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The Essence of Quantum Computing

4.1 Postulates of quantum mechanics

 

The four postulates of quantum mechanics have no resemblance to the postulates of Newtonian mechanics. The quantum postulates are, as given in Nielsen and Chuang23:

Postulate 1: Associated to any isolated physical system is a complex vector space with inner product (that is, a Hilbert space) known as the state space of the system. The system is completely described by its state vector, which is a unit vector in the system’s state space.

Postulate 2: The evolution of a closed quantum system is described by a unitary transformation. That is, the state |Ψ(t1)〉 of the system at time t1 is related to the state |Ψ(t2)〉 of the system at time t1 by a unitary operator U which depends only on the times t1 and t2, i.e.,

Postulate 3: Quantum measurements are described by a collection {Mm} of measurement operators. These are operators, which act on the state space of the system being measured. The index m refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is |Ψ〉 just before the measurement, then the probability that result m occurs is given by

4.2 Comments on the postulates

 

The four postulates are independent of one another. Postulates 1 and 4 describe the mathematical state space in which quantum mechanical systems live and act. It asserts that a quantum system is completely described by its state vector, |Ψ〉 , which is a unit vector in the system’s state space. Postulate 2 describes the evolution of a quantum system as described by its state vector |Ψ〉 . Postulate 3 describes the post-measurement, collapsed state of |Ψ〉.

The evolution of |Ψ〉 under Postulate 2 is completely deterministic, continuous, and smooth. Its value, even using proxies, cannot be determined by any classical measurement system. Its evolution is governed by the Schrödinger wave equation, which provides the rate at which |Ψ〉 changes. In fact, the heart of quantum theory is Postulate 2 once we know how to interpret |Ψ〉 according to Postulate 3. There are no probabilities involved in Postulate 2.

Postulate 3 asserts that any legitimate classical measurement on |Ψ〉, in general, will yield only partial, non-deterministic information about the system’s state immediately prior to the measurement. The very first measurement on a system will irreversibly “collapse” |Ψ〉 and do so in a probabilistic manner. Measurement overrides Postulate 2. It is not possible to know |Ψ〉 of an unknown quantum system completely by making multiple measurements on the same system or even on multiple copies of the system. Since all the probabilities associated with quantum mechanics are contained in Postulate 3, a knowledge of probability is essential to analyze measured data. Measurement is an irreversible process. Once a measurement is made, the previous history of |Ψ〉 is lost. Postulates 2 and 3 appear conflicting. F. Laloë articulates the conflict.

Obviously, having two different postulates for the evolution of the same mathematical object is unusual in physics; the notion was a complete novelty when it was introduced, and still remains unique in physics, as well as the source of difficulties. Why are two separate postulates necessary? Where exactly does the range of application of the first stop in favor of the second? More precisely, among all the interactions – or perturbations – that a physical system can undergo, which ones should be considered as normal (Schrödinger evolution), which ones as a measurement (wave packet reduction)? Logically, we are faced with a problem that did not exist before [in classical physics], when nobody thought that measurements should be treated as special processes in physics.24

Roger Penrose25 too describes the weirdness of the postulates. In his view, in quantum mechanics there exists two completely different postulates those describe the evolution of |Ψ〉. Postulate 2 is totally deterministic, whereas Postulate 3 is probabilistic; Postulate 2 maintains quantum complex superposition, but Postulate 3 grossly violates it; Postulate 2 acts in a continuous way, Postulate 3 is blatantly discontinuous. There is nothing in quantum mechanics that even implies that one can ‘deduce’ Postulate 3 as a complicated instance of Postulate 2. Postulate 3 is simply a different procedure and it contains all the non-determinism found in the theory. It replaces superposition with a definite state. Both postulates are needed for all the marvelous agreements that quantum theory has consistently provided with observational facts. What is missing is a clear rule that would tell us when the probabilistic Postulate 3 should be invoked, in place of Postulate 2.


[23] Nielsen & Chuang (2000). pp. 80-94.
[24] Laloë (2001).
[25] Penrose (1989), p. 323.

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