This paper argues that the requirement of applicableness of quantum linearity to any physical level from molecules and atoms to the level of macroscopic extensional world, which leads to a main foundational problem in quantum theory referred to as the "measurement problem", actually has a computational character: It implies that there is a generic algorithm, which guarantees exact solutions to the Schrodinger equation for every physical system in a reasonable amount of time regardless of how many constituent microscopic particles it comprises. From the point of view of computational complexity theory, this requirement is equivalent to the assumption that the computational complexity classes P and NP are equal, which is widely believed to be very unlikely. As demonstrated in the paper, accepting the different computational assumption called the Exponential Time Hypothesis (that involves P!=NP) would justify the separation between a microscopic quantum system and a macroscopic apparatus (usually called the Heisenberg cut) since this hypothesis, if true, would imply that deterministic quantum and classical descriptions are impossible to overlap in order to obtain a rigorous derivation of complete properties of macroscopic objects from their microstates.
This paper argues that the requirement of applicableness of quantum linearity to any physical level from molecules and atoms to the level of macroscopic extensional world, which leads to a main foundational problem in quantum theory referred to as the "measurement problem", actually has a computational character: It implies that there is a generic algorithm, which guarantees exact solutions to the Schrodinger equation for every physical system in a reasonable amount of time regardless of how many constituent microscopic particles it comprises. From the point of view of computational complexity theory, this requirement is equivalent to the assumption that the computational complexity classes P and NP are equal, which is widely believed to be very unlikely. As demonstrated in the paper, accepting the different computational assumption called the Exponential Time Hypothesis (that involves P!=NP) would justify the separation between a microscopic quantum system and a macroscopic apparatus (usually called the Heisenberg cut) since this hypothesis, if true, would imply that deterministic quantum and classical descriptions are impossible to overlap in order to obtain a rigorous derivation of complete properties of macroscopic objects from their microstates.
This paper argues that the requirement of applicableness of quantum linearity to any physical level from molecules and atoms to the level of macroscopic extensional world, which leads to a main foundational problem in quantum theory referred to as the "measurement problem", actually has a computational character: It implies that there is a generic algorithm, which guarantees exact solutions to the Schr ödinger equation for every physical system in a reasonable amount of time regardless of how many constituent microscopic particles it comprises.From the point of view of computational complexity theory, this requirement is equivalent to the assumption that the computational complexity classes P and NP are equal, which is widely believed to be very unlikely.As demonstrated in the paper, accepting the different computational assumption called the Exponential Time Hypothesis (that involves P =NP) would justify the separation between a microscopic quantum system and a macroscopic apparatus (usually called the Heisenberg cut) since this hypothesis, if true, would imply that deterministic quantum and classical descriptions are impossible to overlap in order to obtain a rigorous derivation of complete properties of macroscopic objects from their microstates.
It is argued that any possible definition of a realistic physics theory – i.e., a mathematical model representing the real world – cannot be considered comprehensive unless it is supplemented with requirement of being computationally realistic. That is, the mathematical structure of a realistic model of a physical system must allow the collection of all the system's physical quantities to compute all possible measurement outcomes on some computational device not only in an unambiguous way but also in a reasonable amount of time. In the paper, it is shown that a deterministic quantum model of a microscopic system evolving in isolation should be regarded as realistic since the NP-hard problem of finding the exact solution to the Schrödinger equation for an arbitrary physical system can be surely solved in a reasonable amount of time in the case, in which the system has just a small number of degrees of freedom. In contrast to this, the deterministic quantum model of a truly macroscopic object ought to be considered as non-realistic since in a world of limited computational resources the intractable problem possessing that enormous amount of degrees of freedom would be the same as mere unsolvable.
Is the dynamical evolution of physical systems objectively a manifestation of information processing by the universe? We find that an affirmative answer has important consequences for the measurement problem. In particular, we calculate the amount of quantum information processing involved in the evolution of physical systems, assuming a finite degree of fine-graining of Hilbert space. This assumption is shown to imply that there is a finite capacity to sustain the immense entanglement that measurement entails. When this capacity is overwhelmed, the system's unitary evolution becomes computationally unstable and the system suffers an information transition (`collapse'). Classical behaviour arises from the rapid cycles of unitary evolution and information transitions. Thus, the fine-graining of Hilbert space determines the location of the `Heisenberg cut', the mesoscopic threshold separating the microscopic, quantum system from the macroscopic, classical environment. The model can be viewed as a probablistic complement to decoherence, that completes the measurement process by turning decohered improper mixtures of states into proper mixtures. It is shown to provide a natural resolution to the measurement problem and the basis problem.
Abstract. Is the universe computable? If yes, is it computationally a polynomial place? In standard quantum mechanics, which permits infinite parallelism and the infinitely precise specification of states, a negative answer to both questions is not ruled out. On the other hand, empirical evidence suggests that NP-complete problems are intractable in the physical world. Likewise, computational problems known to be algorithmically uncomputable do not seem to be computable by any physical algorithms on the one hand, and physical computers on the other, finds a natural explanation if the universe is assumed to be algorithmic; that is, that physical reality is the product of discrete subphysical information processing equivalent to the actions of a probabilistic Turing machine. This assumption can be reconciled with the observed exponentiality of quantum systems at microscopic scales, and the consequent possibility of implementing Shor's quantum polynomial time algorithm at that scale, provided the degree of superposition is intrinsically, finitely upper-bounded. If this bound is associated with the quantum-classical divide (the Heisenberg cut), a natural resolution to the quantum measurement problem arises. From this viewpoint, macroscopic classicality is an evidence that the universe is in BPP, and both questions raised above receive affirmative answers. A recently proposed computational model of quantum measurement, which relates the Heisenberg cut to the discreteness of Hilbert space, is briefly discussed. A connection to quantum gravity is noted. Our results are compatible with the philosophy that mathematical truths are independent of the laws of physics.
Is the universe computable? If yes, is it computationally a polynomial place? In standard quantum mechanics, which permits infinite parallelism and the infinitely precise specification of states, a negative answer to both questions is not ruled out. On the other hand, empirical evidence suggests that NP‐complete problems are intractable in the physical world. Likewise, computational problems known to be algorithmically uncomputable do not seem to be computable by any physical means. We suggest that this close correspondence between the efficiency and power of abstract algorithms on the one hand, and physical computers on the other, finds a natural explanation if the universe is assumed to be algorithmic; that is, that physical reality is the product of discrete sub‐physical information processing equivalent to the actions of a probabilistic Turing machine. This assumption can be reconciled with the observed exponentiality of quantum systems at microscopic scales, and the consequent possibility of implementing Shor’s quantum polynomial time algorithm at that scale, provided the degree of superposition is intrinsically, finitely upper‐bounded. If this bound is associated with the quantum‐classical divide (the Heisenberg cut), a natural resolution to the quantum measurement problem arises. From this viewpoint, macroscopic classicality is an evidence that the universe is in BPP, and both questions raised above receive affirmative answers. A recently proposed computational model of quantum measurement, which relates the Heisenberg cut to the discreteness of Hilbert space, is briefly discussed. A connection to quantum gravity is noted. Our results are compatible with the philosophy that mathematical truths are independent of the laws of physics.
The macro-objectivation problem derives from the fact that the Schrodinger equation is linear and thus requires that a macroscopic system interacting with an entangled state must be entangled as well. However, such a requirement entails that the Schrodinger equation must also be solvable in the macroscopic world in the same way as it is solvable in the microscopic world, which itself is an assumption. In this work, the alternative assumption that the Schrodinger equation is in general an intractable problem is considered.