# Quantum Bayesian Networks

## March 31, 2012

### Dogs is people too, and bluffing a pride of lions

Filed under: Uncategorized — rrtucci @ 11:57 am

This is very technical and nerdy, but I find it really cool.

Recently, Carlen and Lieb proved a very nice inequality for entanglement. Check out

Bounds for Entanglement via an Extension of Strong Subadditivity of Entropy

by Eric A. Carlen, Elliott H. Lieb

Let me try to summarize their main result.

Consider a bipartite density matrix $\rho_{12}$, and define $\rho_1 = {\rm tr_2}\rho_{12}$ and the same with 1 and 2 swapped. Suppose $S_{12}$ is the von Neumann entropy of 12, $S_{1}$ is that of 1 and $S_{2}$ is that of 2. Call the classical counterparts of these entropies $H_{12}$, $H_{1}$ and $H_{2}$. Call $S_{2|1} = S_{12} - S_1$ (respectively, $H_{2|1} = H_{12} - H_1$) the quantum (resp., classical) conditional spread (or conditional variance) of 2 given 1.

One can show that

$H_{2|1}\geq 0$,
$H_{1|2}\geq 0$

so classically, conditional spreads must be positive (or zero). However, in quantum mechanics such spreads can be negative. The Araki-Lieb inequality $S_{12}\geq |S_1-S_2|$ says that

$S_{2|1}\geq -S_2$,
$S_{1|2}\geq -S_1$.

Now, having a negative conditional spread is a bit of a dog, because spreads are not supposed to be negative.

The new Carlen/Lieb inequality teaches us that dogs can count too. If $E$ is entanglement (either entanglement of formation or squashed entanglement), then, according to Carlen/Lieb,

$E\geq max(-S_{1|2}, -S_{2|1}, 0)$

So a dog (i.e, a negative conditional spread $S_{1|2}<0$) forces the entanglement to be greater than zero by $-S_{1|2}$. An example of a highly influential dog.

P.S. I recently wrote some email to Profs. Lieb and Carlen asking them something about their paper. I felt infinitely dumb in their email presence, like a mouse in the presence of lions (As you probably know, The Lion and the Mouse is a beautiful fable by Aesop.) But I believe I successfully muddled and bluffed my way through the conversation. Amazing but true, it is possible to bluff a pride of lions. If you don’t believe me, watch this amazing YouTube video, entitled “BBC – Men stealing meat from lions”. In the video, 3 extremely courageous and also extremely foolish African guys, with scant weapons other than their daunting audacity, intentionally walk right into the middle of a pride of lions that is having dinner.

1. Congrats on your lion encounter skills 🙂 Will have to ponder this for a while before I could attempt a similar feet.

This is an amazing paper. Looks like we can add entanglement to all the other things that won’t work anymore once our universe has run its course (http://en.wikipedia.org/wiki/Heat_death).

Just love how new experimental results and theoretical insights like this are demystifying entanglement.

Comment by Henning Dekant — March 31, 2012 @ 2:37 pm

2. It seems to me that a lot of people are very confused about Hilbert space. They continue to make up, in never ending variations, particular special case results which have to do with highly specialized sub-spaces of general manifolds. First year math fact: all Hilbert spaces over the complex field are isomorphic, in a real sense there is only one Hilbert space of dimension N. Second fact: couple a system of N quantum degrees of freedom to M and you get an N by M tensor product system of which the maximal Hilbert space is of dimension N by M. Wow! Really surprising 🙂 Why then do Physicists continue to waste their time penning one paper after another pretending there is something deep and special to be found in “entanglement”.

They fail to understand that the particular special cases they examine have to do with sub-manifolds of a more general structure which are tied to whatever symmetry group governs the interaction between the chosen systems. The problem is *not* entanglement.

Entanglement is the general condition of any interacting system.

The real problem to understand is “dis-entanglement”. Under what conditions do systems begin to behave in a “dis-entangled” manner.

Bohr was wise enough to know this and said as much when he spoke of isolated quantum systems being describable in “quantum terms”, themselves being probed by weakly interacting systems describable in “classical terms”. Implicitly, the isolated quantum system is very special “dis-entangled state. When you manage to create those, you can then weakly entangle them to understand the elementary interactions.

This is the right way to understand the physics. Now people turn things upside down and are deeply mystified about how to create an entangled system!

The point of all this is quantum information theory is a bit of a canard. The information you get is from an actual experiment, described in Bayesian fashion via the conditional measurement correlation Pr (classical data | quantum state). This you can invert to get Pr ( quantum state | classical data ).

It is remarkable that Physicists don’t get this! Every field known to Man (and I mean *all* of them, from speech recognition, through bioinformatics and machine learning, through classical communication theory and signal processing) understands this.

Only Physicists are special in their supreme ignorance of such matters.

Already, in (1994), I published a general analysis along these lines (including explicit discussion of the dynamical sub-group issue). The paper is at: K.R.W. Jones (1994), Fundamental limits upon the measurement of state-vectors, Phys. Rev. A50, 3682-3699.

After some 18 years, I have formed the distinct impression that my physicist audience may not have understood the basis of the paper for the above reasons.

The general limit has already been established and published. The quest for a “quantum information theory” and “quantum computation theory” etc etc is essentially a Grail Quest to make up for a lack of understanding among physicists of what it means to actually measure something.

I claim … measurement = inference

There is something you know which is correlated to something you don’t know. You use a knowledge of the statistics to infer the unknown.

In statistics you call the “thing you can’t see” a LATENT VARIABLE and you use known observable data to infer the values of that.

This is the basis of quantum inference (a word I coined 23 years ago).

Like I said… EVERY, and I mean EVERY field outside of modern physics has fully grasped this point. It is only the Physicists who are idiots.

The really big BREAKTHROUGH in physics happens when some arrogant up-start re-labels the word LATENT and wins a Nobel prize 🙂

CLUE: Latent variable = Wave function (which happens to be non-local). Non-local hidden variables theories are not excluded.

ANOTHER BIG CLUE: Latent Variable Hidden Variable

Can you hear those pennies drop? Any takers for a genuinely New Physics instead of this sham Quantum XXX caper???

Comment by Kingsley Jones — June 30, 2012 @ 2:32 am

3. Kingsley, don’t think there is anything wrong with the view that inference = measurement and that the pure, isolated disentangled state is the special exception to the rule. But with regards to this paper this is exactly were the rubber hits the road, isn’t it? The intriguing aspect of pure isolated quantum system is that they exhibit an irreversible loss of information when measured. Something that very much ties them to the 2nd law of thermodynamics. In that light I find the Carlen/Lieb inequality rather instructive.

Comment by Henning Dekant — June 30, 2012 @ 3:45 am

4. Kingsley, I think Quantum Information Theory is a giant set that contains many OVERLAPPING subsets, e.g., inference, entanglement, channel capacity, compression, error correction, noise, programming of quantum computers and the computational complexity of such programs, quantum bayesian networks, etc. ,etc.

The same is true with classical information theory. In fact, for each of the quantum topics presented above there is a classical limit (in the case of entanglement, I would say it’s correlations conditioned on a latent variable)

Comment by rrtucci — June 30, 2012 @ 1:39 pm

5. If I am not mistaken this paper claims that this negative conditional spread could be used to construct some sort of quantum Maxwell demon. Are they right or does this involve some sleight of hand?

Pop science write-up can be found here.

Comment by Henning Dekant — August 2, 2012 @ 5:43 pm

6. […] This is very technical and nerdy, but I find it really cool. Recently, Carlen and Lieb proved a very nice inequality for entanglement. Check out Bounds for Entanglement via an Extension of Strong S…  […]

Pingback by Dogs is people too, and bluffing a pride of lions | Quantum Computing | Scoop.it — August 2, 2012 @ 5:45 pm

7. Hi Henning,
I haven’t read the Funo et al paper carefully but it looks very interesting. It will take me a while before I can digest it.

The connection between energy/work and quantum SIT (Shannon information theory) is certainly a fascinating topic. Maxwell’s demon & Landauer’s principle are great gedanken experiments on which to hone one’s understanding of that connection.

The paper doesn’t explicitly use the tools of squashed entanglement or entanglement of formation, which are what the Carlen & Lieb inequality applies to. However, perhaps it can be extended so that it does use these tools.

Comment by rrtucci — August 2, 2012 @ 6:27 pm

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