The n-Category Café

That must be it :)

The link still doesn’t work for me though. I’ll try when I get home.

This is an exciting topic! Matrix product states (tensor network states) arose from the field of quantum information science as a way to efficiently express the states of quantum many-body systems (here’s a review). These methods are important because computer simulations are the backbone of just about everything - quantum systems seem very difficult to simulate using classical computers. No one is totally sure why this is, but that’s another story that tensor networks might help us write.

A key idea behind using a tensor network based computer algorithm to describe a large class of quantum states efficiently is to use the SVD iteratively: skipping details, you simply throw away small singular values (hence truncate the Hilbert space). In many cases this provides highly accurate descriptions of quantum states. My friends and collaborators, Stephen Clark and Dieter Jaksch made some nice slides for our course that outline the current approach:

Lecture 8 Slides (PDF) - Stephen Clark

Background slides by Dieter Jaksch (PPT and PDF)

Lecture 9 Slides (PDF) - Stephen Clark

They look a lot like string diagrams, right?! That’s what got me excited about this topic! There was a bit of a blur and then I met John Baez: after hearing about spin networks I got even more excited and started to understand just how long this stuff had been around!

I wanted to say something more to the tensor networks and quantum information community than “BTW category theory is awesome and you’re already using category theory” this is because I didn’t want to hear: “if I’m already using it, why do I need it?”. Do we need it? Yeah, I think I’ve shown a few reasons why we now do!

The current tensor network approach revolves around ways to approximate quantum states; what I did was create a way to control the type of states created (to zoom in and expose internal structure internal to the network tensors). It’s pretty fun.

Say you have a quantum state, the quantum AND-state: |000>+|010>+|100>+|111> and if we draw this as a network (AND-gate with two-inputs and one-output), and place a |1> on the output and run the network backwards, this leads to the state |00> + |01> + |10>. The idea is we can now “wire” these quantum logic tensors together to create larger and larger tensor network states. Now I’ll say more after I can figure out how to embed figures here!

Hi Todd, do you happen to have a reference for this? I’m still trying to get a handle on what’s been done in related areas, and this sounds interesting. Should be a few nice tricks there to take a look at. Many thanx from Matsudo Japan.

Todd wrote:

One insight is that tensor interchange phenomena can be modeled by homotopies in 2 dimensions, and with that one gains power and speed by visualizing these on a sheet of paper. The 1-dimensional notation masks that.

Jake wrote:

Hi Todd, do you happen to have a reference for this? I’m still trying to get a handle on what’s been done in related areas, and this sounds interesting.

I think you already know the fact Todd is telling you; he’s probably just saying it in a way you find unfamilar.

“Modelling tensor interchange phenomena by homotopies in 2 dimensions” is just a super-groovy way to say that you can wiggle string diagrams around without changing their meaning, most fundamentally like this:

This is a graphical depiction of a particular consequence of the so-called interchange law:

$$(f \otimes g) \circ (f' \otimes g') = (f \circ f') \otimes (g \otimes g')$$

which in turn comes from the fact that $\otimes$ is a functor.

The classic references where all this stuff was made completely precise are these papers by Joyal and Street:

• Andre Joyal and Ross Street, The geometry of tensor calculus I, Advances in Math. 88 (1991) 55-112.
• Andre Joyal and Ross Street, The geometry of tensor calculus II.
• Andre Joyal and Ross Street, Planar diagrams and tensor algebra.

Read these papers and do them proper homage!

This paper is actually about classical stochastic systems, but Section II nicely summarizes the story for quantum systems.

I often think of quantum systems as classical stochastic systems (wick rotated).

If you had said “quantum stochastic system” my head would have probably done a 360 like in the Exorcist.

QUANTUM STOCHASTIC SYSTEM!

Now let’s watch Eric’s head do a wicked rotation.

(Quantum stochastic systems do exist, that’s what that Lindblad equation describes. But I’ve got a lot of questions about that, and so far nobody is answering.)

Hi John,

A little more precisely: the matrix product states are exactly the states with a bounded Schmidt rank across every cut. That is, pick any division of the chain into left and right, do a Schmidt decomposition of the wavefunction across that cut, and the matrix product states are those with bounded Schmidt rank on every such cut (the bound on the Schmidt rank is called the “bond dimension” of the matrix product state, and the larger this number is, the more accurate the results typically are since you have a bigger class of states, but the more CPU time it takes).

This works because in many cases the states we are actually interested in have very rapidly decaying Schmidt coefficients across every cut. So, truncating to the largest few Schmidt coefficients makes small error.

The decay on Schmidt coefficients follows from something called an “area law”, a system-size independent bound on entanglement. I managed to prove an area law for arbitrary local 1 dimensional Hamiltonians with a spectral gap (see http://arxiv.org/abs/0705.2024 ) with no other assumptions, so in fact it is a very general property. Conversely, states without a gap can fail to obey an area law, in that the entanglement can then diverge with system size.

One subtle point (since you mentioned “entanglement drops off with distance”) is the difference between entanglement and correlations. For example, we also know that a local Hamiltonian with a spectral gap has exponentially decaying correlations. So, one might hope to use that to prove an area law. The handwaving argument is: if the correlations decay rapidly then one can only entangle with nearby spins. However, this doesn’t work: there exist families of states, with uniform bounds on the correlation length and Hilbert space dimension on each site, but with arbitrarily large entanglement. This construction is based on “quantum expanders” (introduced by Ben-Aroya and Ta-Shma in a computer science context and by myself in a mathematical physics context) which generalize expander graphs.

Matt

I should mention just a bit more about graphical methods in quantum physics! This stuff has been around for ages, even outside of the tensor network stuff, which is much more recent. Quantum computation reserach took graphical methods in quantum mechanics to a very high level over the last 20+ years in what’s called the the quantum circuit model.

David Deutsch is one of the pioneers here, in 2008 he became a FRS, the Society described Deutsch’s contributions as:

“…the theory of quantum logic gates and quantum computational networks”

Here’s a picture of a quantum logic gate from wikipedia:

Having this theory in place is one of the things that enabled a lot of these new great ideas about tensor network states!

John wrote:

People love to argue about whether to read diagrams up or down, so one man’s “unit” looks just like another woman’s “counit” — but only a monstrous fiend would adopt a convention where a “cap” looks like an upside-down hat and a “cup” looks like a turned-over cup.

Jake: tell me it ain’t so!

it ain’t so: I have no idea how that typo happened! Have had drafts of this paper kicking around for about a year with loads of different plots, used reused etc. so please forgive me as I think the picture was changed without updating the text at some point.

It’s standard to say “canonical basis” for the obvious basis of ℂ n, listed in the obvious order. But if Jake is dealing with an abstract Hilbert space H, he has no right to say “canonical basis” — and it’s definitely not standard to do so.

Yes, again, dealing with the subject, this is perhaps the worst possible typo to have!

On the other hand, it is standard for physicists to screw up math terminology in every conceivable way.

I might quote you on that in the paper - this could be the most profound insight thus far related to this work!

Cheers and thanx!

I mean that say $S$ is some vector like ${|00\rangle} + {|11\rangle}$ and $M$ is some complex number. Then $M\otimes S = M\cdot S$—just multiplication. How would you say this as a hardened category theorist?

Assuming I understand you correctly, I would say it as I did before: “the isomorphism $\mathbb{C}\otimes \mathcal{H} \cong \mathcal{H}$ is given by multiplication.”

To this “hardened category theorist,” at least, $M\otimes S$ and $M\cdot S$ cannot be equal, because they are elements of different vector spaces: the first is an element of $\mathbb{C}\otimes \mathcal{H}$ and the second is an element of $\mathcal{H}$. Recall that for vector spaces $V$ and $W$, their tensor product $V\otimes W$ is a formal construction whose elements can be identified with equivalence classes of formal sums of formal symbols “$v\otimes w$” where $v\in V$ and $w\in W$. The point is that there is a map from $\mathbb{C}\otimes \mathcal{H}$ to $\mathcal{H}$ which takes $M\otimes S$ to $M\cdot S$, and this map is an isomorphism.