A concise introduction to quantum probability, …

1 A concise introduction to quantum probability , quantum mechanics , andquantum computationGreg Kuperberg UC Davis, visiting Cornell University(Dated: 2005) quantum mechanics is one of the most interestingand surprising pillars of modern physics. Its basicprecepts require only undergraduate or early grad-uate mathematics; but because quantum mechanicsis surprising, it is more difficult than these prerequi-sites suggest. Moreover, the rigorous and clear rulesof quantum mechanics are sometimes confused withthe more difficult and less rigorous rules of quantumfield working mathematicians have an excellentintuitive grasp of two parent theories of quantummechanics, namely classical mechanics and probabil-ity theory. The empirical interpretations of each ofthese theories above and beyond their mathemat-ical formalism have been a great source of ideasfor mathematics proper. I believe that more mathe-maticians could and should learn quantum mechan-ics and borrow its interpretation for mathematicalproblems.

2 Two subdisciplines of mathematics thathave assimilated the precepts of quantum mechan-ics are mathematical physics and operator , the prevailing intention of mathematicalphysics is the converse, to apply mathematics toproblems in physics. The theory of operator algebrasis closer to the spirit of this article; in this theory theprecepts of quantum mechanics are sometimes called non-commutative probability .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 (such as testing primality),some problems admit quantum algorithms that arefaster (sometimes much faster) than their classicaland randomized alternatives. These quantum algo-rithms can only run on a new kind of computer calleda quantum computer.

3 As of this writing, convincingquantum computers do not exist. Nonetheless, the-oretical results suggest that quantum computers arepossible rather than impossible. Entirely apart fromits potential as a technology, quantum computationis a beautiful subject that combines mathematics, Electronic and computer article is a concise introduction to quantumprobability theory, quantum mechanics , and quan -tum computation for the mathematically preparedreader. Chapters 2 and 3 depend on Section 1 butnot on each other, so the reader who is interested inquantum computation can go directly from Chap-ter 1 to Chapter article owes a great debt to the textbook onquantum computation by Nielsen and Chuang [20],and to the Feynman Lectures, Vol. III [12]. An-other good textbook written for physics students isby Sakurai [21].ExercisesThese exercises are meant to illustrate how empir-ical interpretations can lead to solutions of mathe-matical 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.

4 By considering randomgraphs, show thatR(n) 2(n 1)/2(2(n!))1 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. Showthat 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. Each square becomes a unitresistor in an electrical network.)21. quantum PROBABILITYThe precepts of quantum mechanics are neithera set of physical forces nor a geometric model forphysical objects.

5 Rather, they are a variant, andultimately a generalization, of classical probabilitytheory. (This is following the standard Copenhageninterpretation; see Section ) quantum proba-bility is usually defined using the matrix mechanicsmodel, which describes vector states (or pure states)and offers a probabilistic interpretation of final mea-surement. We will present this model together withan important extension to mixed states. In physics,wave mechanics is sometimes presented as an alter-nate definition of quantum mechanics ; we will de-scribe it as a special case of pure-state matrix classical probability is a major analogy forus, it is reviewed in Section In short, we canthink of classical probability as a categoryProbwhose objects are measure spaces (or in the finitecase, finite sets) and whose morphisms are stochas-tic maps. (For readers who are not comfortable withthis terminology, Section is a cursory review.)

6 Even though category theory can be very abstract[18], our interpretation of this category is very em-pirical: A measure space is the natural model for aphysical (or otherwise empirical) object that can bein a random state, and stochastic maps are the ac-tions on such objects that are empirically allowed inclassical probability . Stochastic maps also subsumethe notions of events and random variables. Finally(and crucially) the probability categoryProbis atensor category: A Cartesian product of measurespaces, which is in spirit a tensor product, carriesthe joint states of two (or more) separate probabilis-tic will define a categoryQuantfor quan -tum probability which is analogous to the cate-goryProb. The ultimate generalization, discussedin Section , is a categoryvNthat containsbothQuantandProb. Its objects are von Neu-mann algebras, which are sometimes called non-commutative measure spaces.

7 The objects ofQuantare, famously, Hilbert spaces. Until Sec-tion , we will consider only finite-dimensional vec-tor spaces. These are enough to learn from, just asthe finite case is enough to learn most of the empir-ical interpretation of classical Vector states and unitary mapsAlthough it lacks some crucial empirical structure,most of quantum mechanics and much of quantumcomputation relies only on a simpler category (thanQuant) which we will callU. The objects ofUare complex Hilbert spaces and the morphisms areunitary maps. We also add subunitary maps toUto make a moderately larger categoryU . We willalso mostly restrict our attention to the subcategoryU< of finite-dimensional Hilbert that aHilbert spaceis a complex vectorspaceHwith a positive-definiteHermitian innerproducth | i. This means thath | iis a function fromH HtoCthat satisfies these axioms:h 1+ 2| 3i=h 1| 3i+h 2| 3ih 1| 2i=h 2| 1ih 1| 2i= h 1| 2ifor Ch | i>0 for 6= 0.

8 (In the infinite case,Hmust also be complete rela-tive to the norm|| ||=ph | i.)In quantum theory, the traditional notation is| i(a ket ) for andh |(a bra ) for the dual vectorh |= =h | an operator onH, thenh 1|X| 2iis an expression for the inner product of 1withX( 2) . IfX=| 1i h 2|has rank 1, then we can omit the and just writeX=| 1ih 2|.This notation is due to Dirac [10] and is called bra-ket notation. A linear mapU:H1 H2isunitaryif it preserves the inner producth | i; it issubunitaryif it preserves or decreases the attendant norm|| ||.Recall also that a linear map from a Hilbert spaceto itself is called standard finite example of a Hilbert space isthe standard complex vector spaceCnwith the innerproducth~x|~yi=x1y1+x2y2+ + can generalize this to say that for any finite setA, the vector spaceCAis a Hilbert space with stan-dard orthonormal basisA. Every finite-dimensionalHilbert space is isomorphic toCnfor somen, andthereforeCAfor anyAwith|A|= finite quantum mechanics , as in classical prob-ability, we can define a physical object by specifyinga finite setAof independent configurations.

9 In in-formation theory (both quantum and classical), theobject is often called Alice . In the classical case,the set of all normalized states of Alice is the sim-plex Aspanned byAin the vector spaceRA(seeSection ). , a general state has the form =Xa Apa[a]for probabilitiespa 0 that sum to 1. (For unnor-malized states, the sum need not be 1.) The numberpais interpreted as the probability that Alice is instatea. Quantumly, Alice s set ofvector statesis thevector spaceCA. In formulas, a state of this type isa vector| i=Xa A a| state| iisnormalizedifh | i=Xa A| a|2= 1andsubnormalizedif the left side is at most 1. Thecoefficient ais called theamplitudeof the quantumstate|aiand the square norm| a|2is interpreted asthe probability that Alice is in state|ai. The phaseof a( , its argument or angle as a complex num-ber) has no direct probabilistic interpretation, but itwill be immediately relevant when we consider op-erations on| i.

10 More precisely, the relative phaseof two coordinates aand a is indirectly measur-able. It will turn out that the global phase of| iis not empirical; Section discusses a change informalism that eliminates state| iis also called aquantum superpo-sition, anamplitude function, or awave last name, perhaps the most common term inphysics, is motivated by the fact that| itypicallysatisfies a wave equation in infinite quantum me-chanics (Example and Section ). It also pre-dates the Copenhagen interpretation and arguablydistracts from the configuration sets of two quan -tum systems ( Alice and Bob ), then, as we said,an empirical transition from Alice s state to Bob sstate is a unitary (or subunitary) mapU:CA requirement thatUbe linear is thequantumsuperposition principle. It contradicts the similar-looking classical superposition principle: if ampli-tudes add, then probabilities usually do not.