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.
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.
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.
We investigate the reason for the quantum speedup (quantum algorithms require fewer computation steps than their classical counterparts). We extend the representation of the quantum algorithm to the process of setting the problem, namely choosing the function computed by the black box. The initial measurement selects a setting at random, Bob (the problem setter) unitarily changes it into the desired one. With reference to the observer dependent quantum states of relational quantum mechanics, this representation is with respect to Bob and any external observer, it cannot be with respect to Alice (the problem solver). It would tell her the function computed by the black box, which to her should be hidden. To Alice, the projection of the quantum state due to the initial measurement is retarded at the end of her problem solving action, so that the algorithm input state remains one of complete ignorance of the setting. By black box computations, she unitarily sends it into the output state that, for each possible setting, encodes the corresponding solution, acquired by the final measurement. Mathematically, we can ascribe to the final measurement the selection of any fraction R of the random outcome of the initial measurement. This projects the input state to Alice on one of lower entropy where she knows the corresponding fraction of the problem setting. Given the appropriate value of R, the quantum algorithm is a sum over classical histories in each of which Alice, knowing in advance one of the R-th parts of the setting, performs the black box computations still required to identify the solution. Given a quantum algorithm, this retrocausality model provides the value of R that explains its speed up; in the major quantum algorithms, R is 1/2 or slightly above it. Conversely, given the problem, R=1/2 always yields the order of magnitude of the number of black box computations required to solve it in an optimal quantum way. Quanta 2016; 5: 34–52.
In former work, we showed that a quantum algorithm is the sum over the histories of a classical algorithm that knows in advance 50% of the information about the solution of the problem - each history is a possible way of getting the advanced information and a possible result of computing the missing information. We gave a theoretical justification of this 50% advanced information rule and checked that it holds for a large variety of quantum algorithms. Now we discuss the theoretical justification in further detail and counter a possible objection. We show that the rule is the generalization of a simple, well known, explanation of quantum nonlocality - where logical correlation between measurement outcomes is physically backed by a causal/deterministic/local process with causality allowed to go backward in time with backdated state vector reduction. The possible objection is that quantum algorithms often produce the solution of the problem in an apparently deterministic way (when their unitary part produces an eigenstate of the observable to be measured and measurement produces the corresponding eigenvalue - the solution - with probability 1), while the present explanation of the speed up relies on the nondeterministic character of quantum measurement. We show that this objection would mistake the nondeterministic production of a definite outcome for a deterministic production.
Recent works in foundations of quantum (field) theory and relativistic quantum information try to better grasp the interplay between the structure of quantum correlations and the constraints imposed by causality on physical operations. Chapter 1 is dedicated to the study of the conceptual implications of quantum nonlocality, a concept that subsumes that of entanglement in a certain way. We detail the recent information-theoretic approaches to understanding the structure of nonlocal correlations, and the issues the latter raise concerning the ability of local observers to isolate a system from its environment. Chapter 2 reviews in what sense imposing Poincare invariance affects entanglement detection and quantification procedures. This invariance ultimately forces a description of all quantum systems within the framework of quantum field theory, which leads to the impossibility of localized finite-energy states and to the divergence of all entanglement measures for local observers. We provide a solution to these two problems by showing that there exists a generic equivalence between a finite spatial resolution of the measurement apparatus and the exclusion of high-energy degrees of freedom from the definition of the observed system. This equivalence allows for an epistemic interpretation of the standard quantum formalism describing nonrelativistic localized systems and their correlations, hence a clarification of the origin of the finite measures of entanglement between such systems. Chapter 3 presents a recent theoretical framework that predicts the existence of correlations with indefinite causal order. In analogy to the information-theoretic approaches to nonlocal correlations, we introduce some principles that constrain the structure of such correlations, which is a first step toward a clear understanding of their physical origin.
We explain the mechanism of the quantum speed-up -quantum algorithms requiring fewer computation steps than their classical equivalentfor a family of algorithms.Bob chooses a function and gives to Alice the black box that computes it.Alice, without knowing Bob's choice, should find a character of the function (e. g. its period) by computing its value for different arguments.There is naturally correlation between Bob's choice and the solution found by Alice.We show that, in quantum algorithms, this correlation becomes quantum.This highlights an overlooked measurement problem: sharing between two measurements the determination of correlated (thus redundant) measurement outcomes.All is like Alice, by reading the solution at the end of the algorithm, contributed to the initial choice of Bob, for half of it in quantum superposition for all the possible ways of taking this half.This contribution, back evolved to before running the algorithm, where Bob's choice is located, becomes Alice knowing in advance half of this choice.The quantum algorithm is the quantum superposition of all the possible ways of taking half of Bob's choice and, given the advanced knowledge of it, classically computing the missing half.This yields a speed-up with respect to the classical case where, initially, Bob's choice is completely unknown to Alice.