Charles Akemann and Nik Weaver note in their 2008 PNAS paper that confirmation of the Kadison-Singer conjecture implies (S1) and (S2) are equivalent. The statement (S1) is that ‘‘every pure state on $\mathcal{B}(H)$ restricts to a pure state on some atomic masa’’, and (S2) is Anderson’s conjecture that every pure state on $\mathcal{B}(H)$ is diagonalizable, that is, of the form $f(A) =lim_{U}\langle Ae_{n}, e_{n}\rangle$ for some orthonormal basis $(e_{n})$ and some ultrafilter $U$ over $\mathbf{N}$. Prior to showing that the Continuum Hypothesis (CH) refutes (S1), they remark: “It seems likely that the statement ‘‘every pure state on $\mathcal{B}(H)$ restricts to a pure state on some atomic masa’’ is also consistent with standard set theory.”

Has there been any further progress on whether (S1) (and hence now (S2)) is relatively consistent with ZFC, e.g. under a forcing axiom? Or has CH been eliminated or weakened?

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Abstract:$UNIQUE \ SAT$ is the problem of deciding whether a given Boolean formula has exactly one satisfying truth assignment. The $UNIQUE \ SAT$ is $coNP-hard$. We prove the

$UNIQUE \ SAT$ is in $NP$, and therefore, $NP = coNP$. Furthermore, we prove if $NP = coNP$, then some problem in $coNPC$ is in $P$, and thus, $P = NP$. In this way, the $P$

versus $NP$ problem is solved with a positive answer.

I solved the second riddle in a slightly different way:

If Joe happen to sit in his own seat then everyone’s happy; if Joe happen to sit in Jim’s seat, then Jim will not have his seat at the end. Those two events happen with the same probability of 1/100. If none of the two seats are taken then whenever someone finds their seat occupied, the following two events will end the story: 1) if he takes Joe’s seat, we have a closed loop of cyclic permutation of serval people, all the later passengers, including Jim, will take their own seat; 2) if he takes Jim’s seat, then we are done because Jim will not have his seat. 1) and 2) are equally likely to occur at any given time. Hence at the end it will be exactly 1/2 chance that Jim’s seat is taken.

“In the car with 10^{100} horse-powers example, this means that if your imperfect car has 100 horsepower it is physically impossible to distinguish between a description of an imperfect 10^{100} horse-powered car and an imperfect 100 horsepower car and we should prefer the second explanation.”

Dear Gil, I doubt, it could be accepted without additional explanations – I suppose, if we could see some car with 10^100 horse-power engine, we could very simply distinguish that from car with 100 horsepower after simple research – in some way we may distinguish ship with power plant and sailing ship even if they have the same speed and size, because we may see that the power plant is working and how the ship is using only tiny fraction of the power.

In the same way, if we are supposing usual model of quantum computer, we know that the model (“power plant”) may factor integers in principle and the model with noise is an analogue of slowly moving ship with power plant.

In such example your argumentation looks like idea, that the power plant is only illusion and, in fact, both ships are moving due to wind.

But such a case looks like you could suggest instead of usual model of quantum computation some model that may be simulated classically even without noise – then you indeed would say that due to noise we may not distinguish such models and to claim quantum supremacy.

” One idea which seems implicit in Alexander’s thinking is that nature has superior quantum computational powers “

No, I did not make any claim about real computational power of Nature. I simply meant that idea about possibility restoration of ECT due to noise (found in your exchange elsewhere) seems wrong due to thought experiment I mentioned.

]]>We have a system consisting of a quantum computer + noise. This combined system is not less powerful than the quantum computer alone. So if the quantum computer disprove ECT some *additional* effect may not restore ECT!

Alexander mentioned the analogy with horse-power: If your car’s engine has horsepower but by some imperfection you only uses 100 horse-power, it still has horsepower!

One idea which seems implicit in Alexander’s thinking is that nature has superior quantum computational powers (quantum supremacy, for short) even if noise does not allow us to make use of these powers. I think that this point of view is not correct for the following reasons. (The BosonSampling previous comment and paper exemplify these points quite well.)

1) Quantum computers are hypothetical and general quantum evolutions that quantum computers can perform are also hypothetical.

The -horsepower car is only in our imagination.

2) Noise sensitivity means that realistic noisy quantum systems depend on exponential input size.

When you or nature implement a state which seemingly requires exponential time computation(in some parameter , robust outcomes depends on exponentially many parameters in describing the noise.

3) It is unuseful and incorrect to think about mixed-state (say bosonic or anyonic states) evolutions as “really” representing some pure state evolution “known” to nature but unknown to us. (This is also related to Nick Read parenthetical comment displayed above. ) A main reason is that the expression of pure states and mixtures is not unique.

(Analogy: it is unuseful and probably incorrect to think that at any given moment there are two elephants in your office which, having opposite signs, cancel each other.)

In the car with horse-powers example, this means that if your imperfect car has 100 horsepower it is physically impossible to distinguish between a description of an imperfect horse-powered car and an imperfect 100 horsepower car and we should prefer the second explanation.

Of course the crucial hypothesis is

4) Realistic noisy quantum systems can be simulated in BPP.

Correct asymptotic modeling of realistic noisy quantum evolutions man-made or natural gives us an evolution that can be simulated efficiently on classical computers.

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