I received a very interesting link to a popular article (see this) explaining a recently discovered deep result in mathematics having implications also in physics. The article (see this) by Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen has a rather concise title " MIP*=RE". In the following I try to express the impressions of a (non-mainstream) physicist about the result. In the first posting I discuss the finding and the basic implications from the physics point of view. In the second posting the highly interesting implications of the finding in the TGD framework are discussed.
The result expressed can be expressed by using the concepts of computer science about which I know very little at the hard technical level. The results are however told to state something highly non-trivial about physics.
The finding, which is characterized as shocking, is that all problems solvable by a Turing computer belong to this class: RE= MIP*. All problems solvable by computer would reduce to problems in the class MIP*! Quantum computation would indeed add something genuinely new to the classical computation.
- RE (recursively enumerable languages) denotes all problems solvable by computer. P denotes the problems solvable in a polynomial time. NP does not refer to a non-polynomial time but to "non-deterministic polynomial acceptable problems" - I hope this helps the reader- I am a little bit confused! It is not known whether P = NP is true.
- IP problems (P is now for "prover" that can be solved by a collaboration of an interrogator and prover who tries to convince the interrogator that her proof is convincing with high enough probability. MIP involves multiple l provers treated as criminals trying to prove that they are innocent and being not allowed to communicate. MIP* is the class of solvable problems in which the provers are allowed to entangle.
1. Two physically interesting applications
There are two physically interesting applications of the theorem which are interesting also from the TGD point of view and force to make explicit the assumptions involved.
1.1 About the quantum physical interpretation of MIP*
To proceed one must clarify the quantum physical interpretation of MIP*.
1.2 Connes embedding problem and the notion of finite measurement/cognitive resolution
- Quantum measurement requires entanglement of the observer O with the measured system M. What is basically measured is the density matrix of M (or equivalently that of O). State function reduction gives as an outcome a state, which corresponds to an eigenvalue of the density matrix. Note that this state can be an entangled state if the density matrix has degenerate eigenvalues.
- Quantum measurement can be regarded as a question to the measured system: "What are the values of given commuting observables?". The final state of quantum measurement provides an eigenstate of the observables as the answer to this question. M would be in the role of the prover and Oi would serve as interrogators.
In the first case multiple interrogators measurements would entangle M with unentangled states of the tensor product H1⊗ H2 for O followed by a state function reduction splitting the state of M to un-entangled state in the tensor product M1⊗ M2.
In the second case the entire M would be interrogated using entanglement of M with entangled states of H1⊗ H2 using measurements of several commuting observables. The theorem would state that interrogation in this manner is more efficient in infinite-D case unless HFFs are involved.
- This interpretation differs from the interpretation in terms of computational problem solving in which one would have several provers and one interrogator. Could these interpretations be dual as the complete symmetry of the quantum measurement with respect to O and M suggests? In the case of multiple provers (analogous to accused criminals) it is advantageous to isolate them. In the case of multiple interrogators the best result is obtained if the interrogator does not effectively split itself into several ones.
Alain Connes formulated what has become known as Connes embedding problem. The question is whether infinite matrices forming factor of type II1 can be always approximated by finite-D matrices that is imbedded in a hyperfinite factor of type II1 (HFF). Factors of type II and their HFFs are special classes of von Neumann algebras possibly relevant for quantum theory.
This result means that if one has measured of a complete set of for a product of commuting observables acting in the full space, one can find in the finite-dimensional case a unitary transformation transforming the observables to tensor products of observables associated with the factors of a tensor product. In the infinite-D case this is not true.
What seems to put alarms ringing is that in TGD there are excellent arguments suggesting that the state space has HFFs as building bricks. Does the result mean that entanglement cannot help in quantum computation in TGD Universe? I do not want to live in this kind of Universe!
1.3. Tsirelson problem
Tsirelson problem (see this) is another problem mentioned in the popular article as a physically interesting application. The problem relates to the mathematical description of quantum measurement.
Three systems are considered. There are two systems O1 and O2 representing observers and the third representing the measured system M. The measurement reducing the entanglement between M and O1 or O2 can regarded as producing correspondence between state of M and O1 or O2, and one can think that O1 or O2 measures only obserservables in its own state space as a kind of image of M. There are two manners to see the situation. The provers correspond now to the observers and the two situations correspond to provers without and with entanglement.
Consider first a situation in which one has single Hilbert space H and single observer O. This situation is analogous to IP.
Consider now the analog of the MIP* involving more than one observer. For simplicity consider two observers.
- The state of the system is described statistically by a density matrix - not necessarily pure state -, whose diagonal elements have interpretation as reduction probabilities of states in this bases. The measurement situation fixes the state basis used. Assume an ensemble of identical copies of the system in this state. Assume that one has a complete set of commuting observables.
- By measuring all observables for the members of the ensemble one obtains the probabilities as diagonal elements of the density matrix. If the observable is the density matrix having no- degenerate eigenvalues, the situation is simplified dramatically. It is enough to use the density matrix as an observable. TGD based quantum measurement theory assumes that the density matrix describing the entanglement between two subsystems is in a universal observable measure in state function reductions reducing their entanglement.
- Can one deduce also the state of M as a superposition of states in the basic chosen by the observer? This basis need not be the same as the basis defined by - say density matrix if the system has interacted with some system and this ineracton has led to an eigenstate of the density matrix. Assume that one can prepare the latter basis by a physical process such as this kind of interaction.
The coefficients of the state form a set of N2 complex numbers defining a unitary N× N matrix. Unitarity conditions give N conditions telling that the complex rows and unit vectors: these numbers are given by the measurement of all observables. There are also N(N-1) conditions telling that the rows are orthogonal. Together these N+N(N-1)=N2 numbers fix the elements of the unitary matrix and therefore the complex coefficients of the state basis of the system can be deduced from a complete set of measurements for all elements of the basis.
Are these manners to see the situation equivalent? Tsirelson demonstrated that this is the case for finite-dimensional Hilbert spaces, which can indeed be decomposed to a tensor product of factors associated with O1 and O2. This means that one finds a unitary transformation transforming the entangled situation to an unentangled one and to tensor product observables.
- Assume that the state space H of M decomposes to a tensor product H=H1⊗ H2 of state spaces H1 and H2 such that O1 measures observables X1 in H1 and O2 measuresobservables X2 in H2. The observables X1 and X2 commute since they act in different tensor factors. The absence of interaction between the factors corresponds to the inability of the provers to communicate. As in the previous case, one can deduce the probabilities for the various outcomes of the joint measurements interpreted as measurements of a complete set of observables X1 ⊗ X2.
- One can also think that the two systems form a single system O so that O1 and O2 can entangle. This corresponds to a situation in which entanglement between the provers is allowed. Now X1 and X2 are not in general independent but also now they must commute. One can deduce the probabilities for various outcomes as eigenstates of observables X1 X2 and deduce the diagonal elements of the density matrix as probabilities.
For the infinite-dimensional case the situation remained open. According to the article, the new result implies that this is not the case. For hyperfinite factors the situation can be approximated with a finite-D Hilbert space so that the situations are equivalent in arbitrary precise approximation.
2. What MIP*= RE could mean in TGD Universe?
The MIP*=RE means that quantum entanglement has a crucial role in quantum computation. One the other hand, HFFs would not allow to use quantum entanglement to quantum computational purposes. This looks at first a bad news from the TGD point of view, where HFFs are highly suggestive. One must be however very careful with the basic definitions.
2.1. The notion of finite measurement resolution in TGD
Measurement resolution is one of the basic notions of TGD.
For hyperfinite factors the analog of MIP*=RE cannot hold true. Doesn't the TGD Universe allow a solution of all the problems solvable by Turing Computer? There is however a loophole in this argument
- There are intuitive physicist's arguments demonstrating that in TGD the operator algebras involved with TGD are HFFs provides a description of finite measurement resolution. The inclusion of HFFs defines the notion of resolution: included factor represents the degrees of freedom not seen in the resolution used (see this) and this).
Hyperfinite factors involve new structures like quantum groups and quantum algebras reflecting the presence of additional symmetries: actually the "world of classical worlds" (WCW) as the space of space-time surfaces as maximal group of isometries and this group has a fractal hierarchy of isomorphic groups imply inclusion hierarchies of HFFs. By the analogs of gauge conditions this infinite-D group reduces to a hierarchy of effectively finite-D groups. For quantum groups the infinite number of irreps of the corresponding compact group effectively reduces to a finite number of them, which conforms with the notion of hyper-finiteness.
It looks that the reduction of the most general quantum theory to TGD-like theory relying on HFFs is not possible. This would not be surprising taking into account gigantic symmetries responsible for the cancellation of infinities in TGD framework and the very existence of WCW geometry.
- Second TGD based approach to finite resolution is purely number theoretic (see this) and involves adelic physics as a fusion of the real physics with various p-adic physics as correlates of cognition. Cognitive representations are purely number theoretic and unique discretizations of space-time surfaces defined by a given extension of rationals forming an evolutionary hierarchy: the coordinates for the points of space-time as a 4-surface of the imbedding space H=M4× CP2 or of its dual M8 are in the extension of rationals defining the adele. In the case of M8 the preferred coordinates are unique apart from time translation. These two views would define descriptions of the finite resolution at the level of space-time and Hilbert space. In particular, the hierarchies of extensions of rationals should define hierarchies of inclusions of HFFs.
- The point is that for the hierarchy of extensions of rationals also Hilbert spaces have as a coefficient field the extension of rationals!. Unitary transformations are restricted to matrices with elements in the extension. In general it is not possible to realize the unitary transformation mapping the entangled situation to an un-entangled one! The weakening of the theorem would hold true for the hierarchy of adeles and entanglement would give something genuinely new for quantum computation!
- A second deep implication is that the density matrix characterizing the entanglement between two systems cannot in general be diagonalized such that all diagonal elements identifiable as probabilities would be in the extension considered. One would have stable or partially stable entanglement (could the projection make sense for the states or subspaces with entanglement probability in the extension). For these bound states the binding mechanism is purely number theoretical. For a given extension of p-adic numbers one can assign to algebraic entanglement also information measure as a generalization of Shannon entropy as a p-adic entanglement entropy (real valued). This entropy can be negative and the possible interpretation is that the entanglement carries conscious information.
2.2 What about the situation for the continuum version of TGD?
At least the cognitively finitely representable physics would have the HFF property with coefficient field of Hilbert spaces replaced by an extension of rationals. Number theoretical universality would suggest that HFF property characterizes also the physics of continuum TGD. This leads to a series of questions.
Also fermions are highly interesting in the recent context. In TGD spinor structure can be seen as a square root of Kähler geometry, in particular for the "world of classical worlds" (WCW). Fermions are identified as correlates of Boolean cognition. The continuum case for fermions does not follow as a naive limit of algebraic picture.
- Does the new theorem imply that in the continuum version of TGD all quantum computations allowed by the Turing paradigm for real coefficients field for quantum states are not possible: MIP*⊂ RE? The hierarchy of extensions of rationals allows utilization of entanglement, and one can even wonder whether one could have MIP*= RE at the limit of algebraic numbers.
- Could the number theoretic vision force change also the view about quantum computation? What does RE actually mean in this framework? Can one really assume complex entanglement coefficients in computation. Does the computational paradigm makes sense at all in the continuum picture?
Are both real and p-adic continuum theories unreachable by computation giving rise to cognitive representations in the algebraic intersection of the sensory and cognitive worlds? I have indeed identified real continuum physics as a correlate for sensory experience and various p-adic physics as correlates of cognition in TGD: They would represent the computionally unreachable parts of existence.
Continuum physics involves transcendentals and in mathematics this brings in analytic formulas and partial differential equations. At least at the level of mathematical consciousness the emergence of the notion of continuum means a gigantic step. Also this suggests that transcendentality is something very real and that computation cannot catch all of it.
- Adelic theorem allows to express the norm of a rational number as a product of inverses of its p-adic norms. Very probably this representation holds true also for the analogs of rationals formed from algebraic integeres. Reals can be approximated by rationals. Could extensions of all p-adic numbers fields restricted to the extension of rationals say about real physics only what can be expressed using language?
These considerations relate in an interesting manner to consciousness. Quantum entanglement makes in the TGD framework possible telepathic sharing of mental images represented by sub-selves of self. For the series of discretizations of physics by HFFs and cognitive representations associated with extensions of rationals, the result indeed means something new.
- The quantization of the induced spinors in TGD looks different in discrete and continuum cases. Discrete case is very simple since equal-time anticommutators give discrete Kronecker deltas. In the continuum case one has delta functions possibly causing infinite vacuum energy like divergences in conserved Noether charges (Dirac sea).
- I have proposed (see this) how these problems could be avoided by avoiding anticommutators giving delta-function. The proposed solution is based on zero energy ontology and TGD based view about space-time. One also obtains a long-sought-for concrete realization for the idea that second quantized induce spinor fields are obtained as restrictions of second quantized free spinor fields in H=M4× CP2 to space-time surface. The fermionic variant of M8-H-duality (see this) provides further insights and gives a very concrete picture about the dynamics of fermions in TGD.
2.3 What about transcendental extensions?
During the writing of this article an interesting question popped up.
- Also transcendental extensions of rationals are possible, and one can consider the generalization of the computationalism by also allowing functions in transcendental extensions. Could the hierarchy of algebraic extensions could continue with transcendental extensions? Could one even play with the idea that the discovery of transcendentals meant a quantum leap leading to an extension involving for instance e and π as basic transcendentals? Could one generalize the notion of polynomial root to a root of a function allowing Taylor expansion f(x)= ∑ qn xn with rational coefficients so that the roots of f(x)=0 could be used define transcendental extensions of rationals?
- Powers of e or its root define and infinite-D extensions having the special property that they are finite-D for p-adic number fields because ep is ordinary p-adic number. In the p-adic context e can be defined as a root of the equation xp-∑ pn/n!=0 making sense also for rationals. The numbers log(pi) such that pi appears a factor for integers smaller than p define infinite-D extension of both rationals and p-adic numbers. They are obtained as roots of ex-pi=0.
- The numbers (2n+1)π (2nπ) can be defined as roots of sin(x)=0 (cos(x)=0. The extension by π is infinite-dimensional and the conditions defining it would serve as consistency conditions when the extension contains roots of unity and effectively replaces them.
- What about other transcendentals appearing in mathematical physics? The values ζ(n) of Riemann Zeta appearing in scattering amplitudes are for even values of n given by ζ(2n)= (-1)n+1 B2n (2π)2n/2(2n+1)!. This follows from the functional identity for Riemann zeta and from the expression ζ(-n)= (-1)n Bn+1/(n+1) ( (B(-1/2)=-1/2) (see this). The Bernoulli numbers Bn are rational and vanish for odd values of n. An open question is whether also the odd values are proportional to πn with a rational coefficient or whether they represent "new" transcendentals.
2.4. What does one mean with quantum computation in TGD Universe?
The TGD approach raises some questions about computation.
See the chapter Evolution of Ideas about Hyper-finite Factors in TGD or the article MIP*= RE: What could this mean physically?.
- The ordinary computational paradigm is formulated for Turing machines manipulating natural numbers by recursive algorithms. Programs would essentially represent a recursive function n→ f(n). What happens to this paradigm when extensions of rationals define cognitive representations as unique space-time discretizations with algebraic numbers as the limit giving rise to a dense in the set of reals.
The usual picture would be that since reals can be approximated by rationals, the situation is not changed. TGD however suggests that one should replace at least the quantum version of the Turing paradigm by considering functions mapping algebraic integers (algebraic rational) to algebraic integers.
Quite concretely, one can manipulate algebraic numbers without approximation as a rational and only at the end perform this approximation and computations would construct recursive functions in this manner. This would raise entanglement to an active role even if one has HFFs and even if classical computations could still look very much like ordinary computation using integers.
- ZEO brings in also time reversal occurring in "big" (ordinary) quantum jumps and this modifies the views about quantum computation. In ZEO based conscious quantum computation halting means "death" and "reincarnation" of conscious entity, self? How the processes involving series of haltings in this sense differs from ordinary quantum computation: could one shorten the computation time by going forth and back in time.
- There are many interesting questions to be considered. M8-H duality gives justifications for the vision about algebraic physics. TGD leads also to the notion of infinite prime and I have considered the possibility that infinite primes could give a precise meaning for the dimension of infinite-D Hilbert space. Could the number-theoretic view about infinite be considerably richer than the idea about infinity as limit would suggest (see this).
The construction of infinite primes is analogous to a repeated second quantization of arithmetic supersymmetric quantum field theory allowing also bound states at each level and a concrete correspondence with the hierarchy of space-time sheets is suggestive. For the infinite primes at the lowest level of the hierarchy single particle states correspond to rationals and bound states to polynomials and therefore to the sets of their roots. This strongly suggests a connection with M8 picture.