An introduction to quantum probability, quantum …

1 An introduction to quantum probability , quantum mechanics , and quantumcomputationGreg Kuperberg UC Davis(Dated: October 8, 2007) quantum mechanics is one of the most surprisingsides of modern physics. Its basic precepts requireonly undergraduate or early graduate mathemat-ics; but because quantum mechanics is surprising,it is more difficult than these prerequisites , the rigorous and clear rules of quantummechanics are sometimes confused with the more dif-ficult and less rigorous rules of quantum field working mathematicians have an excellentintuitive grasp of two parent theories of quantummechanics, namely classical mechanics and proba-bility theory. The empirical interpretations of bothof these theories, above and beyond their mathe-matical formalism, have been a great source of ideasin mathematics, even for many questions that havenothing to do with physics or practical statistics.

2 Forexample, the probabilistic method of Erd os and oth-ers [?] is a fundamental method in combinatoricsto show the existence of combinatorial objects. Inprinciple, the precepts of quantum mechanics couldbe similarly influential; there could easily be oneor more kind of quantum probabilistic method .But in practice the precepts of quantum mechan-ics are not very familiar to most subdisciplines of mathematics that have assim-ilated these precepts are mathematical physics andoperator algebras. However, much of the intentionof mathematical physics is the converse of our pur-pose, to apply mathematics to problems in theory of operator algebras is close to the spiritof this article; in this theory what we call quantumprobability is often called non-commutative proba-bility.

3 Recently quantum computation has entered as anew reason for both mathematicians and computerscientists to learn the precepts of quantum mechan-ics. Just as randomized algorithms can be moder-ately faster than deterministic algorithms for somecomputational problems, quantum algorithms canbe moderately faster or sometimes much faster thantheir classical and randomized alternatives. Quan-tum algorithms can only run on a new kind of com-puter called a quantum computer. As of this writ-ing, convincing quantum computers do not exist. Electronic theoretical results suggest that quan-tum computers are possible rather than impossi-ble. Entirely apart from technological implications, quantum computation is a beautiful subject thatcombines mathematics, physics, and computer article is an introduction to quantum prob-ability theory, quantum mechanics , and quan-tum computation for the mathematically preparedreader.

4 Chapters??and??depend on Section 1but not on each other, so the reader who is inter-ested in quantum computation can go directly fromChapter 1 to Chapter??.This article owes a great debt to the textbook onquantum computation by Nielsen and Chuang [4],and to the Feynman Lectures, Vol. III [2]. An-other good textbook written for physics students isby Sakurai [5].ExercisesThese exercises are meant to illustrate how empir-ical interpretations can lead to solutions of problemsin pure probabilistic method: The Ramsey num-berR(n) is defined as the leastRsuch that ifa simple graph hasRvertices, then either itor its complement must have a complete sub-graph withnvertices. By considering randomgraphs, show thatR(n) 2(n 1)/2(2(n!))

5 1/n.(The proof can be described as a couting ar-gument. However, a solution phrased in termsof probabilitistic existence is more in the spiritof these notes.) momentum: LetSbe a smooth sur-face of revolution about thez-axis inR3, andlet~p(t) be a geodesic arc onS, parameterizedby length, that begins at the point (1,0,0) att= 0. Show that~p(t) never reaches any pointwithin 1/|p y(0)|of the vertical s laws: Suppose that a unit square istiled by finitely many smaller squares. Show2that the edge lengths are uniquely determinedby the combinatorial structure of the tiling,and that they are rational. (Hint: Build theunit square out of material with unit resistivitywith a battery connected to the top and bot-tom edges. Cut slits along the vertical edges ofthe tiles and affix zero-resistance wires to thehorizontal edges.)

6 Each square becomes a unitresistor in an electrical network.)1. quantum PROBABILITYThe precepts of quantum mechanics are neithera set of physical forces nor a geometric model forphysical objects. Rather, they are a generalizationof classical probability theory that modifies the ef-fects of physical forces. If you have firmly acceptedclassical probability , it is tempting to suppose thatquantum mechanics is a set of probabilistic objects,in effect a special case of probability rather than ageneralization. But this is not true in any reasonablesense; quantum probability violates certain inequal-ities that hold in classical probability (Section??).It is also tempting to view quantum mechanics asa a deterministic dynamical system that producesclassical probabilities and is otherwise hidden.

7 Thisinterpretation is not reasonable physics courses, quantum mechanics is usu-ally defined in terms of operators acting on Hilbertspaces. A state of a system is a vector of its Hilbertspace, the vector evolves by unitary operators, thevector is measured by Hermitian operators, and themeasured values have probability we will discuss the vector-state model,we will emphasize the non-commutative probabilitymodel from operator algebras. In this model, a sys-tem can be fully quantum , or fully classical, or thingsin between. The fully quantum case corresponds tothe vector-state model, but even in this case, thegeneral state is described by an operator rather thana vector. The states that can be described by vectorsare called pure; the others are mixed vector-state model of quantum mechanicswas originally known as matrix mechanics and isdue to Heisenberg.

8 The historical alternative isSchr odinger s wave mechanics . Wave mechanics isbest understood as a special case of matrix mechan-ics, and we will describe it this way. The probabilis-tic interpretation of quantum mechanics is due toMax Born and is known as the Copenhagen inter-pretation (Section??).Since classical probability is a major analogy forus, it is reviewed in Section??. The point is that aclassical probabilistic system (or measurable space)is an algebra of random variables that satisfies rel-evant axioms. One of the restrictions on the alge-bra is commutativity: Ifxandyare two real- orcomplex-valued random variables, thenxyandyxare the same random variable. In quantum prob-ability, this commutative algebra is replaced by anon-commutative algebra called a von Neumann al-gebra.

9 The remaining definitions stay as much thesame as will mostly consider finite-dimensional quan-tum systems. These are enough to show most ofthe basic ideas of quantum probability , just as finiteor combinatorial probability is enough to show mostof the basic ideas of classical probability . Infinite-dimensional quantum systems are discussed in Sec-tion??.To summarize, quantum probability is the mostnatural non-commutative generalization of classicalprobability. In this author s opinion, this descriptiondoes the most to demystify quantum probability andquantum quantum superpositionsWe will begin by discussing part of the pure-statemodel of quantum mechanics in order to show theinadequacy of classical pure state of a quantum mechanical system canbe described as a vector of a complex vector spaceH.

10 If the system is finite, then we can say that thevector space isCn. It will be convenient to labelthe basis of this vector space by an arbitrary finitesetArather than by the numbers from 1 ton; wecan then denote the vector spaceCA. The generalstate spaceHis not just a vector space but a Hilbertspace, meaning that it has a positive-definite Hermi-tian inner producth | i. WhenHisCnorCA, thenit has the standard inner producth | i= a A a quantum theory, the traditional notation is| i(a ket ) for a vector andh |(a bra ) for thecorresponding dual vectorh |= =h | notation is due to Dirac [1] and is called bra-ket notation. Recall also that a linear map from aHilbert space to itself is called finite quantum mechanics , as in classical proba-bility, we can define a physical object by specifying afinite setAof independent configurations.