A Concise Introduction to Quantum Probability, Quantum Mechanics, and Quantum Computation

A concise introduction to quantum probability, quantum mechanics, and quantum computation Greg Kuperberg∗ UC Davis, visiting Cornell University (Dated: 2005) Quantum mechanics is one of the most interesting physics, and computer science. and surprising pillars of modern physics. Its basic This article is a concise introduction to quantum precepts require only undergraduate or early grad- probability theory, quantum mechanics, and quan- uate mathematics; but because quantum mechanics tum computation for the mathematically prepared is surprising, it is more difficult than these prerequi- reader. Chapters 2 and 3 depend on Section 1 but sites suggest. Moreover, the rigorous and clear rules not on each other, so the reader who is interested in of quantum mechanics are sometimes confused with quantum computation can go directly from Chap- the more difficult and less rigorous rules of quantum ter 1 to Chapter 3. field theory. This article owes a great debt to the textbook on Many working mathematicians have an excellent quantum computation by Nielsen and Chuang [20], intuitive grasp of two parent theories of quantum and to the Feynman Lectures, Vol. III [12]. An- mechanics, namely classical mechanics and probabil- other good textbook written for physics students is ity theory. The empirical interpretations of each of by Sakurai [21]. these theories — above and beyond their mathematical formalism — have been a great source of ideas Exercises for mathematics proper. I believe that more mathematicians could and should learn quantum mechanics and borrow its interpretation for mathematical These exercises are meant to illustrate how empir- problems. Two subdisciplines of mathematics that ical interpretations can lead to solutions of mathe- have assimilated the precepts of quantum mechan- matical problems. ics are mathematical physics and operator algebras. 1. The probabilistic method: The Ramsey num- However, the prevailing intention of mathematical ber R(n) is defined as the least R such that if physics is the converse, to apply mathematics to a simple graph Γ has R vertices, then either it problems in physics. The theory of operator algebras or its complement must have a complete sub- is closer to the spirit of this article; in this theory the graph with n vertices. By considering random precepts of quantum mechanics are sometimes called graphs, show that “non-commutative probability”. 2(n 1)/2 Recently quantum computation has entered as a − R(n) 1/n . new reason for both mathematicians and computer ≥ (2(n!)) scientists to learn the precepts of quantum mechan- 2. Angular momentum: Let S be a smooth sur- ics. Just as randomized algorithms can be moder- face of revolution about the z-axis in R3, and ately faster than deterministic algorithms for some let ~p(t) be a geodesic arc on S, parameterized computational problems (such as testing primality), by length, that begins at the point (1, 0, 0) at some problems admit quantum algorithms that are t = 0. Show that ~p(t) never reaches any point faster (sometimes much faster) than their classical within 1/ p′ (0) of the vertical axis. and randomized alternatives. These quantum algo- | y | rithms can only run on a new kind of computer called 3. Kirchoff’s laws: Suppose that a unit square is a quantum computer. As of this writing, convincing tiled by finitely many smaller squares. Show quantum computers do not exist. Nonetheless, the- that the edge lengths are uniquely determined oretical results suggest that quantum computers are by the combinatorial structure of the tiling, possible rather than impossible. Entirely apart from and that they are rational. (Hint: Build the its potential as a technology, quantum computation unit square out of material with unit resistivity is a beautiful subject that combines mathematics, with a battery connected to the top and bot- tom edges. Cut slits along the vertical edges of the tiles and affix zero-resistance wires to the horizontal edges. Each square becomes a unit ∗Electronic address: [email protected]@math. resistor in an electrical network.) cornell.edu 2 1. QUANTUM PROBABILITY Quant) which we will call U. The objects of U are complex Hilbert spaces and the morphisms are The precepts of quantum mechanics are neither unitary maps. We also add subunitary maps to U a set of physical forces nor a geometric model for to make a moderately larger category U’. We will physical objects. Rather, they are a variant, and also mostly restrict our attention to the subcategory U< of finite-dimensional Hilbert spaces. ultimately a generalization, of classical probability ∞ theory. (This is following the standard Copenhagen Recall that a Hilbert space is a complex vector space with a positive-definite Hermitian inner interpretation; see Section 1.6.) Quantum proba- H bility is usually defined using the matrix mechanics product . This means that is a function from h·|·ito C that satisfies theseh·|·i axioms: model, which describes vector states (or pure states) H → H and offers a probabilistic interpretation of final mea- ψ + ψ ψ = ψ ψ + ψ ψ surement. We will present this model together with h 1 2| 3i h 1| 3i h 2| 3i an important extension to mixed states. In physics, ψ1 ψ2 = ψ2 ψ1 wave mechanics is sometimes presented as an alter- h | i h | i ψ1 αψ2 = α ψ1 ψ2 nate definition of quantum mechanics; we will de- h | i h | i for α C scribe it as a special case of pure-state matrix me- ∈ chanics. ψ ψ > 0 for ψ =0. h | i 6 Since classical probability is a major analogy for (In the infinite case, must also be complete rela- us, it is reviewed in Section 1.10. In short, we can H think of classical probability as a category Prob tive to the norm whose objects are measure spaces (or in the finite ψ = ψ ψ .) case, finite sets) and whose morphisms are stochas- || || h | i tic maps. (For readers who are not comfortable with In quantum theory, the traditionalp notation is ψ (a this terminology, Section 1.11 is a cursory review.) “ket”) for ψ and ψ (a “bra”) for the dual vector| i Even though category theory can be very abstract h | [18], our interpretation of this category is very em- ψ = ψ∗ = ψ . pirical: A measure space is the natural model for a h | h |·i physical (or otherwise empirical) object that can be If X is an operator on , then in a random state, and stochastic maps are the ac- H tions on such objects that are empirically allowed in ψ1 X ψ2 classical probability. Stochastic maps also subsume h | | i the notions of events and random variables. Finally is an expression for “the inner product of ψ1 with (and crucially) the probability category Prob is a X(ψ2)”. If tensor category: A Cartesian product of measure spaces, which is in spirit a tensor product, carries X = ψ1 ψ2 | i ⊗ h | the joint states of two (or more) separate probabilistic objects. has rank 1, then we can omit the “ ” and just write ⊗ We will define a category Quant for quan- X = ψ ψ . tum probability which is analogous to the cate- | 1ih 2| gory Prob. The ultimate generalization, discussed in Section 1.8, is a category vN that contains This notation is due to Dirac [10] and is called “bra- both Quant and Prob. Its objects are von Neu- ket” notation. A linear map U : 1 H2 is unitary if it preserves the inner product H ;→ it is subunitary mann algebras, which are sometimes called “non- h·|·i commutative measure spaces”. The objects of if it preserves or decreases the attendant norm . Recall also that a linear map from a Hilbert space|| · || Quant are, famously, Hilbert spaces. Until Sec- tion 1.7, we will consider only finite-dimensional vec- to itself is called an operator. The standard finite example of a Hilbert space is tor spaces. These are enough to learn from, just as Cn the finite case is enough to learn most of the empir- the standard complex vector space with the inner ical interpretation of classical probability. product ~x ~y = x y + x y + + x y . h | i 1 1 2 2 · · · n n 1.1. Vector states and unitary maps We can generalize this to say that for any finite set A, the vector space CA is a Hilbert space with stan- Although it lacks some crucial empirical structure, dard orthonormal basis A. Every finite-dimensional most of quantum mechanics and much of quantum Hilbert space is isomorphic to Cn for some n, and computation relies only on a simpler category (than therefore CA for any A with A = n. | | 3 In finite quantum mechanics, as in classical prob- will eventually be reconciled.) The entries of U ability, we can define a physical object by specifying are also called amplitudes, just as the entries of a a finite set A of independent configurations. In in- stochastic map are themselves probabilities. The formation theory (both quantum and classical), the unitary condition is interpreted as conservation of 2 object is often called “Alice”. In the classical case, probability. Since we have posited that αa is a the set of all normalized states of Alice is the sim- probability, U conserves total probability if| and| only A plex ∆A spanned by A in the vector space R (see if Section 1.10). I.e., a general state has the form Uψ = ψ || || || || µ = pa[a] for all ψ CA. If U is allowed to extinguish the a A X∈ state ψ, then∈ in general for probabilities pa 0 that sum to 1. (For unnor- ≥ Uψ ψ malized states, the sum need not be 1.) The number || || ≤ || || p is interpreted as the probability that Alice is in a for all ψ CA, i.e., U is subunitary.

Load more