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arXiv:2103.03173

The quantum mechanical notion of unobservable causal loop and the anthropic principle

Giuseppe Castagnoli

Published Mar 4, 2021• 0 citations
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Abstract

It can be argued that the ordinary description of the reversible quantum process between two one-to-one correlated measurement outcomes is incomplete because, by not specifying the direction of causality, it allows causal structures that violate the time symmetry that is required of a reversible process. This also means that it can be completed simply by time-symmetrizing it, namely by requiring that the initial and final measurements evenly contribute to the selection of their correlated pair of outcomes. This leaves the description unaltered but shows that it is the quantum superposition of unobservable time-symmetrized instances whose causal structure is completely defined. Each instance consists of a causal loop: the final measurement that changes backwards in time the input state of the unitary transformation that leads to the state immediately before it. In former works, we have shown that such loops exactly explain the quantum computational speedup and quantum nonlocality. In this work we show that they lead to a completion of the anthropic principle that allows a universe evolution with quantum speedup.

DOI
10.48550/arxiv.2103.03173

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It can be argued that the ordinary description of the reversible quantum process between two one-to-one correlated measurement outcomes is incomplete because, by not specifying the direction of causality, it allows causal structures that violate the time symmetry that is required of a reversible process. This also means that it can be completed simply by time-symmetrizing it, namely by requiring that the initial and final measurements evenly contribute to the selection of their correlated pair of outcomes. This leaves the description unaltered but shows that it is the quantum superposition of unobservable time-symmetrized instances whose causal structure is completely defined. Each instance consists of a causal loop: the final measurement that changes backwards in time the input state of the unitary transformation that leads to the state immediately before it. In former works, we have shown that such loops exactly explain the quantum computational speedup and quantum nonlocality. In this work we show that they lead to a completion of the anthropic principle that allows a universe evolution with quantum speedup.
The ordinary quantum description of the reversible process between two one-to-one correlated measurement outcomes mathematically describes the correlation but leaves the causal structure that physically ensures it free. Time-symmetrizing this description leaves it unaltered but shows that it is the quantum superposition of unobservable time-symmetrization instances whose causal structure is completely defined. Each instance hosts a causal loop: a quantum measurement that retrocausally changes the input state of the unitary transformation that led to the state immediately before it. This form of retrocausality is immune to the paradoxes that plague the usual form: it vanishes in the quantum description and therefore cannot change the past as described by it. The causal loops explain the physical structure of the process in a more complete way than its ordinary quantum description. In the quantum speedup, the two correlated measurement outcomes are the setting and the solution of the problem. In each time-symmetrization instance, the causal loop is as follows. All is as if the problem solver knew in advance half of the information about the solution she will produce in the future and could use this knowledge to produce the solution with fewer computation steps. In quantum nonlocality, the two outcomes are those of measuring two maximally entangled observables in a state of spatial separation of the respective subsystems. In each time-symmetrization instance, the measurement on either subsystem retrocausally and locally changes the state of both subsystems when the two were not yet spatially separate.
We consider the reversible processes between two one-to-one correlated measurement outcomes which underlie both problem solving and quantum nonlocality. In the former case, the two outcomes are the setting and the solution of the problem; in the latter, they are those of measuring a pair of maximally entangled observables whose subsystems are space separate. We argue that the quantum description of these processes mathematically describes the correlation but leaves the causal structure that physically ensures it free of violating the time symmetry required of the description of a reversible process. It would therefore be incomplete and could be completed by time symmetrizing it. This is done by assuming that the two measurements evenly contribute to selecting the pair of correlated measurement outcomes. Time symmetrization leaves the ordinary quantum description unaltered but shows that it is the quantum superposition of unobservable time-symmetrization instances whose causal structure is completely defined. Each instance is a causal loop: Causation goes from the initial to the final measurement outcome and then back from the final to the initial outcome. In the speedup, all is as if the problem solver knew in advance half of the information about the solution she will produce in the future and could use this knowledge to produce the solution with fewer computation steps. In nonlocality, the measurement on either subsystem retrocausally and locally changes the state of both subsystems when the two were not yet spatially separate. This locally causes the correlation between the two future measurement outcomes.
Modern approaches to causal modeling give a central role to interventions, which require the active input of an observer and introduces an explicit `causal arrow of time'. Causal models typically adopt a mechanistic interpretation, according to which the direction of the causal arrow is intrinsic to the process being studied. Here we investigate whether the direction of the causal arrow might be a contribution from the observer, rather than an intrinsic property of the process. Working within a counterfactual and non-mechanistic interpretation of causal modeling developed in arXiv:1806.00895, we propose a definition of a `quantum observational scheme' that we argue characterizes the observer-invariant properties of a causal model. By restricting to quantum processes that preserve the maximally mixed state (unbiasedness) we find that the statistics is symmetric under reversal of the time-ordering. The resulting model can therefore accommodate the idea that the causal arrow is observer-dependent, indicating a route towards reconciling the causal arrow with time-symmetric laws of physics.
It is brought forward that viable theories of the physical world that have no variable at all that can play the role of time, do not exist; some notion of time is one of the very first ingredients a candidate theory should possess. Almost by definition, time has an arrow. In contrast, time reversibility, or even the possibility to run the equations of motion backwards in time, is not at all a primary requirement. This means that the direction of the arrow of time may well be uniquely defined in the theory, even locally. It is explained that a rigorous definition of time, as well as a formulation of the causality and locality concepts, can only be given when one has a model for the physical phenomena described. The only viable causality condition is one that is symmetric under time reversal. We explain these statements in terms of the author’s favoured deterministic cellular automaton interpretation of quantum mechanics, also to be referred to as 'vector space analysis', and expand on these ideas. It is also summarised how our more rigorous causality condition affects Bell’s theorems. What distinguishes quantum systems from classical ones is our fundamental inability to control the microscopic details of the initial state when phenomena are studied in the light of some theoretical model.
The standard formulation of quantum theory assumes a predefined notion of time. This is a major obstacle in the search for a quantum theory of gravity, where the causal structure of space-time is expected to be dynamical and fundamentally probabilistic in character. Here, we propose a generalized formulation of quantum theory without predefined time or causal structure, building upon a recently introduced operationally time-symmetric approach to quantum theory. The key idea is a novel isomorphism between transformations and states which depends on the symmetry transformation of time reversal. This allows us to express the time-symmetric formulation in a time-neutral form with a clear physical interpretation, and ultimately drop the assumption of time. In the resultant generalized formulation, operations are associated with regions that can be connected in networks with no directionality assumed for the connections, generalizing the standard circuit framework and the process matrix framework for operations without global causal order. The possible events in a given region are described by positive semidefinite operators on a Hilbert space at the boundary, while the connections between regions are described by entangled states that encode a nontrivial symmetry and could be tested in principle. We discuss how the causal structure of space-time could be understood as emergent from properties of the operators on the boundaries of compact space-time regions. The framework is compatible with indefinite causal order, timelike loops, and other acausal structures.
Reversible algorithms play a crucial role both in classical and quantum computation. While for a classical bit the only nontrivial reversible operation is the bit-flip, nature is far more versatile in what it allows to do to a quantum bit. The reversible operations that a quantum computer can perform on a qubit are group of linear unitary transformations. However, laws of quantum mechanics prohibit implementation of anti-linear anti-unitary gates, even though they are perfectly reversible. Here we show that such a restriction on possible set of reversible operations is, remarkably, a fundamental constraint of spacetime structure. In particular, it will be shown that construction of any anti-linear anti-unitary gate will lead to violation of a fundamental causal primitive which, as we shall argue, is fundamentally different from the principle of relativistic causality.