### Quantum computations without definite causal structure and TGD

I encountered a link to a interesting popular article

*"Causal Witness" Provides First Experimental Evidence Of Indefinite Causal Order*(see this). The article tells about an article

*Experimental verification of an indefinite causal order*by Rubio et al (see this). In the following are my first impressions. Probably I mest update impressions. The findings are however so fascinating that I cannot resists the temptation to

post them in blog.

**1. Indefinite causal order**

The abstract of the article Rubio et al might give some idea about what is involved.

* Investigating the role of causal order in quantum mechanics has recently revealed that the causal relations of events may not be a priori well-defined in quantum theory. Although this has triggered a growing interest on the theoretical side, creating processes without a causal order is an experimental task. We report the first decisive demonstration of a process with an indefinite causal order. To do this, we quantify how incompatible our setup is with a definite causal order by measuring a "causal witness">. This mathematical object incorporates a series of measurements that are designed to yield a certain outcome only if the process under examination is not consistent with any well-defined causal order. In our experiment, we perform a measurement in a superposition of causal ordersâ€”without destroying the coherenceâ€”to acquire information both inside and outside of a causally non-ordered process. Using this information, we experimentally determine a causal witness, demonstrating by almost 7 SDs that the experimentally implemented process does not have a definite causal order.*

The experiment thus shows that they are are processes which cannot have a definite causal order in the sense that they are superposition of opposite causal orders. The notion of causal order is discussed in the article "Quantum correlations with no causal order" by Oreshkov et al (see this). Unfortunately, I do not have prerequisites to say anything interesting about the experiment itself. Since causal order is fixed by that associated with space-time in standard physics, the implications of the experiment could be world view changing.

Second basic notion is called SWITCH. One has two events A and B, which can be connected by a time-like curve. One can tell whether A precedes B or vice versa. SWITCH is a quantum computer operation switching the causal order. The obvious manner to do this would permute A and B and would require "time travel" not allowed in standard physics.

If superpositions of causal orders are possible, one can have a situation in which causal order is indefinite. Also this is something which does not conform the ordinary view about physics in fixed space-time but is allowed by postulates for quantum computation and there even exist an example of quantum computation impossible with a fixed causal order.

The notions of causal order and superposition of causal orders are revolutionary ideas and the article claims that they have been experimentally verified. Therefore there are excellent motivations to find whether they could be be understood in TGD framework.

** 2. ZEO and discrete symmetries for twistor lift of TGD**

First some background.

- Zero Energy Ontology (ZEO) is the cornerstone of TGD and TGD inspired theory of consciousness. Zero energy states appear as two variants and correspond to different WCW spinor fields (WCW for "world of classical worlds"). I have proposed that they correspond also to WCW spinor fields localized to different sectors of WCW but this might be un-necessarily strong assumption.

Zero energy states for given basis have been subject to a state function reduction at either boundary of CD - passive boundary. Neither the passive boundary nor the members of state pairs at it appearing in the superposition of state pairs are affected in repeated state function reductions. One has what I have called generalized Zeno effect.

At the opposite boundary the states evolve: every state function reduction is followed by a unitary time evolution ending to a localization of the active boundary of CD. The temporal distance between the tips of CD increases in this process and gives to clock time and experienced flow of time.

Eventually the first reduction to the opposite boundary occurs and the roles of active and passive boundary of CD are changed. One can say that time reversed zero energy state is obtained and begins to evolve. The first reduction to the opposite boundary of CD would mean death of the conscious entity defined by the sequence of state function reductions at the same boundary of CD and generation of time reversed re-incarnation of self.

- The twistor lift of TGD has led to a rather detailed understanding of discrete symmetries CP, P,T. If M
^{4}factor of imbedding space - or more precisely CD - is endowed with a generalized*self-dual*Kähler form J(M^{4}) (analogs of magnetic and electric fields of same magnitude and direction), new violations of CP, P and T occurring in long scales and having no counterpart in standard model emerge. The reason is that CP, P and T do not respect the self-duality of M^{4}Kähler form. The violations of Poincare invariance are avoided if one assumes moduli space for CDs containing the Lorentz boosts and translations of CD.

**3. Two views about SWITCH and superposition of causal orders**

One can imagine two approaches to the identification of SWITCH operation and superposition of causal orders.

** 3.1 Option I: SWITCH as time reversal (as "time travel")**

- Time reversal operation T applied to the 3-surfaces at the boundaries of CD would naturally change the causal order for the zero energy state. If T is an exact symmetry, the two causal orders would correspond to the time evolution of 3-surfaces with the 3-D boundaries of space-time surface at opposite boundaries of CD permuted.

This operation would also permute the values fermionic states at the boundaries of CD. This would correspond to "time travel" option. If T is violated, the connecting space-time surfaces for the causal orders would not be be simply time reflections of each other with respect to the center point of CD. This implies differences visible in the dynamics of spinors.

If T is violated it is not at all obvious that all pairs of 3-surfaces at opposite boundaries of CD can appear as boundaries of preferred extremals. It is also quite possible that only the 3-surface at the passive boundary can be taken by T to the opposite boundary of CD as such. The 3-surface at opposite boundary would change in the process by T-violation for preferred extremals. It could be that the sub-WCWs formed by pairs and the time reversals and disjoint and form different sectors of WCW.

- One could also have a superposition of zero energy states such with fixed set of 3-surfaces at either boundary related to T leaving middle point of CD invariant. These two zero energy states in the superpositions would have different causal order in a well-defined sense. It it is possible to realize this variant of SWITCH as almost T symmetry, then also the measurement of causal order should be possible by state function reduction.

- Violation of T would also imply asymmetry between selves and their time reversals. For instance, the average duration for the state reduction sequences keeping boundary fixed could be different and second causal order could dominate giving rise to a dominating arrow of time. Since these reduction sequences are identified as correlates for conscious selves, the time reversed re-incarnations would live much shorter time. Biological systems might be an exception: in TGD inspired theory of consciousness sensory perception and motor action are time reversals of each other.

- What about the situation for a union of CDs? Different CDs should be able to have their own arrow of time: for instance, there are reasons to think that in living matter one can have subsystems with non-standard arrow of time. Also phase conjugate laser ray could be also example of this. This requires that WCW spinor fields associated with a union of CDs form a tensor product. Causal order need not be same for all CDs but characterizes the sub-WCW associated with CD forming a Cartesian factor of WCW so that WCW spinors for the CDs form tensor product. Note that sub-WCWs with opposite causal orders do not correspond to Cartesian product space with additional coordinate but there could be a binary digit telling causal order. The state function for this binary digit could fix the causal order and quantum computation should be able to generate entanglement with this bit.

** 3.2 Option II: SWITCH as the first state function reduction to the opposite boundary of CD**

Could SWITCH be realized as the first state function reduction to the opposite boundary - death of self followed by a re-incarnation as time-reversed self? At least the analogy of SWITCH is in questions. This SWITCH does not correspond to "time travel".

What about the superposition of self and time reversed self? Schrödinger cat would be transformed to a superposition of cat and re-incarnated cat! This does not look plausible idea. I have assumed that the first reduction corresponds to a localization or to a basis of WCW spinor fields for which the parts of fermionic states at second boundary are state function reduced.

Contrary to what I have proposed, this need not require localization to a different - time reversed sector of WCW. This is however possible: if the initial an final zero energy states are orthogonal (inner product for zero energy states would involve both boundaries of CD): the sectors would have no common pairs of 3-surfaces at the boundaries of CD. The violation of T symmetry could cause this.

If the CDs with different arrow of time correspond to different sectors of WCW, then different causal orders could correspond to states localized in these these sectors. If so, a superposition of causal orders would correspond to WCW spinor field having component in both these sectors. In state function reduction to opposite boundary the causal orders of these components would change. If state function reduction takes place at the level of entire WCW, as seems natural, it should occur simultaneously in both sectors of WCW. There would be also a state function reduction measuring causal order and therefore performing a localization in either sector. SWITCH would switch these sectors.

The cautious conclusion is that Option I seems is more plausible.

** 4. Higher level quantum computations and ZEO**

From the article of Rubio et al one ends up to an article * Quantum computations without definite causal structure* by Chiribella et al (see this). The article considers a rather far reaching generalization of quantum computation. Ordinary quantum computation is a time evolution of quantum states followed by a state function reduction. Since the outcome of state function reduction halting the quantum computer program is non-deterministic, the extraction of the result involves statistical averaging over a large enough number of quantum computations to get the outcome, say prime factorization or a period of periodic function.

The notion of classical computation is however generalized by Church. The computation need not be a function but can assign function to a function. One can continue this abstraction hierarchy indefinitely and it is realized formally in terms of so called Λ calculus (see this). Could this hierarchy be extended to quantum computations? The quantum computation in question would assign to quantum computation a quantum computation, it would be a kind of super-computation.

In TGD this kind of hierarchies emerge naturally. At space-time level there is hierarchy of space-time sheets: space-time sheet is (topologically) condensed to a larger space-time sheet and contains smaller space-time sheets condensed at it. The hierarchy of infinite primes corresponds to an infinite hierarchy of second quantizations and could relate to this hierarchy (see this). In each scale space-time sheets would be particles consisting of smaller particles consisting of... Even galaxy could be seen as elementary particle in some scale and at the galactic space-time sheet.

The analog of quantum computational hierarchy emerges quite concretely in ZEO. The simplest zero energy states have positive and negative energy states with opposite total quantum numbers at the opposite boundaries of CD (intersection of future and past directed light-cones). One can have CDs within CDs. Furthermore, the positive/negative energy states assignable to the boundaries of CD could be also zero energy states associated with smaller CDs near the boundaries of CD. The simplest zero energy states correspond to quantum evolution representing ordinary quantum computation. Higher level zero energy stats would represent time evolution assigning to a quantum computation represented as zero energy state at the boundary of CD a second quantum computation at opposite boundary of CD.

For a summary of earlier postings see Latest progress in TGD.