- Yes, I agree that their result shows that ontic states unambiguously determines the pure state. But I wonder why this is trivial and just restates QM? It is like calling equilibrium thermodynamics a trivial statistical mechanics.

OK, I think I finally understand your analogy to thermodynamics—and the answer is that what people had wanted to do with psi-epistemic theories was different. If the ontic state uniquely determined the pure state, then each ψ would correspond to a *disjoint* collection of ontic states. The ontic states for a given ψ might behave differently under measurement (e.g., each one could be deterministic), but they’d behave exactly like ψ as an ensemble. Call that a “thermodynamic hidden-variable theory.”

The above can obviously be done, and for reasons that have nothing to do with the structure of quantum mechanics! Any time we have any random outcome, we can always reify whatever the source of the randomness is as a “hidden variable” (though of course, our “hidden variable” might fail to satisfy further natural properties, like locality).

But we might ask whether, for quantum mechanics in particular, we can do something better / stronger / less obvious, and have different pure states correspond to *overlapping* collections of ontic states. And that turns out to be a mathematically interesting question.

The context for the PBR theorem—which for me, played a absolutely crucial role in understanding its motivation—is that if you give up PBR’s tensor-product axiom, then you *can* get psi-epistemic theories where the same ontic state can correspond to multiple pure states (i.e., the “ontic distributions can overlap”). In 3 or more Hilbert space dimensions, the theories are ugly, but in 2 dimensions, there’s such a theory that’s simple and natural enough that you could almost believe it! See my paper with Bouland, Chua, and Lowther for more.

Yes, I agree that their result shows that ontic states unambiguously determines the pure state. But I wonder why this is trivial and just restates QM?

It is like calling equilibrium thermodynamics a trivial statistical mechanics.

(1) Unfortunately, Lenny’s talk was blackboard-only, and I don’t think it was recorded.

(2) I either don’t understand or don’t agree with your interpretation of PBR. To avoid contradicting the PBR theorem (assuming their tensor-product axiom), it’s not enough to stipulate that the classical preparation setup unambiguously determines the pure state. Rather, you need that the *ontic state* unambiguously determines the pure state! And that’s a much stronger requirement, since it implies that the only epistemic theory you can get is a “trivial” one that essentially just restates quantum mechanics.

One related and one off-topic question:

1) I am interested in Susskind’s talk that you mentioned. Is there a chance that I can have his slides?

2) On the issue of psi-epistemic vs psi-ontic.

PBR’s paper showed that even non-orthogonal states must correspond to non-overlapping distributions on some ontic states, and this is what they claim that pure quantum states cannot be epistemic distributions over some ontic states.

However, I think the correct implication of their proof is that pure states correspond to objective epistemic distributions over some ontic states – they still can represent our ignorance about the true ontic state, but in an objective way. Objective, in the sense that the (classical) preparation setup unambiguously determines the pure state.

In fact this is nothing new, in classical thermodynamics a two canonical distributions almost do not overlap when N is very large, yet they represent our ignorance about the microstates.

]]>Metaphorically, the transition from Hilbert-state-space dynamics to Kähler-state-space dynamics is like moving from civilized Paris of the 1850s to the California Gold Fields. A great many civilized conveniences of Hilbert state-manifolds (exact superposition, exact relativistic invariance, and exact spectral theorems) become mere approximations on Kähler state-manifolds. Yet there are compensations … 21st century quantum systems engineers increasingly appreciate that *exact* superposition, relativistic invariance, and spectral purity, are in practice not required of our dynamical theories … it is entirely feasible — even advantageous! — for dynamical theorists to content themselves with local approximation as contrasted with global exactitude (as the transition from Euclidean to Riemannian geometry has previously shown us).

Having experienced the excitement of California’s wild frontier in the 1850s, many young people never returned to Paris. Perhaps the same will prove true of the 21st century’s young quantum theorists, in regard to the exponentially increasing practical STEM attractions of pulled-back Kählerian dynamics!

]]>I don’t get it. You don’t need a Lagrangian to use the Hilbert space formalism, only a Hamiltonian.

]]>**Mateus Araújo** asks [#4] “Why do you consider it [hidden-variable theories] to be a fruitful dead end?”

Mateus, this is an good question precisely because *it has more than one good answer*.

To set the stage, please let me comment to *Shtetl Optimized* readers the (high-ranked) *MathOverflow* question “**Where does a math person go to learn quantum mechanics?**” to which the highest-rated answer is

“I think it is difficult to learn quantum mechanics without first learning classical mechanics.”

To this you yourself contributed the (perspicacious) comment:

There’s a quick review of what I consider a good approach to classical mechanics in Terry Tao’s blog: “From Bose-Einstein condensates to the nonlinear Schrodinger equation (which begins with

)“A quick review of classical mechanics“

Rather like reading *The Feynman Lectures on Physics*, reading Terry Tao’s weblog is such an outstandingly good idea (for students especially) that it can also be an utterly *terrible* idea (for students). In this regard, a paradigmatically problematic passage is Terry’s starting assertion:

Classical mechanics can be formulated in a number of essentially equivalent ways: Newtonian, Hamiltonian, and Lagrangian.

which is followed by multiple occurrences of phrases like “a typical Hamiltonian”, and “a typical example of a symmetric Hamiltonian”, and “a typical Hamiltonian in this case”, and “a typical model”, and “a typical Hamiltonian in this setting” (the latter appears in three places).

Here a distinction is falling-into-the-cracks — the cracks between math and physics, that is! — which is (arguably) crucial to fundamental quantum information theory, namely, the cases that Terry calls “typical” are (at least in the context of QIT) markedly *atypical*.

Specifically, QIT is very largely concerned with Hamiltonian dynamical systems whose symplectic state-manifold (aka “quibits”) does not have the fiber-bundle topology that is physically associated to particle-potential systems. To appreciate that the set of Lagrangian dynamical systems is a restricted subset of Hamiltonian systems — and arguably not the most mathematically interesting or even the most physically important subset — just try to associate a (classical) Lagrangian to the (classical) Bloch equations!

We thus appreciate that the 20th centory’s mathematical toolset for doing classical and quantum field theory — including staples of undergraduate education such as Hilbert’s state-space and Feynman’s path integrals — is strikingly ill-suited to QIT physics … to such a high degree, that too-early over-familiarity with early-and-mid 20th century quantum formalisms makes it difficult (for students especially) to grasp the ever-increasing power of more modern (and mathematically broader) geometric dynamical formalisms.

**Summary** Increasingly in the quantum 21st century, as in the -4th classical century, ** “None But Geometers Enter Here“!**Or as Robert Frost expressed it:

]]>When I was young my teachers were the old.

I gave up fire for form till I was cold.

I suffered like a metal being cast.

I went to school to age to learn the past.Now I am old my teachers are the young.

What can’t be molded must be cracked and sprung.

I strain at lessons fit to start a suture.

I go to school to youth to learn the future.

some modular forms (cusp forms), again non trivial work of Deligne. These estimates were used by Lubotzky-Phillips and Sarnak to prove that their graph constructions (the LPS graphs) are Ramanujan and in particular expanders, yielding the first explicit (algorithmic) construction of these types of graphs which have proved to be so useful in TCS (See the survey paper of Hoory-Linial-Wigderson). More elementary

constrcutions like the zigzag were given (much) later on. ]]>