Quantum mechanics is the basic framework of modern subatomic physics. In quantum mechanics, the world unfolds through a combination of two basic ingredients. One is a smooth, fully deterministic wave function: a mathematical expression that conveys information about a particle in the form of numerous possibilities for its location and characteristics. The second is something that realizes one of those possibilities and eliminates all the others. Opinions differ about how that happens, but it might be caused by observation of the wave function or by the wave function encountering some part of the classical world.
Many physicists accept this picture at face value in a conceptual kludge known as the Copenhagen interpretation, authored by Niels Bohr and Werner Heisenberg in the s. But the Copenhagen approach is difficult to swallow for several reasons. Among them is the fact that the wave function is unobservable, the predictions are probabilistic and what makes the function collapse is mysterious.
Hugh Everett second from right originated the many-worlds theory. What are we to make of that collapsing wave?
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Carroll outlines several alternatives to the Copenhagen interpretation, along with their advantages and disadvantages. Another tack, named quantum Bayesianism, or QBism, by Christopher Fuchs, regards the wave function as essentially subjective. Thus it is merely a guide to what we should believe about the outcome of measurements, rather than a name for a real feature of the subatomic world.
Late in his life, Heisenberg proposed that we have to change our notion of reality itself. Carroll argues that the many-worlds theory is the most straightforward approach to understanding quantum mechanics. It accepts the reality of the wave function. In fact, it says that there is one wave function, and only one, for the entire Universe. Further, it states that when an event happens in our world, the other possibilities contained in the wave function do not go away. Instead, new worlds are created, in which each possibility is a reality.
For physicists, the theory is attractive because it explains many puzzles of quantum mechanics. Carroll also shows that the theory offers simpler explanations of certain complex phenomena, such as why black holes emit radiation. Furthermore, the theory might help to develop still-speculative ideas about conundrums such as how to combine quantum mechanics with relativity theory. Something Deeply Hidden is aimed at non-scientists, with a sidelong glance at physicists still quarrelling over the meaning of quantum mechanics.
Carroll brings the reader up to speed on the development of quantum physics from Max Planck to the present, and explains why it is so difficult to interpret, before expounding the many-worlds theory. This interlude, between a philosophically sensitive physicist and a scientifically alert philosopher, is designed to sweep away intuitive reservations that non-scientists might have.
Like many physicists, he assumes that reality is whatever a scientific theory says it is. But what gives physicists a lock on this concept, and the right to say that the rest of us not to mention, say, those in extreme situations such as refugees, soldiers and people who are terminally ill are living through a less fundamental reality? However, the main ideas can be understood using the finite-dimensional spectral theorem.
One semantics of quantum logic is that quantum logic is the logic of boolean observables in quantum mechanics, where an observable p is associated with the set of quantum states for which p when measured is true with probability 1 this completely characterizes the observable. From there,. Thus, expressions in quantum logic describe observables using a syntax that resembles classical logic. This occurs because measurement affects the system, and measurement of whether a disjunction holds does not measure which of the disjuncts is true.
For an example, consider a simple one-dimensional particle with position denoted by x and momentum by p , and define observables:. Now, position and momentum are Fourier transforms of each other, and the Fourier transform of a square-integrable nonzero function with a compact support is entire and hence does not have non-isolated zeroes. Such interference is key to the richness of quantum logic and quantum mechanics.
The so-called Hamiltonian formulations of classical mechanics have three ingredients: states , observables and dynamics. In the simplest case of a single particle moving in R 3 , the state space is the position-momentum space R 6. We will merely note here that an observable is some real-valued function f on the state space. Examples of observables are position, momentum or energy of a particle. For classical systems, the value f x , that is the value of f for some particular system state x , is obtained by a process of measurement of f. The propositions concerning a classical system are generated from basic statements of the form.
Figure 8 from Ju l 2 00 5 Quantum logic . A brief outline - Semantic Scholar
It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to that of some Boolean algebra of subsets of the state space. By logic in this context we mean the rules that relate set operations and ordering relations, such as de Morgan's laws. These are analogous to the rules relating boolean conjunctives and material implication in classical propositional logic. For technical reasons, we will also assume that the algebra of subsets of the state space is that of all Borel sets.
The set of propositions is ordered by the natural ordering of sets and has a complementation operation. We summarize these remarks as follows: The proposition system of a classical system is a lattice with a distinguished orthocomplementation operation: The lattice operations of meet and join are respectively set intersection and set union. The orthocomplementation operation is set complement. In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some possibly unbounded densely defined self-adjoint operator A on a Hilbert space H.
A has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of R. In particular, for any bounded Borel function f on R , the following extension of f to operators can be made:. In case f is the indicator function of an interval [ a , b ], the operator f A is a self-adjoint projection, and can be interpreted as the quantum analogue of the classical proposition. This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics.
Henceforth we identify elements of Q with self-adjoint projections on the Hilbert space H.
The structure of Q immediately points to a difference with the partial order structure of a classical proposition system. In the classical case, given a proposition p , the equations. In these equations I refers to the atomic proposition that is identically true and 0 the atomic proposition that is identically false. In the case of the lattice of projections there are infinitely many solutions to the above equations any closed, algebraic complement of p solves it; it need not be the orthocomplement.
Having made these preliminary remarks, we turn everything around and attempt to define observables within the projection lattice framework and using this definition establish the correspondence between self-adjoint operators and observables: A Mackey observable is a countably additive homomorphism from the orthocomplemented lattice of the Borel subsets of R to Q. Effectively, then, a Mackey observable is a projection-valued measure on R.
There is a bijective correspondence between Mackey observables and densely defined self-adjoint operators on H. This is the content of the spectral theorem as stated in terms of spectral measures. Imagine a forensics lab that has some apparatus to measure the speed of a bullet fired from a gun. Under carefully controlled conditions of temperature, humidity, pressure and so on the same gun is fired repeatedly and speed measurements taken. This produces some distribution of speeds.
Though we will not get exactly the same value for each individual measurement, for each cluster of measurements, we would expect the experiment to lead to the same distribution of speeds. This leads naturally to propose that under controlled conditions of preparation, the measurement of a classical system can be described by a probability measure on the state space.
This same statistical structure is also present in quantum mechanics. The following highly non-trivial theorem is due to Andrew Gleason :. Suppose Q is a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measure P on Q there exists a unique trace class operator S such that.
The operator S is necessarily non-negative that is all eigenvalues are non-negative and of trace 1. Such an operator is often called a density operator. Physicists commonly regard a density operator as being represented by a possibly infinite density matrix relative to some orthonormal basis. For more information on statistics of quantum systems, see quantum statistical mechanics. Now we use a theorem of Richard V. Kadison :. Then there is an operator U on the Hilbert space that is either linear or conjugate-linear, preserves the inner product and is such that.
In the first case we say U is unitary, in the second case U is anti-unitary. This note is included for technical accuracy only, and should not concern most readers. The operator U is not quite unique; if r is a complex scalar of modulus 1, then r U will be unitary or anti-unitary if U is and will implement the same automorphism.
In fact, this is the only ambiguity possible.
Mathematical Model of Quantum Computing
It follows that automorphisms of Q are in bijective correspondence to unitary or anti-unitary operators modulo multiplication by scalars of modulus 1. Moreover, we can regard automorphisms in two equivalent ways: as operating on states represented as density operators or as operating on Q.
In non-relativistic physical systems, there is no ambiguity in referring to time evolution since there is a global time parameter. Moreover, we assume. Now the square of an anti-unitary is a unitary, so that all the U t are unitary. Considering the statistical state of system as specified by lab conditions used for its preparation, the convex combination S can be regarded as the state formed in the following way: toss a biased coin with outcome probabilities p 1 , p 2 and depending on outcome choose system prepared to S 1 or S 2.
Density operators form a convex set. The convex set of density operators has extreme points ; these are the density operators given by a projection onto a one-dimensional space. To see that any extreme point is such a projection, note that by the spectral theorem S can be represented by a diagonal matrix; since S is non-negative all the entries are non-negative and since S has trace 1, the diagonal entries must add up to 1.
Now if it happens that the diagonal matrix has more than one non-zero entry it is clear that we can express it as a convex combination of other density operators.
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The extreme points of the set of density operators are called pure states. Consider a quantum mechanical system with lattice Q that is in some statistical state given by a density operator S.
This essentially means an ensemble of systems specified by a repeatable lab preparation process. The result of a cluster of measurements intended to determine the truth value of proposition E , is just as in the classical case, a probability distribution of truth values T and F.
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Perhaps the most fundamental difference between classical and quantum systems is the following: regardless of what process is used to determine E immediately after the measurement the system will be in one of two statistical states:. We leave to the reader the handling of the degenerate cases in which the denominators may be 0. We now form the convex combination of these two ensembles using the relative frequencies p and q. We thus obtain the result that the measurement process applied to a statistical ensemble in state S yields another ensemble in statistical state:.