLY with a change to the probabilities, but ALSO with a change in dimension.

In a neural network, critical states are almost always accompanied by a temporary increase in dimension. This is what is being described in the links in the previous post. The links show how the information matrix becomes singular at the point of phase transition.

Translating to the quantum case, this can tell us when and why entanglement occurs. Consider: we know of several ways of creating entanglement. (Parametric down parametrization is one of the current favorites). We also know that it occurs naturally with certain forms of quantum "splitting", which is to say radiation or other forms of emission, like when a particle splits into other particles. Why exactly this happens, is what we'd like to know. Because the correlations are propagated NON LOCALLY, they can't be described "at all" with the Ricci calculus.

The Watanabe approach gives us a way to track the non local correlations. It doesn't exactly tell us the underlying physical mechanism but it accurately describes the behavior, which is the first step in understanding the underlying physics. Clearly, it has to do with INFORMATION, and Watanabe suggests we need to extend the description of free energy this way.

See the DSLT-4 (last) link for an animated version of the picture in the previous post.

 

[scruffy]

This link shows the relationship between information geometry and catastrophe theory, in terms of the Hessian matrix referenced in the Watanabe papers.

Catastrophe Theory

jfeldbrugge.github.io

Catastrophe theory: "The main thesis of the theory is that the parameter space of the system is a low-dimensional projection of the state parameters and state relationships of the system, which are summarized as higher-dimensional, smooth manifolds."

https://www.sciencedirect.com/topics/earth-and-planetary-sciences/catastrophe-theory

There are some constraints on catastrophes in infinite dimensional spaces. Here is an example:

Infinite Dimensions and the Fold Catastrophe

A simple example of the manner of breakdown of Thom’s theorem in infinite dimensions, but for which catastrophe theoretic methods remain illuminating, is discussed both informally and analytically. The co-dimension 1 character of the set of jets on a Hilbert...

link.springer.com

Understanding these constraints is of primary importance when we're talking about forming and breaking symmetries, which is what the picture and animation are showing you in the previous post.

Entanglement is a symmetry in the information space, and it is broken by measurement, which results in the splitting of the wave function into two (or more) parts.

A measurement is the opposite of what you see in the animation - in the case of a measurement, one peak becomes two.

 

[scruffy]

The physics of Bose Einstein condensates is illuminating to understand the information geometry more fully.

""Mixing of quantum states" in a Bose-Einstein condensate (BEC) refers to the process where different quantum states within the condensate begin to overlap and interact with each other, effectively creating a new, mixed quantum state, often achieved by manipulating the interactions between atoms within the BEC, allowing for the exploration of complex quantum phenomena like spin mixing or the formation of quantum droplets"

QUANTUM DROPLETS are self bound clusters of atoms stabilized by a balance of attractive and repulsive forces.

There are also SOLITONS, which are wave packets that propagate without dispersing. Both of these characteristics become important in entanglements.

Bose Einstein condensates have been extensively studied. There are nonlinear nonlocal Bose Einstein condensates.

"Nonlinear nonlocal Bose-Einstein condensate" refers to a state of matter where a Bose-Einstein condensate (BEC) exhibits both nonlinear interactions between particles (meaning the interaction strength depends on the particle density) and nonlocal interactions, where the influence of one particle extends beyond its immediate vicinity, affecting particles further away in the system; essentially, the wavefunction of the condensate is affected by a non-local interaction term, leading to complex and unique behaviors not seen in standard BECs"

The behavior of a nonlocal nonlinear BEC is modeled by a modified Gross-Pitaevskii equation, which incorporates additional terms accounting for the nonlocal interactions.

Gross–Pitaevskii equation - Wikipedia

en.wikipedia.org

Examples of nonlocal interactions include atoms with dipole moments and periodic potentials created by optical lattices.

Bose-Einstein condensation of photons with nonlocal nonlinearity in a dye-doped graded-index microcavity

We consider a microcavity made by a graded-index glass, doped with dye molecules, placed within two planar mirrors and study Bose-Einstein condensation of photons. The presence of the mirrors leads to an effective photon mass, and the index grading provides an effective trapping frequency; the...

link.aps.org

Non-Local Theory of Bose-Einstein Condensate and Stopping of Light

Explore the fascinating world of Bose-Einstein condensates (CBE) and their transport processes. Discover the space-temporal evolution of CBE in a gravitational field, including black hole evolution and flickering. Uncover the quantization of black hole flickering and its implications for...

www.scirp.org

Radware Bot Manager Captcha

https://opg.optica.org/oe/fulltext.cfm?uri=oe-31-20-33518&id=539662

Nonlinear localized modes in dipolar Bose-Einstein condensates in optical lattices

Modulational instability and discrete matter wave solitons in dipolar BECs, loaded into a deep optical lattice, are investigated analytically and numerically. The process of modulational instability of nonlinear plane matter waves in a dipolar nonlinear lattice is studied and the regions of...

link.aps.org

- and -

https://www.sciencedirect.com/science/article/pii/S2211379723000232

 

[scruffy]

Here:

Entanglement asymmetry as a probe of symmetry breaking - Nature Communications

A measure of symmetry breaking in a quantum many-body system could provide insight into its dynamics. Ares et al. introduce a subsystem measure of symmetry breaking dubbed entanglement asymmetry and apply it to quantum quench dynamics in spin chains, revealing a quantum analogue of the Mpemba...

www.nature.com

And here:

Entanglement—A Higher Order Symmetry

Can we accurately model the spin state of a quantum particle? If so, we should be able to make identical copies of such a state and also obtain its mirror image. In quantum mechanics, many subatomic particles can form entangled pairs that are mirror images of each other, although the state of an...

www.mdpi.com

And here:

Symmetry classification of typical quantum entanglement

Entanglement entropy of typical quantum states, also known as the Page curve, plays an important role in quantum many-body systems and quantum gravity. However, little has hitherto been understood about the role of symmetry in quantum entanglement. Here, we establish the comprehensive...

link.aps.org

Entanglement asymmetry seeks to quantify "how much" the symmetry is breaking. You can see the close relationship to the entanglement entropy, which is an information measure.

Referring back to Watanabe's SLT, neural networks typically push memories into the corners of hypercubes which is fundamentally a symmetry related behavior. When they don't, we end up with the surfaces shown in the DSLT links, which can be thought of as "entangled" in one or more parameters.

Mpemba effect (discussed in 1st link):

Mpemba effect - Wikipedia

en.wikipedia.org

 

[scruffy]

Here's an interesting one.

Let me see if I can explain what these guys did.

They're quantum computer types. They strung a bunch of quantum logic gates in series. The gates actually live right next to each other in the quantum computer, so they were able to entangle the entire chain. They call it an inverter-chsin link.

So, this chain is a multi dimensional information space, it's discrete because there are only a finite number of inverters. Each qubit (inverter) is continuous within the probability space, it's represented by a magnitude and a pair of angles on a Bloch sphere

What they're going to look at, is the phase space for the entangled chain. What they found is, the phase space can be decomposed into prime number subspaces related topologically by Hadamard transforms.

Hadamard transforms are like the square wave equivalents of Fourier transforms, instead of sines and cosines you have square waves.

So then, they did the math using the Bell basis states and discovered that the output of the inverter chain exactly matches the matrix algebra for the discrete topology. In other words, in the chain there is pairwise entanglement, triplet entanglement, and so on. These entanglements increase in dimension every time you add an inverter to the chain. Each new inverter ends up taking you one step deeper in the Hadamard space. At the output of the inverter chain, the value is either the inverse of the input, or its identity, but you can MEASURE the entanglement caused by the chain.

The spectacular result is the input and output can be separated by arbitrarily large distances (length of inverter sequence) and still remain entangled, exactly according to the discrete topology. What happens is you can entangle a pair of qubits, then add another one at one end and keep adding them, - and when you do that, the second will entangle the third, and the third will entangle the fourth, and so on. The end result is a "transmission of entanglement" from one end of the chain to the other. What comes out, is either a replica of the first qubit's entanglement, or its inverse. You can make the chain as long as you want, the same behavior results.

The beautiful thing is the whole setup can be represented by a series of Pauli matrices. They go on to show how this can be used for quantum teleportation and superdense coding.

So what they've demonstrated here, is the conservation of information through multiple layers of entanglement. The Pauli matrices serve as the symmetry underlying the conservation law. With this scheme you can DERIVE the entangled states for any number of multi partite entanglements.

The authors conclude that this behavior requires extra dimensionality in spacetime itself, to account for the perfect relationship between neighbors. They speculate that entanglement may be akin to the endpoints of a vortex tube in spacetime (kind of like a microscopic wormhole at the Planck scale).

This is a bigger deal than it sounds. It's not just a bunch of inverters in a row. The next step is to create directed acyclic graphs from this setup, which will require them to split the entanglement along two or more paths. The question is, can you split an entanglement the same way you split energy?

The mathematics of Noetherian rings plays into this big time. Ultimately, when the number of inverters goes to infinity you have a continuous ring which is the foundation for a FIELD. In information field theory there is the concept of an "information Hamiltonian" which allows you to use the methods of physics on the information itself (including perturbation and the methods of variations). Which would be kind of like the Holy Grail of information theory.

Paper:

An inverter-chain link implementation of quantum teleportation and superdense coding

Energy splitting:

Energy level splitting - Wikipedia

en.wikipedia.org

Why it matters:

Information field theory - Wikipedia

en.wikipedia.org

Background on rings:

Ring (mathematics) - Wikipedia

en.wikipedia.org

 

[scruffy]

And FURTHERMORE ( )

Here is a direct relationship between the quantum, relativistic, and information Hamiltonians.

There is an Unruh effect, and an anti-Unruh effect.

The Unruh effect, is when an accelerating observer sees a heat bath (an increased temperature) in the vacuum.

The anti-Unruh effect, is when the accelerator's measuring device cools down.

They're due to two different mechanisms.

In the first, the ground state of the observer becomes a mixed state in thermal equilibrium with the vacuum.

Unruh effect - Wikipedia

en.wikipedia.org

In the second, entanglement increases with acceleration.

Influence of acceleration on multibody entangled quantum states

We study the influence of acceleration on the twin-Fock state which is a class of specific multibody entangled quantum state and was already realized experimentally with high precision and sensitivity. We show that the multibody quantum entanglement can be increased with the acceleration...

link.aps.org

The increase in entanglement is a form of "squeezing", where the normally circular phase space changes into an ellipse.

Squeezed coherent state - Wikipedia

en.wikipedia.org

The net effect of Unruh and anti-Unruh cases acts "as if" heat is being sucked out of the detector into the vacuum, in exchange for information about the vacuum.