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 bottom 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. state a. Quantumly, Alice’s set of vector states is the ∈ vector space CA. In formulas, a state of this type is a vector i/2 i/2 ψ = α a . | i a| i i/2 a A X∈ i/2 The state ψ is normalized if i/2 | i −i/2 ψ ψ = α 2 =1 h | i | a| a A X∈ Figure 1: An idealized two-slit experiment. and subnormalized if the left side is at most 1. The coefficient αa is called the amplitude of the quantum It is traditional to illustrate the quantum super- 2 state a and the square norm αa is interpreted as position principle in an idealized setting called the the probability| i that Alice is in| state| a . The phase “two-slit experiment” (or a more general diffraction | i of αa (i.e., its argument or angle as a complex num- experiment). Figure 1 shows the basic idea: A laser ber) has no direct probabilistic interpretation, but it emits photons that can travel through either of two will be immediately relevant when we consider op- slits in a grating and then may (or may not) reach erations on ψ . More precisely, the relative phase a detector. The source has a single state (the state | i of two coordinates αa and αa′ is indirectly measur- set A has one element), while the grating has two able. It will turn out that the global phase of ψ states and there are two detectors (B and C each is not empirical; Section 1.4 discusses a change| ini have two elements). The transitions for each pho- formalism that eliminates it. ton, as it passes from A to B to C, are described by The state ψ is also called a quantum superpo- two subunitary matrices sition, an amplitude| i function, or a wave function. U : CA CB V : CB CC . This last name, perhaps the most common term in → → physics, is motivated by the fact that ψ typically The matrices are satisfies a wave equation in infinite quantum| i mechanics (Example 1.7.1 and Section 2.1). It also pre- i i i 2 2 2 dates the Copenhagen interpretation and arguably U = i V = i i , distracts from it. 2 2 − 2 If A and B are the configuration sets of two quan- and tum systems (“Alice” and “Bob”), then, as we said, 1 an empirical transition from Alice’s state to Bob’s VU = 2 . −0 state is a unitary (or subunitary) map U : CA CB. The total amplitude of the photon reaching the top → 1 1 detector is 2 and the probability is 4 ; this case The requirement that U be linear is the quantum is called constructive− interference. The total ampli- superposition principle. It contradicts the similar- tude reaching the bottom detector is 0, so the photon looking classical superposition principle: if ampli- never reaches it; this case is called destructive in- tudes add, then probabilities usually do not. (They terference. On the other hand, if one of the slits of 4

blocked, then we can discard one of the states in B , quantum superpositions are: with the result that each detector is reached with| | 1 0 + 1 0 1 probability 16 . The classical superposition principle + = | i | i = | i − | i 1 | i √2 |−i √2 would dictate a probability of 8 for each detector with both slits open; thus it is violated. 1 Both of these states have probability 2 of being in either configuration 0 or 1 , but they are different states. This is demonstrated| i | i by the effect of a i/2 i/2 unitary operator H called the Hadamard gate:

1 1 H = . 1 1 i/2 ±i/2 − It exchanges 0 with + and 1 with . | i | i 1 | i |−i The spin state of a spin- 2 particle is a two-state system which is important in physics. (Electrons, Figure 2: An angle-dependent detector in the two-slit 1 experiment. protons, and neutrons are all spin- 2 particles.) The conventional orthonormal basis is (“spin up”) and (“spin down”). The names of|↑i the states refer to A natural reaction to the violation of classical su- the|↓i property of the electron spinning (according to perposition is to try to determine which slit the pho- the right-hand rule) about a vertical axis in these ton went through. One way to do so is to use a de- two states. Even though a rotated electron is still tector which is sensitive to the angle that the photon an electron, this conﬁguration set for it does not ro- comes in, as in Figure 2. But then this detector rep- tate to itself; neither does any other. The resolution resents two distinct states rather than one. Thus the of this paradox is that rotated states appear as su- ﬁnal state vector is perpositions. For example, the states “spin left” and 1 “spin right” are analogous to + and : 4 | i |−i ψ = − 1 | i 4 + ± = |↑i |↓i = |↑i − |↓i. and its total probability is |→i √2 |←i √2

1 ψ ψ = ψ 2 = , Exercises h | i || || 8 regardless of the phases of path segments to and 1. Suppose that the lengths of the entries of a from the slits. The broader lesson is that amplitudes complex matrix U are all fixed, but the phases of different trajectories of an object only add when are all chosen uniformly randomly. (If you like, there is no evidence of which trajectory it took; oth- you can also suppose that for any choice of the erwise the probabilities add. If we want to see quan- amplitudes, U is subunitary.) Show that on tum superposition, it is not enough to wittingly or average, each entry of U ψ satisfies the clas- unwittingly ignores such evidence. Rather, if the two sical superposition principle.| i trajectories induce different states of the universe, so that some observer could in principle distinguish 2. If U is a matrix, then the matrix them, then they obey classical superposition. More- M = U 2 over, the effect is not the result of interaction be- ab | ab| tween photons; photons do not interact with each can be called dephasing of U. A dephasing of 1 other . Indeed, the laser could be tuned to shoot a unitary matrix is always doubly stochastic, only one photon at a time. Of course, our two-slit meaning that the entries are non-negative and “experiment” is only an idealization of a real exper- the rows and columns add to 1. Find a 3 iment; but see Sections 1.3 and 1.6. 3 doubly stochastic matrix which is not the× dephasing of any unitary matrix. Examples 1.1.1. A qubit is a two-state quantum object with configuration set 0, 1 . Two of their 3. Show that every n k subunitary matrix U { } can be extended to an× (n+k) (n+k) unitary matrix V : ×

1 U More precisely, detecting photon-photon interactions re- V = ∗ . quires enormous particle accelerators. ∗ ∗ 5

Show that V cannot usually have order less where Pλ is the orthogonal projection onto the than n + k. eigenspace of λ. (Note that this probability does not depend on the global phase of ψ .) Moreover, if the 4. If U1,U2,...,Un are unitary operators, then value λ is measured, the conditional| i state afterward each entry of their product is

U = Un ...U2U1 Pλ ψ ψ′ = | i . | i ψ P ψ can be expressed as a sum of products of en- h | λ| i tries of the factors: Conditioning on a measurementp is also called “state collapse” or “wave function collapse”. a U ...U U a h n| n 2 1| 0i This abstract definition of a measurement, and the = an Un an 1 . . . a2 U1 a1 a1 U1 a0 . references to abstract Hilbert spaces, can be moti- h | | − i h | | ih | | i a0,a1,...,an vated by the more concrete discussion in Section 1.1, X and they lead to a better presentation of unitary Such an expansion is interpreted as path sum- quantum probability. In Section 1.1, we tacitly ac- mation; it is the same idea as a sum over his- cepted that if = CA is Alice’s state space, then tories in classical probability. one kind of a validH measurement is whether Alice is For example, let n = 4 and let each in the configuration a A, and we said that its probability of this is the square∈ amplitude α 2. More | a| 1 1 1 generally, if D : A S is some function, then Uk = . → √2 1 1! 2 − P [D = s]= αa ; | | Find the amplitudes of the 16 paths and group DX(a)=s them according to how they sum. this was implied by the discussion about distinct and R 5. In general for a spin- 1 particle, the state identical states. But, taking S = , the function 2 D uniquely extends to a Hermitian operator on ~v = α + β which is diagonal in the basis A. At the same time,H | i |↑i |↓i we posited that unitary operators represent the em- spins in the direction pirical operations on Alice. Since every Hermitian ~v = (Re αβ, Im αβ, α 2 β 2). operator X is diagonalized by a unitary operator, | | − | | 1 Check that this is a unit vector when ~v is X = U − DU, |Ri3 normalized, and that every unit vector in is we can think of a general measurement X as a mea- achieved. This formula is therefore a surjective surement of Alice’s configuration a A after Alice 3 C2 function from the unit 3-sphere S to the is prepared by the transition map U∈. 2-sphere S2 R3. What is its usual⊂ name in ⊂ 1 mathematics? Example 1.2.1. Consider a spin- 2 particle and let

0 1 1 0 Jx = Jz = 1.2. Measurements and basis independence 1 0 0 1 ! − ! Suppose that (or = CA) is the Hilbert space be two Hermitian operators, given as matrices in the H H standard basis , . These operators measure the of a quantum object, and that the object is in the {↑ ↓} state ψ . A measurement or real-valued quan- particle’s spin in horizontal and vertical directions. tum random| i ∈ H variable is a Hermitian operator X on If Jz is definite, then the spin state is either or . Both of these states are superpositions of|↑i . The eigenvalues of X are interpreted as its range |↓i |←i asH a random variable. (Since we are assuming that and , so if Jz is definite, Jx is not; rather, it has a| →i1 chance of being either 1 or 1. If J is is finite-dimensional, X admits a complete set of 2 − x orthogonalH eigenvectors. For the infinite case see measured, then the particle’s state becomes one of the two conditional states or , after which Section 1.7.) The assertion that X = λ as a random |←i |→i variable is interpreted as the condition that ψ is an Jz is no longer definite; its old value is forgotten. eigenvector of X with eigenvalue λ. More generally,| i This example illustrates that every state of a for any ψ , the probability that X = λ is given by quantum system is a source of randomness; every the formula| i state is indefinite. The popular paraphrase of Ein- stein, “God does not play dice with the universe,” P [X = λ]= ψ P ψ , refers to this principle. h | λ| i 6

By the same token, if is the Hilbert space of a The outcome X = s corresponds to the orthogonal H quantum object, we can think of any orthonormal projection Ps onto the summand s. Its probability basis A of as its conﬁguration set. Two com- of occurrence in the state ψ is H pletely diﬀerentH orthonormal bases can be equally | i empirical; a very important part of empirical think- ψ Ps ψ , ing in quantum theory is to be able to change from h | | i one orthonormal basis to another. In physics such a which is also the squared length of the projected change of description is often called a “duality”. For vector Ps ψ . The corresponding conditional state | i example, one form of particle-wave duality (namely, is second quantization of bosons) is very similar to an Ps ψ orthonormal change of basis (Section 2.6). ψs = | i . | i ψ Ps ψ Example 1.2.2. We can now have a second un- h | | i derstanding of a qubit as a quantum object with a One common case is that of several random vari- two-dimensional Hilbert space . We can label any ables X ,...,X . If they commute, then they have H 1 n orthonormal basis 0 and 1 , or we can choose not a common diagonalization, and they induce an or- | i | i to distinguish any particular basis. For example, thogonal decomposition of with S = Rn. If two one person’s 0 and 1 may be another person’s measurement operators X andH X do not commute, | i | i 1 2 + and . One important quantum algorithm, then the set of states for which they are both def- | i |−i the Grover search algorithm (Section ??) alternates inite does not span . As in Example 1.2.1, there H between (dilated) classical computations in the two is often no state for which X1 and X2 are both def- bases. inite; they do not share an eigenvector. In words, 1 A spin- 2 particle illustrates the same point more two such variables are mutually uncertain; they are geometrically. As it happens, every orthonormal ba- not simultaneously measurable. sis of its spin state space consists of the positive and negative spin states in some direction. But the 1 model of a qubit as a spin- 2 particle is ultimately Exercises misleading. Particle spin has been successfully em- ployed as a qubit, but some other qubit devices have 1. Verify that if X and Y are commuting Her- much more complicated states. Figure 3 shows one mitian operators, then X + Y and XY corre- example. spond, as measurements, to adding and mul- tiplying the outcomes of the measurements X and Y .

2. Let = CZ/n be a state space whose basis is the cyclicH group Z/n. Deﬁne operators X and Z by

X k = k +1 Z k = e2πi/n k | i | i | i | i Conﬁrm that X has the same eigenvalues as Z. Find the eigenvalues of X + Z.

3. Show that if X is an anti-Hermitian operator, Figure 3: A Josephson junction qubit: superconducting it represents an imaginary random variable; aluminum on a silicon chip [17]. that if X is unitary, it represents a random variable with values in the unit circle S1 C; and that if X commutes with its adjoint,⊂ it A Boolean measurement or quantum random vari- represents a complex random variable. In the able can be represented as a Hermitian operator P last case, X is called a normal operator. whose eigenvalues are 0 (for “no”) and 1 (for “yes”). I.e., P is an orthogonal projection on . More gen- H 4. Suppose that ψ and φ are two states in erally, a random variable X that takes (discrete) val- the same Hilbert| i space | i, and suppose that ues in a set S can be represented by an orthogonal a physical object is in stateH ψ . Show that decomposition the probability that it is in state| i φ is | i = . H Hs φ ψ 2. s S M∈ |h | i| 7

5. Following Exercise 1.1.5, show that every or- If A and B are finite configuration sets for two 1 thonormal basis of the spin- 2 Hilbert space classical systems, then the configuration set for the consists of two spin states that point in op- joint system is the Cartesian product A B. Equiv- posite directions. alently, the state space of the joint system× is a tensor product: 6. Show that a state ψ which is simultaneously | i RA RB R definite for two Hermitian operators X and Y ∼= [A B]. lies in the kernel of the commutator ⊗ × This definition extends to the quantum case: If two [X, Y ]= XY Y X. quantum systems have state spaces A and B, − H H then the joint system has state space A B. In Show that these states span ker[X, Y ] when X particular if A and B are orthogonalH bases⊗ H of HA and Y commute with [X, Y ], but not in gen- and B (i.e., configuration sets for Alice and Bob), eral. thenHA B is a joint basis, just as in the classical case. (But× see Section 2.4.) 7. Show that if a measurement X is performed on If a quantum object were somehow a cloud of am- a state ψ, its expectation (or average value) is plitudes or probabilities, we would expect Alice and given by: Bob to have independent states ψA and ψB , at least if they were physically separated.| i When| i this E[X]= ψ X ψ . h | | i happens, their joint state is ψA ψB ; this is also called a product state. But most| i states ⊗ | arei not prod- 8. Suppose that X and Y are Hermitian opera- uct states; these states are called entangled. Entan- tors on a Hilbert space with a state ρ. Recall H gled quantum states are evidently similar to corre- that if X is a classical random variable, lated classical states. V [X]= E[X2] E[X]2 Examples 1.3.1. Since a qubit has the configura- − tion set 0 and 1 , a system of n qubits has config- denotes the variance of X. Prove the general- uration set| i 0, 1| ni. Thus the general state for this ized Heisenberg uncertainty relation: system has{ 2n amplitudes;} for example the general three-qubit state is E[i[X, Y ]]2 V [X]V [Y ] . ≥ 4 ψ = a 000 + a 001 + a 010 + a 011 | i 000| i 001| i 010| i 011| i (Hint: After subtracting the means from X + a100 100 + a101 101 + a110 110 + a111 111 . | i | i | i | i and Y , show that the 2 2 matrix × It may look as if an n-qubit state carries an exponential amount of information, namely its 2n ampli- E[X2] E[XY ] 2 tudes, but this is only true in a weak sense. With E[YX] E[Y ] ! respect to a reasonable definition of information (see Exercise 1.4.5 and Section 1.8), a quantum superpo- is positive semi-definite. The expectation for- sition is not a record of its list of amplitudes, just as mula in Exercise 1.2.7 is reasonable for arbi- 52 a hand of poker is not a record its 5 probabilities. trary operators, not just normal ones.) One important product state on n qubits is the constant state: 1.3. Joint states n/2 ψ = ++ . . . + =2− s . | i | i | i s 0,1 n ∈{X} Up until this point, a skeptic could still view quantum “probability” as kind of a cloud model and not One important entangled state is the cat state (as really a modification of probability theory itself. If in “Schrödinger’s cat”): a configuration set A of a particle is a set of po- 00 . . . 0 + 11 . . . 1 sitions, then perhaps the particle is merely diffuse, ψ = | i | i. like a cloud. Quantum superposition, measurement, | i √2 and equivalence between different orthonormal bases As another example, an EPR pair (see Sec- are all surprising, but they are not quite show stop- tion 1.6.2) is a pair of electrons or other elementary pers. The topic of this section, namely the correct particles in the entangled spin model of joint quantum states, radically contradicts the cloud interpretation. (Section 1.6 has a more ψ = |↑↓i − |↓↑i. conclusive result in this direction.) | i √2 8

It is similar to the cat state with n = 2. In general 5. Even though measurements are a source of any state one two qubits of the form randomness, the category Ucannot express classical randomness. For example, if the spin a,b + c, d ψ = | i | i state of an electron is prepared by randomly | i √2 choosing between and , what is its state? The model of probability|↑i |↓i distributions on the with manifold of vector states of an object is sus- pect, and in the end, redundant. a c = b d =0 h | i h | i Sections 1.4 and 1.5 describe another model of is called a Bell state or a Bell pair. quantum probability, the category Quant, that ad- Unitary transitions and Hermitian measurements dresses most of these problems. Section 1.8 describes a final model, the category vN, that settles them on a joint system ψA ψB which affect only Alice (respectively Bob)| takei⊗| the formi X I (respectively more completely. I X), where X is unitary or Hermitian.⊗ This is exactly⊗ analogous to the classical case. Such opera- Exercises tions are also called local to Alice or Bob. The combined model of unitary transitions, Her- mitian measurements, and tensor products for joint 1. Another description of the EPR state in Ex- states describes an isolated quantum object whose amples 1.3.1 is via measurement. Let have the spin basis and and define theH oper- state is measured after a period of evolution. It is |↑i |↓i the standard description of quantum mechanics in ators many physics courses. It also describes a unitary J tot = J I + I J J tot = J I + I J quantum computer that is alternately manipulated x x ⊗ ⊗ x z z ⊗ ⊗ z and interrogated by a classical controller. But it on , where Jx and Jz are defined as H ⊗ H tot tot also has shortcomings and omissions which confuse in Example 1.2.1. Show that Jx and Jz its interpretation, namely: have one common eigenstate, for which both eigenvalues vanish. 1. Hermitian measurements are missing from the unitary category U. In physical terms, the 2. Show that if is a Hilbert space of dimension H model does not include observers, even though at least 2, there does not exist a linear map observers can also be observed. (But note U : that a Boolean measurement P is subunitary; H→H⊗H conditioning without normalization does lie in that takes every state ψ to a state equivalent U’.) to ψ ψ . (Recall that| i two states are equivalent| i if ⊗ they | i differ by a global phase.) This is 2. Many physical objects, including typical ob- the simplest of a series of no cloning theorems servers, are effectively classical, even if they are for quantum states. A harder version: Show prima facie quantum. These are also missing that such a map U is not even approximately from the category U. linear. 3. The category U is only weakly connected: 3. Show if ψ and φ are two Bell states shared there is no strictly unitary map from to A by Alice| andi Bob,| i then there is a unitary op- when H HB erator local to Alice (i.e., of the form U I) ⊗ dim > dim . which takes ψ to φ : HA HB | i | i (U I) ψ = φ . The subunitary category U’ is strongly con- ⊗ | i | i nected, but a subunitary map from A to B Thus all Bell states are equivalent. then includes extinction. In other words,H inH the 4. Show that if ψ is a vector state category U’, if Alice has more states than Bob, | i ∈ HA ⊗ HB she cannot transfer her state to Bob without and the possibility that the world ends. dim dim , HA ≤ HB 4. There is no notion of marginals: If Alice and then it has the form Bob are in an entangled state, there is no vector state for Alice alone. In particular, Alice ψ = α a f(a) | i a| i ⊗ | i can entangle with the environment (“Eve”). a A X∈ 9

for some orthonormal bases A and B and some then an operator state ρ becomes a matrix; it can function f : A B. This presentation is be written called a Schmidt→ decomposition of ψ . Show | i 2 that the unordered set of numbers αa is ρ = p ′ a a′ . | | a,a | ih | uniquely determined by ψ . a,a′ A | i X∈

5. A common error in quantum probability is to The diagonal entry pa,a is the probability of the con- mistake the direct sum A B for the joint ﬁguration a . Thus the positivity condition ρ 0 H ⊕ H state space of Alice and Bob. Provide an em- asserts that| thei probability of any conﬁguration≥ (in pirical interpretation for direct sums which, any basis) is non-negative. The normalization con- among other properties, would also work in dition asserts that the total probability in any basis classical probability. is 1:

6. Show that the cat states from Examples 1.3.1 Tr(ρ)= pa,a = 1; are entangled. a A X∈ evidently this condition is basis-independent. Be- 1.4. Operator states cause the diagonal entries of ρ are probabilities, it is often called a density matrix or a density operator Let be the (finite-dimensional) Hilbert space in physics. H The geometric features of +,1( ) and ∆ are of a quantum object, Alice. We define an opera- M H S tor state of Alice (or more simply a state) to be a also similar, albeit with some important differences positive semi-definite Hermitian operator ρ on . as well. Both regions are convex in order to allow (Positive semi-definiteness is denoted ρ 0.) TheH classical superpositions. More precisely, if ρ1 and ρ2 state ρ is normalized if Tr(ρ) = 1 and subnormalized≥ are two states and 0

1 [1] |−i |+i

0 [0] |1i (a)Deterministic bit (b)Randomized bit (c)Quantum bit

Figure 4: The space of states for three diﬀerent types of bits.

Example 1.4.1 hints at a more general fact: Ev- These rules for probabilities and conditional states ery mixed state is a mixture of pure states in many also apply to set-valued measurements, using the different ways. A mixed state encodes all of the sta- projection Ps corresponding to an outcome s S. tistical information that can be extracted by mea- ∈ Recall that if A and B are the Hilbert spaces surements and other operations. Thus a probability of Alice and Bob,H then theirH joint Hilbert space is distribution on pure states is a highly redundant de- A B . If Alice and Bob have independent states scription of a mixed state. H ⊗ H ρA and ρB, then their joint state is ρA ρB, a prod- Also following Example 1.4.1, if has d states, ⊗ H uct state. General non-product joint states are a then the state ρ = I/d, where I is the identity op- non-trivial mutual generalization of classical corre- erator, is called the uniform state. It is the uniform lation and quantum entanglement, and their nomen- mixture of all configurations in any orthonormal ba- clature reflects some of their surprising properties. sis, hence a strong analogue of the uniform distribu- A joint state ρ is called separable if it is a mixture tion on a finite set in classical probability. of independent states. Non-independent separable All of the operations defined for vector states read- states are roughly analogous to classical correlated ily extend to operator states. If states, but even these have some interesting quantum properties [5]. If ρ is not separable, then it is U : A B H → H entangled. In some crucial respects, entanglement is a unitary or subunitary operator, its induced ac- of mixed states is a weaker condition than entangle- tion on operators is given by ment of pure states. Current research is devoted to relating these two forms of entanglement. (ρ)= UρU ∗. U As promised, there is a way to express marginals If U is unitary, then of joint states using operator states. If ρ is a joint state on A B, its marginal states on A and +,1 +,1 H ⊗ H H ( ( 1)) ( 2), are defined as partial traces: U M H ⊂M H HB while if U is subunitary, then ρA = TrB(ρ) ρB = TrA(ρ). ( +,1( )) +,1( ). U M H1 ⊂M H2 If More explicitly, suppose that A and B are orthonormal bases of and . Then X : HA HB H → H is a Hermitian measurement, then its expectation ρ = Tr (ρ)= a a,b ρ a′,b a′ value is defined as A B | ih | | ih | a,a′ A;b B ∈X ∈ Eρ[X] = Tr(ρX). ρ = Tr (ρ)= b a,b ρ a,b′ b′ . B A | ih | | ih | a A;b,b′ B, If H = P is a Hermitian projection, i.e.a Boolean ∈ X∈ measurement, then the probability of P is defined as Example 1.4.2. Suppose that Alice and Bob are Pρ[P ] = Tr(ρP ) qubits in the entangled vector state and the conditional state is P ρP 00 + 11 ρ P = . ψ = | i | i. | Tr(ρP ) | i √2 d 11

The operator form of this state is then 6. Verify that a pure state conditioned on a measurement is still pure. More generally, show 1 0 0 1 that measurement does not increase entropy: 1 0 0 0 0 For any projection P and any state ρ, ρ = ψ ψ =   . | ih | 2 0 0 0 0   S(P ρP ) S(ρ).  1 0 0 1  ≤     7. Show that the each marginal of a pure joint Both of its marginals are the uniform state: state ψ A B is pure if and only if ψ is unentangled.| i ∈ H ⊗ H | i 1 1 0 ρA = ρB = . 8. A purification of a state ρ on is a pure state 2 0 1! H on a joint system ′ whose left marginal H ⊗ H Whereas in classical probability, a marginal of a def- is ρ. Show that every state on has a purifi- H inite state is definite, in quantum probability the cation in , and that it is unique up to a H ⊗ H marginal of a pure state need not be pure. unitary operator local to the second factor. In general a linear map 9. The support of a state ρ on is its image in as a linear operator. ShowH that if ρ has full H : ( A) ( B) support, then every outcome of a projective E M H →M H measurement has non-zero probability. is called a superoperator. If we interpret a unitary map (including both dimension-preserving operators 10. Show that the uniform state on is the only and dimension-increasing embeddings) as a super- one which is invariant under all unitaryH opera- operator, and we have described a partial trace as tors on . Show, following Exercise 1.1.5, that H 1 another kind of superoperator. Both of these op- the uniform spin- 2 state is the only state that erations are empirical, and we can naively consider is not direction-dependent. the category that they generate inside the category of all superoperators. For the moment we will call 11. Show that every state in an open neighborhood of the uniform state on is separable. it Quant; in the next section we will show that it HA ⊗ HB includes all maps of states that could reasonably be 12. Given a joint Hilbert space A B, compute empirical. the dimension (in terms ofH the⊗H dimensions of and ) of: HA HB Exercises a) the space of all joint states on . HA ⊗ HB b) the space of all joint pure states. 1. Verify that a local measurement X I applied c) the space of all product states. ⊗ to a state ρ on a joint system A B has d) the space of all pure product states. the same probabilities as the measurementH ⊗ H X applied to the marginal state TrB(ρ), and that 13. This exercise requires knowledge of some the conditioned states are also consistent. topology and differential geometry. Show that the space of pure states of an n-dimensional 2. Prove Proposition 1.4.1. Hilbert space is a 2n 2-dimensional man- H − n 1 +,1 ifold, explicitly the manifold CP − . Show 3. Show, as Example 1.4.1 claims, that 2 is a round 3-dimensional ball and that pureM states that the Riemannian metric that it inherits from its embedding in ( ) is two-point ho- are orthonormal if and only if they are antipo- M H dal. Show that the probability of any Boolean mogeneous, meaning that isometries act tra- measurement on a state ρ is proportional to sitively on unit tangent vectors. Show that the the displacement of ρ from some hyperplane Riemannian metric (which is called the Fubini- passing through the center of +,1. Study metric) is positively curved. M2 +,1 4. Show that every state ρ n is a convex combination of at most n pure∈ M states that have 1.5. Quantum operations the same diagonal entries as ρ. This section is mathematically more challenging 5. The entropy S(ρ) of a state ρ is defined as than previous sections in Chapter 1. Our goal is to Tr(ρ(log ρ)). Show that the uniform state I/d +,1 characterize all maps maximizes the entropy S on n . Show that a state is pure if and only ifM it has no entropy. : ( ) ( ) E M HA →M HB 12

that satisfy relatively weak conditions that we might Theorem 1.5.1 (Stinespring,Kraus). Let want from empirical operations. A map that satis- ﬁes them will be called a “quantum operation”. We : A B E M →M will abbreviate ( A) as A for Alice’s operators, M H M be a superoperator. Then is completely positive if B for Bob’s, etc. E MBy the classical superposition principle, an empir- and only if there exist operators ical map E ,...,E : 1 N HA → HB : E MA →MB such that

should ﬁrst be a linear map, i.e., a superoperator. If N is linear, it is called positive if (ρ)= E ρE∗. (2) E E k k k=1 ρ 0 = (ρ) 0 X ≥ ⇒ E ≥ Equivalently factors as and trace-preserving if E TrC U , Tr(ρ)=1 = Tr( (ρ))=1. MA →MB ⊗MC → MB ⇒ E where Thus the condition that

(ρ)= DρD∗ ( +,1( )) ( +,1( )) D E M HA ⊂E M HB and TrC is a partial trace. says that is positive and trace-preserving or TPP. The map is trace-preserving if and only if (Likewise E is positive if it preserves all states and E E positive, sub-trace-preserving or PSTP if it preserves N subnormalized states.) By analogy with stochastic E∗E = I , (3) k k ∈MA matrices, it is tempting to propose TPP maps as k=1 quantum operations. However, the tensor product of X two TPP maps need not be positive, so the category in which case D is unitary. of TPP maps is not compatible with joint states as Often Theorem 1.5.1 is called Stinespring’s the- they are defined in Section 1.3. orem [23]. Equation (2) is called the operator-sum A map : is called completely posi- A B representation or the Kraus decomposition [16]. The tive (CP)E if forM every→M quantum system C, the map operation , or the corresponding operator D, is a D : dilation of the CP map . E ⊗ I MA ⊗MC →MB ⊗MC Theorem 1.5.1 justifiesE the quantum superposition principle as a consequence of the classical superpo- is positive, where is the identity on C . I M sition principle and complete positivity. These two Example 1.5.1. The transpose map : ρ ρT on assumptions alone imply that every quantum oper- T 7→ n for n 2 is positive but not completely positive. ation is a sum (or classical superposition) of subuni- M ≥ taries (which are quantum superpositions.) In this Completely positive, trace-preserving (TPCP) sense, the radical element of quantum probability is maps do form a tensor category which for the mo- not quantum superposition itself, but rather replac- ment we will call Quant. Every quantum operation ing the simplex of states ∆A with the Bloch region should be TPCP; the category Quant should con- +,1( ). tain the empirical class Quant of quantum opera- MThisH point of view is further supported by the fol- tions. (If extinction is allowed, then every quantum lowing corollary. Say that a CP map is coherent if it operation should be STPCP.) In Section 1.4, we de- is a single Kraus term. In particular, a unitary map fined a category Quant generated by unitary maps is coherent. and partial traces; it should be contained in the empirical class Quant. The important result is that Corollary 1.5.2. If a CP map : A B takes pure states to pure states, thenE eitherM → it is M ei- Quant = Quant, ther coherent, or all states in its image are proportional. If is invertible in the category Quant, then which justifies either one as a definition of Quant. it is unitary.E We can likewise define Quant′ as the category of STPCP maps and Quant+ as the category of CP Note that in physics, an invertible process is usu- maps. ally called reversible. 13

We will prove these results at the end of this or other particle can measure another one; a quan- section; we ﬁrst consider some particular classes of tum computer can measure some of its qubits and quantum operations. place the outcome in other qubits; etc. Whenever A state ρ on a Hilbert space can be interpreted two objects become entangled, we can say that each as a quantum operation fromH the 1-state Hilbert one is measuring the other. We can also say that space C to . In the other direction, the trace map decoherence is generally equivalent (by dilation) to H entanglement with the environemtn. In the limit, Tr : ( ) C M H → one description of a non-quantum physical object is that it is a quantum object which is constantly being is also a quantum operation. These two operations measured, or becoming entangled with, its environ- can be thought of as creation and destruction of ment. states. The composition ρ Tr can be thought of ◦ as initializing an object in the state ρ. A partial Proof of Theorem 1.5.1. The proof here is based on trace a characterization of CP maps due to Jamio lkowski Tr : ( ) ( ) ( ) and Choi [7, 15, 22]. First, any superoperator A M HA ⊗M HB →M HB : is also completely positive. E MA →MB Suppose that an orthogonal decomposition can be interpreted as an element = H Hs s S X A B = ( A B). M∈ E ∈M ⊗M M H ⊗ H represents a set-valued measurement. We noted in The point is that is a tensor with four indices (Sec- E the previous section that the probability of the out- tion 1.11), two for A and two for B. In indices, come s is given by its usual interpretationH as a map is givenH by the expression: Tr(ρPs) ′ b ba a (ρ) ′ = ′ ρ ′ . and that the conditional state is E b Eb a a

PsρPs But we can also pair the indices differently as follows: ρ Ps = . | Tr(ρP ) ′ ′ s a ba a X (χ)b′ = b′a χb . If we imagine a hiddend observer, Eve, performing E E this measurement, she will effect the operation Here χ A B. In reference to the alternate pairing| ofi ∈ indices, H ⊗ Hwe will call X the sideways action E (ρ)= PsρPs (4) of . P E s S We claim that is completely positive as a super- X∈ operator if and onlyE if X 0 as a Hermitian opera- on the state ρ. This is evidently a quantum oper- tor. This identification isE known≥ as the Jamio lkowski ation, one that expresses blind or hidden measure- criterion or (in greater generality) the Choi isomor- ment. It has an explicit dilation phism. We will rephrase the completely positivity D : CS condition to establish the logical equivalence. The H→H⊗ map is completely positive if and only if for any , E given by the formula MC

Dψ = s SPsψ s . ( )(ρ) 0 ⊕ ∈ ⊗ | i E ⊗ I ≥

We can interpret this dilation as a visible measure- for all states ρ A C. The lemma that ment, because the factor CS could belong to Eve and (and therefore ∈ M) preserves⊗M the Hermitian prop-E does record the measurement outcome. erty of ρ if andE only ⊗ I if X is Hermitian is left to Ex- The main shortcoming of the dilation D as a ercise 1.5.2. The more interestingE positive semideﬁ- model of of measurement is that Eve must possess niteness condition says that quantum memory — she cannot be a classical computer or a human being. Section 1.8 discusses a bet- ψ ( )(ρ) ψ 0 ter model with both quantum and classical objects. h | E ⊗ I | i≥ Nonetheless the model is very useful. An object for all vectors ψ B C . This numerical in- can be measured by its environment; one electron equality is linear| i in ∈ρ H, so⊗ we H may assume that ρ is 14

extremal, i.e., pure. Thus by Proposition 1.4.1, com- By hypothesis, every vector in the image of D must plete positivity may be written more symmetrically have this form. Now let as ψ = ψB ψC ψ′ = ψB′ ψC′ ψ ( )( φ φ ) ψ 0 | i | i ⊗ | i | i | i ⊗ | i h | E ⊗ I | ih | | i≥ be two inequivalent states (i.e., non-proportional for all vectors) in the image of D. If the sum ψ + ψ′ is also a product state, then either the left| i factors| i ψ B C φ A C . | i ∈ H ⊗ H | i ∈ H ⊗ H ψB and ψB′ or the right factors ψC and ψC′ |arei proportional| i — but not both, because| i then| ψi In indices, | i and ψ′ would be proportional. If this relationship ′ ′ ′ | i bc ab a c holds for every inequivalent pair of states in im D, ψ ψ ′ ′ ′ φ φ 0. b c Ea b ac ≥ then they must all have either the same left factor If or the same right factor. If all vectors in im D have the same left factor, dim min(dim , dim ), respectively the same right factor, then HC ≥ HA HB then D ψ = ψ (E ψ ), | i | B i⊗ | i a bc χb = ψ φac respectively is an arbitrary vector in . With this abbre- A B D ψ = (E ψ ) ψC , viation, complete positivityH ⊗ of H is the condition | i | i ⊗ | i E ′ ′ for some linear map E. In the ﬁrst case, states in the b ab a χ ′ ′ χ 0 a Ea b b ≥ image of = TrC are proportional to ψB ψB . In the secondE case,◦ D | ih | for all χ. This is precisely positivity of X . The operator X is extremal among positiveE op- = ψC ψC , erators if and onlyE if it has rank 1, i.e., E h | iD hence it is coherent. X = E E E | ih | If is invertible, then it must send extremal points E+,1 +,1 for some E . In indices, of A to B . (This is generally true of any in- ∈ HA ⊗ HB vertibleM mapM in the category of linear maps between ′ ′ ab a b convex bodies.) I.e., it must send pure states to pure ′ = E ′ E . Ea b b a states. In this case E is unitary for two independent In operator form, this says that reasons: is invertible, and preserves trace. E E

(ρ)= EρE∗. E Exercises In other words, is a single Kraus term if (and only if) it is extremalE among CP maps. Therefore the 1. Show directly from the deﬁnition of complete general CP map is a sum of such terms. positivity that every state ρ on a Hilbert space The further assertions when is trace-preserving E is ρ(1) for a completely positive map are left to Exercise 1.5.3. H E : C ( ). Proof of Corollary 1.5.2. Let be a dilation of , Eρ →M H and let D E Show that dilation of is equivalent to puriﬁ- E D : cation of ρ. HA → HB ⊗ HC 2. Establish a missing step of Theorem 1.5.1: The be its operator form. If ψ B C is a vector state, then its marginal | i ∈ H ⊗ H map commutes with the Hermitian adjoint operationE if and only if X is Hermitian. E Tr ( ψ ψ ) C | ih | 3. Establish the other missing step of Theo- rem 1.5.1: is TPCP if and only if Equa- is pure if and only if ψ is a product state (Exer- E cise 1.4.7): | i tion (3) holds, if and only if D is unitary. Mod- ify Equation (3) to the case when is STPCP, ψ = ψ ψ . and show that in this case D is subunitary.E | i | Bi ⊗ | C i 15

4. Show that if 3. Evolution: After an observer performs a measurement, the new state of reality is given by : E MA →MB projecting its state. More generally the state is STPCP, then there is an STPCP map of reality evolves by quantum operations.

: A C 4. Statistics: An observer’s experiences are in- F M → terpreted as independently repeatable experi- such that ments. The probability of a measured value is : A B C the fraction of times that it occurs in repeated E⊕F M →M ⊕ trials of the experiment. is TPCP. Compare with Exercises 1.1.3 and ??. This interpretation is exactly parallel to the one for classical probability theory at the end of Sec- 5. Find Kraus elements for a partial trace map tion 1.10. The ﬁrst person to understand it clearly

TrB : A B A. was Max Born in 1926 [6], an insight for which he M ⊗M →M eventually won the Nobel Prize. (Our presenta- 6. Show that every blind measurement quantum tion with mixed states is due to von Neumann and operation (4) can be expressed as a convex Hellwig-Kraus [13, 14, 16, 24].) It is intellectually combination of unitary quantum operations. healthy to have trouble accepting the Copenhagen interpretation. It is not healthy to reject it outright, 7. Show that the uniform state on is sent to it- H even though this fate befell two disappointed parents self by every blind measurement quantum op- of the interpretation, Einstein and Schrödinger. In eration (4), and that it is the only state with this section we will discuss some of the overwhelm- this property. ing physical evidence for this interpretation, and a 8. A quantum operation is doubly stochastic if mathematical result in support of its radical nature. and only if it is both trace-preservingE and pre- First the evidence: serves the uniform state. For example, uni- 1. Quantum probability and quantum mechan- tary quantum operations are doubly stochas- ics were originally developed to understand tic. Doubly stochastic quantum operations for molecular, atomic, and subatomic structure a fixed Hilbert space form a convex region, and processes. It is a vast edifice that makes and unitary quantumH operations are extremal quantum probability truly irrefutable. For ex- points (check). Show that if dim = 2, then ample, the structure of a hydrogen molecule all extremal doubly stochastic quantumH oper- is grossly arbitrary is understood in great de- ations are unitary, but that this is not true tail This is in the same sense that an ex- when dim > 2. Compare with Exercise ??. H pert game of backgammon can be understood in great detail with classical probability, but 1.6. Empiricism seems grossly arbitrary without it. 2. A variety of real experiments and demonstra- 1.6.1. Interpretation and evidence tions match the thought experiments of quantum probability. This includes the examples in Having defined the category Quant of quantum this article. For one, the two-slit “experiment” operations, we can now state its empirical interpre- in Section 1.1 is a qualitatively correct model tation: of laser speckle (scattering interference) and holography (photographic interference). Laser 1. State: Every observer in the universe can speckle is familiar as the twinkle in the dot of a model external reality as a quantum system laser pointer; see Figure 5. At the same time, with a Hilbert space that carries some par- light is composed of discrete, non-interacting ticular state in the BlochH region +,1( ) at M H photons. This is an unavoidable aspect of low- each point in time. intensity X-ray photography, as shown in Fig- 2. Independence: Reality decomposes into ap- ure 6. proximately disjoint subsystems whose joint Photons are not the only particles that ex- Hilbert spaces are tensor products such as hibit quantum superposition; in principle ev- A B. An observer is an approximately ery physical object does. Figure 7 shows is an independentH ⊗ H subsystem whose residual non- image of electrons obeying quantum superpo- independence is described by measurements sition. Recently it has been demonstrated for such as Hermitian operators. C60 carbon molecules (buckyballs) [1]. 16

Figure 5: Laser speckle [25].

Figure 7: Quantum interference of individual electrons [19].

Even if quantum probability is irrefutable, is it necessary? Classical probability theory is (to some extent) unnecessary in the sense that it can be repro- duced by hidden determinism. On the other hand, there is no reasonable reduction from quantum probability to classical probability or hidden determinism; see Section 1.6.2. Some entirely new theory Figure 6: An X-ray image of Venus comprised of discrete could conceivably arise to “explain” quantum prob- photons (from the Chandra telescope) [9]. ability, but there is no reason to expect that a hypo- thetical successor would spare anyone from disbelief. Even irrefutable scientific facts can be refined; but 3. As mentioned in Section 1.3, and as discussed if they are irrefutable, there is no turning back3. As further in Section 1.6.2, the radical aspects of it happens, the other known fundamental laws of quantum probability require entangled joint physics do not modify quantum probability at all: states. Entanglement has also been demon- relativity is geometric, while quantum field theory strated by a variety of experiments; see Exer- postulates specific physical forces. cise 1.6.3. If quantum probability is true and necessary, why is most macroscopic experience (on biological length scales and above) classical? In fact, length does not 4. The known fundamental laws of physics are re- directly determine whether a physical system is de- versible, or in the quantum language, unitary. scribed by classical or quantum probability. Rather, Unitary quantum probability does not encom- the system’s relevant attribute is the number of pass determinism or classical probability as a accessible states. If the system has many states, special case. Thus if the laws of physics are then different evolutionary paths in the sense of Sec- reversible and any physical objects are quan- tion 1.1 are likely to arrive at different final states, tum, then the entire universe must be quan- whence total probability is given by classical rather tum. Among fundamental forces, the only one than quantum path summation. In quantum theory, without a satisfactory quantum model is grav- ity2.

3 For example, if you do not want to believe that the Earth orbits the sun, it does not help to learn that its orbit is an 2 And string theory is a promising attempt at such a model. ellipse rather than a Copernican circle. 17

“microscopic” and “macroscopic” properly refer to possess classically implausible correlations. Their amounts of entropy rather than to distances. argument was sharpened by John Bell [3]. He es- Another way to say it is that macroscopic objects tablished a simple inequality in classical probability, typically evolve by highly decoherent quantum oper- Bell’s theorem, that is violated by quantum mea- ations. They therefore constantly export entangle- surements on the EPR state. (Bell was also unsat- ment to the environment. This is why the mixed- isfied with the Copenhagen interpretation [4].) state model is useful for empirical interpretations. We first give an informal description of Bell’s the- The macroscopic world consists of physical systems orem. Suppose that Alice and Bob are two sus- whose quantum state is strongly coupled to a com- pects in prison together who are taken apart for mon sea of thermal entropy, but which retain ap- separate questioning. In questioning, they are al- proximately independent classical states. lowed to use notes and even electronic organizers, In particular the “paradox” of Schrödinger’s cat, but they are not supposed to communicate by any which Schrödinger offered as a criticism of the means. Each of the suspects is given a sequence of Copenhagen interpretation, is misleading. (But it questions (which may continue for several interroga- is a useful antecedent of the notion of a cat state; tion sessions). There are only three distinct ques- see Examples 1.3.1.) The claim is that if a cat is tions, “X”,“Y ”, and “Z”, and only two answers, at risk of death from a vial of poison that is con- say “yes” and “no”. The suspects are not expected trolled by a radioactive decay, then the cat is in a to give consistent answers, but the authorities still quantum superposition of life and death. But for hope to glean some information from the pattern of thermal reasons, any room-temperature state of a the answers. For simplicity, the questions are ran- cat is massively mixed, and typical superpositions dom and independent. are effectively classical. Such mixed states are also Suppose that the authorities notice that if the nth unaffected by typical blind measurement operations question posed to Alice and Bob is the same, they (Exercise 1.5.6). Only a frozen cat could be pre- always give the same answer; but when the nth question posed is different, they only give the same an- pared in a pure state well enough to demonstrate 1 non-commutativity of measurements. swer 4 of the time. Can they conclude that Alice and Bob are secretly communicating during the in- Finally we caution against over-interpreting quan- terrogations, or that they have advance access to the tum probability. The best reason to believe or inter- question lists, despite efforts to isolate them? If they pret anything in science is to understand it better. are classical entities, then they must be cheating. If The basic statistical interpretation — the Copen- they always give the same answer when asked the hagen interpretation — is very helpful for under- same question in the nth round, then they must have standing quantum mechanics and almost manda- prepared common answers lists to all three questions tory for understanding quantum computation. It in advance. But if the nth question differs, then at is useful in the theory of operator algebras and po- least two of the three prepared answers are equal, if tentially useful in some other areas of mathemat- they are not all equal, so the probability of giving ics. One claimed alternative, the Everett “many the same answer is at least 1 . worlds” interpretation, is narrowly relevant to path 3 But if Alice’s and Bob’s electronic organizers can summation (Exercise 1.1.4). Another alternative, store entangled EPR pairs, then they can reduce the the Bohm interpretation, makes the narrow point rate of agreement for distinct questions to 1 . It is that quantum probability can be viewed as a non- 4 convenient to re-express the EPR pair as the qubit local deterministic system. (Non-local means that cat state the model sacrifices any notion of independence in 00 + 11 joint systems.) These alternative interpretations are ψ = | i | i. not broadly useful. | i √2 Alice and Bob can each answer one the three “questions” by performing the corresponding measure- 1.6.2. Entanglement paradoxes ments

X = J2π/3 Y = J 2π/3 Z = J0, − Einstein was a more inspired critic of the Copen- where hagen interpretation than Schr¨odinger. In a joint paper with Podolsky and Rosen [11], he noted that cos(θ) sin(θ) Jθ = commuting measurements on an EPR pair, sin(θ) cos(θ) − ! Let ψ = |↑↓i − |↓↑i, | i √ XA = X I XB = I X 2 ⊗ ⊗ 18

be the corresponding factor measurements for Alice of any genuine interaction between the photons in and Bob, and likewise for Y and Z. Then (Exer- these demonstrations, much less non-quantum inter- cise 1.6.1): actions that would present an illusion of quantum non-interaction. 1. Each of the six variables is an unbiased 1- ± The original purpose of the Bell-EPR paradox was valued random variable. the simple conclusion that quantum operations do not admit a deterministic or classically random sim- 2. The variables XA and XB (and likewise for Y and Z) agree with probability 1. ulation that preserves locality. In hindsight, it is a first step in the direction of quantum algorithms 3. The variables XA and YB (and likewise the (Section ??) and especially quantum security (Sec- 1 other pairs) agree with probability 4 . tion ??), since these can also be viewed as entanglement paradoxes. The problem of communication (Note that each pair of questions converts an EPR security is for two parties (Alice and Bob) to share pair to a product state; the EPR pair cannot be information with some confidence that there is no reused.) If the interrogators witness these classically eavesdropper (Eve). The shared information is ide- impossible correlations, they might be tempted to ally random, because it can then be used to mask seize Alice’s and Bob’s electronic devices and try to arbitrary messages. In Bell’s protocol, the same ar- use them to communicate with each other. But they gument that Alice’s and Bob’s answers are classi- would not succeed, because no quantum operation cally impossible also shows that there cannot be an on Alice’s qubits affects the marginal state on Bob’s Eve who knows their answers in advance. Thus the qubits, or vice-versa. shared answers are also shared secrets. More formally, Bell’s theorem is an inequality con- The relation to quantum algorithms is less for- cerning correlations of two-valued classical random mal. Intuitively, quantum algorithms exploit entan- variables which does not hold for quantum random glement as a kind of communication. For example, variables: the result of Grover’s search algorithm (Section ??) Theorem 1.6.1 (Bell). If X, Y , and Z are three can be described as a guessing game: If Alice thinks classical random variables taking values in 1 , of a number from 1 to N and only responds “yes” or then {± } “no” depending one whether Bob guesses correctly, then Bob can guess it with O(√N) guesses, provided E[XY ]+ E[XZ]+ E[YZ] 1. that he can guess in quantum superposition and Al- ≥− ice’s consideration of each guess is unitary. Grover’s algorithm is a classically impossible form of communication afforded by quantum entanglement. Proof. We would like to show, equivalently, that

E[XY + XZ + YZ] 1. ≥− Exercises It is easy to check that 1. Establish that Bell’s operators XA, YA, ZA, 3 if X = Y = Z X , Y , and Z applied to a Bell state violate XY + XZ + YZ = . B B B 1 otherwise Bell’s inequality. (− Since the random variable XY +XZ +YZ is always 2. The quantum violation of Bell’s theorem can at least 1, its expectation is at least 1. be called a “no hidden variables” theorem: − − Quantum operations cannot be simulated by A variant of the Bell-EPR paradox (with a dif- hidden structure which is deterministic or clas- ferent set of classically impossible correlations) was sically random. One rigorous (but possibly famously demonstrated in an experiment by Aspect limited) interpretation of this principle can be et al [2], and since then by others. In the experi- phrased as category theory: There does not ment the two halves of Bell-state photon pairs were exist a non-trivial linear tensor functor from interrogated at almost simultaneously, so that there the category Quant

for 1-valued classical random variables, due an integral with respect to an operator-valued mea- ± to Clauser, Horne, Shimony, and Holt [8]. sure µP whose value on any interval is a projection (This avoids the assumption in Bell’s theorem that commutes with X: that when Alice and Bob perform the same measurement, they will agree with probability X = λdµP . (5) R one.) Prove this inequality, and then find a Z violation using the operators Jθ for four par- If we pair the measure µP with the state ρ, the result ticular values of θ. is the desired scalar-valued measure on R, indeed on the spectrum of X. A (projective) measurement X is again defined as 1.7. Infinite systems a direct sum decomposition

The immediate way to extend the finite-state the- = s H ∼ H ory to infinite quantum systems is to allow the s S M∈ Hilbert space to be infinite-dimensional (but usu- for some outcome set S, which may now be infinite. ally separable).H Section 1.8 discusses a better and Probabilities and conditional states have the same more general extension due to von Neumann, but formulas: much can be learned from the this less creative ap- \ PsρPs proach. P [X = s] = Tr(P ρ) ρ = . We can use various definitions from operator the- ρ s |X=s P [X = s] ory [? ] to adapt various objects such as states, random variables, and quantum operations to in- Not every real-valued random variable defines a mea- finite Hilbert spaces. Once these are defined, we surement of this type. Rather, the spectral theorem can define the category Quant to be the category of says that a Hermitian operator X has a point spec- Hilbert spaces (both finite and infinite) with TPCP trum and a continuous spectrum. Only the point maps as the morphisms or quantum operations. spectrum possesses eigenspaces, so X must have a First, a (normal) state ρ is defined as a positive pure-point spectrum in order to define a measure- semi-definite trace-class operator with trace 1. In ment. However, there are various ways to approxi- other words, the Bloch region +,1( ) is defined as mately measure a continuous-spectrum values of an B H operator. The point spectrum is usually discrete, the trace 1 subspace of t( ), the algebra of trace- class operators. The spectralB H theorem for compact meaning that the eigenvalues are isolated, while the operators implies that such a state ρ can be ex- continuous spectrum usually consists of intervals, pressed as: which in quantum mechanics are called bands. But there are other possibilities for both parts of the spectrum. ρ = ps s s | ih | An unbounded random variable is defined as a s S X∈ self-adjoint unbounded operator, although such an for some orthonormal basis S of . Thus as in the operator is an artifice in the sense that it is only finite case, pure states (by definitionH the extremal defined on a dense subset of . By definition it is elements of the Bloch region +,1( )) correspond a densely defined function whoseH graph is a closed to vector states (by definitionM unit vectorsH in ) up vector subspace of which is invariant under to a global phase. H switching the two summands.H ⊕ H The definition is cho- A real-valued, bounded random variable X on sen so that self-adjoint operators satisfy the spectral is defined as a self-adjoint bounded operator. ThisH theorem. If X is a self-adjoint operator, then matches the definition of states in that ( ), the itX algebra of bounded operators, is the BanachB H space U(t)= e dual of ( ). This duality means that for any state Bt H is a one-parameter group of unitary operators, and ρ and any bounded variable X, the trace Tr(ρX) is conversely every strongly continuous one-parameter well-defined as a finite real number. Thus we can group of unitary operators defines a possibly un- define the expectation bounded operator. Either the spectral theorem or def the unitary operator model could be taken as an Eρ[X] = Tr(ρX) alternate definition of an unbounded self-adjoint operator. as before. More generally, a state ρ and a real-valued random variable X produce a probability measure Example 1.7.1. The function spaces L2(Rd), with on R, the distribution of X, by the spectral theorem 1 d 3, are very common in quantum mechan- for bounded operators. This theorem expresses X as ics.≤ Pure≤ states are naturally referred to as wave or 20

amplitude functions. Technically they are half den- suitable axioms so that we can deﬁne states, random sities, meaning that the square norm of a wave func- variables, and quantum operations. Von Neumann tion is a probability density function with a volume deﬁned two types of algebras for this purpose, C∗- form factor. For example, the wave function algebras and W ∗-algebras; the latter are now called

2 von Neumann algebras. Von Neumann algebras are e x /2 ψ(x)= − √dx, actually just C∗-algebras with a stronger topological π1/4 closure property. here written as a half density, is called a coherent The algebras ( ) of all bounded operators on a Hilbert spaces MareH one class of von Neumann alge- state on = L2(R) (notwithstanding that elsewhere H every pureH state is called coherent). It will appear bras that happen to contain all von Neumann algebras as subalgebras. But considering only ( ) is later as the ground state of the harmonic oscillator. M H To give an example of a mixed state on L2(R), let a very restricted view of the theory of operator algebras, just as considering only symmetric groups is a a 1 and let ≥ very restricted view of finite group theory. Quantum − 1 a(x y)2+a 1(x+y)2)/4 physics has also drifted towards considering specific ρ(x, y)= e − √πa algebras of operators, although not usually with von Neumann’s axioms. be a kernel (in the sense of integration, not null A C∗-algebra is, first, a complex vector space spaces). The corresponding operator with an associativeA and bilinear multiplication law. It also has an abstract anti-linear, product-reversing ρ(f)(x)= ρ(x, y)f(y)dy adjoint operation denoted “ ”: R ∗ Z is trace-class with trace 1 and is called a quasifree (X + Y )∗ = X∗ + Y ∗ (λXY )∗ = λY ∗X∗. state; when a = 1 it is the coherent state. Finally is also a Banach space a norm that A || · || Example 1.7.2. If f(x) is a continuous (or even satisfies the relation integrable) function on R, then multiplication by f is 2 X∗X = X . an operator on which is given the same name. For || || || || H example, x is an unbounded, continuous-spectrum This last axiom, the “C∗ axiom” is coy and has many operator whose distribution with respect to the pure consequences for the structure of . Among other state ψ(x) has the density function ψ(x) 2. Another A | | things, it means that the norm is completely example is the operator determined by the algebra structure|| · || of and that A ∂ p = i X∗ = X . − ∂x || || || || Intuitively consists of bounded operators and X which has the same spectrum as x, namely all of R A || || as a continuous spectrum. They are both Gaussian behaves as the spectral radius of X. For simplicity we will assume that every C∗-algebra has a unit, random variables with respect to coherent and quasi- A free states. We will see later that the operator even though non-unital C∗-algebras are also an interesting class. p2 + x2 Possessing a unit is traditionally an optional ax- H = 2 iom for C∗-algebras; we will assume it for simplicity. 1 has discrete spectrum Z 0 + . In the standard co- Theorem 1.8.1 (Gelfand,Naimark). If is a (uni- ≥ 2 A 1 tal) commutative C∗-algebra, then it is isomorphic to herent state, H is definite with value 2 , while in a standard quasi-free state, it has a discrete exponen- an algebra of continuous functions C(A) on a com- tial distribution. pact Hausdorff topological space A.

By Theorem 1.8.1, and since C(A)isa C∗-algebra for every compact Hausdorff space A, a C∗-algebra Exercises can be thought of as a “non-commutative topological space”. In particular if A is a finite set, then 1.8. Operator algebras C(A) = CA is exactly the model of finite probability described in Section 1.10 — its set of normalized Following von Neumann, we can represent a quan- states is ∆A. tum object not as a Hilbert space , but as an ab- Theorem 1.8.1 also implies that if X sa (the stract algebra whose elements canH be called “op- self-adjoint subspace of ) and f : R ∈R Ais a con- erators”. SuchA an operator algebra should satisfy tinuous function, then thereA is a well-defined→ element 21

f(X) sa (Exercise ??). For example, sin X and and are von Neumann algebras, there a von Neu- X are∈ A well-defined. We will need a slight gener- mannN algebra which is a further topological | | M ⊗ N alization of this principle: An element X is completion than the C∗-algebra completion. ∈ Asa positive, or X 0, if X = Y ∗Y for some Y . If After accepting these preliminaries (perhaps on ≥ f : R 0 R is continuous and X 0, then f(X) is faith), we can proceed to define the basic structures ≥ → ≥ well-defined; if f 0 as a function, then f(X) 0. of quantum probability. A C∗-algebra or a von ≥ ≥ A A representation of a C∗-algebra is a homomor- Neumann algebra can be assigned. The algebra A M phism from to the C∗-algebra ( ) of bounded is termed its algebra of observables, because the self- A B H operators on a Hilbert space. (A homomorphism be- adjoint elements ( sa or sa) will be interpreted as A M tween two C∗-algebras is a linear map that respects real-valued random variables. If is a C∗-algebra, #A multiplication, , and is continuous with respect to a state is a dual vector ρ which is positive, ∈ A the Banach norm.)∗ Crucially, the algebra ( ) has meaning that other topologies besides the one coming fromB H its Ba- X 0 = ρ(X) 0. nach norm, namely the strong and weak operator ≥ ⇒ ≥ topologies. If is a C∗-algebra which is closed The state ρ is normalized if with respect toM the weak operator topology in some ρ(I)=1. faithful representation , then is a von Neumann algebra. H M Previously we took ρ to be an operator rather than a dual vector; this is not really different if we define Theorem 1.8.2. If is a commutative von Neu- mann algebra, then itM is isomorphic to the algebra ρ(X) = Tr(ρX). L∞(M) for some σ-field M. In particular we can call ρ(I) the “trace” of ρ. A state ρ on a von Neumann algebra is normal By Theorem 1.8.2, and since L∞(M) is a von Neu- if it lies in the pre-dual # . The commutativeM case mann algebra for many natural σ-fields M, a von illustrates the reason to takeM states from the dual of a Neumann algebra can be thought of as a “non- M C∗-algebra but from the pre-dual of a von Neumann commutative measure space”. algebra. If = C(A), then by the ??? theorem, Theorem 1.8.2 also implies that if X sa (the states are equivalentA to finite Borel measures on A. self-adjoint subspace of ) and f : R ∈R Mis a mea- A → If = L∞(M), then general states are equivalent surable function, then there is a well-defined element toM finitely additive, finite measures on M; normal f(X) a. This closure property is called “func- ∈Ms states are equivalent to countably additive measures tional calculus”. and are hence more empirical. Note also that the The traditional definition of von Neumann alge- states of the C∗-algebra are the normal states of bra via a faithful action on a Hilbert space is A ## H the von Neumann algebra . contrary to our intention of emphasizing operators A pair ( ,ρ) consistingA of a von Neumann al- over vectors. Happily there are other characteriza- gebra Mand a normalized, normal state ρ is tions of von Neumann algebras within the class of also calledM a quantum probability space or a non- C∗-algebras: commutative probability space. Each random variable X sa has a well-defined spectrum Spec X Theorem 1.8.3 (???). A C∗ algebra is a von R ∈M ⊂ M (which depends crucially on the structure of ), Neumann algebra if and only if it has a pre-dual and each state ρ induces a probability distributionM # as a Banach space. The pre-dual, if it exists, M on Spec X. If ( ) is defined as an alge- is unique up to isometry. bra of operatorsM⊆B on a HilbertH space, one way to construct the spectrum and distribution of X is by In particular, the algebra ( ) of bounded opera- Equation (5). The point is that # is a quotient of tors on a Hilbert space isB a vonH Neumann algebra ( ), and we can use any lift ofMρ to a trace-class with pre-dual ( ). TheoremH 1.8.3 interplays with t t operatorB H on . the general factB thatH every Banach space embeds As usual, ifH and are Alice’s and Bob’s algebras isometrically in its second dual ##. ThusB the pre- of observables,A theirB joint algebra of observables is dual # can be viewed as subspaceB of the dual #. M M . Also, by a construction of ???, if is a C∗-algebra, A⊗B ## A If and are C∗-algebras, then a linear map its second dual has the natural structure of a A B von Neumann algebra,A the universal enveloping von : Neumann algebra of . E B → A A is positive if it takes positive elements of to posi- If and are two C∗ algebras, there is a natural A A B tive elements of ; it is completely positive if tensor product C∗-algebra which is a topologi- B cal completion of the algebraicA⊗B tensor product. If : M E ⊗ I B⊗C→A⊗C 22

is positive for every C∗-algebra ; and it is unital if where each E 0 and C a ≥ (I)= I. E = I. E a a A If and are von Neumann algebras, a linear X∈ mapM N This structure is also known concretely as a positive, operator-valued measure, or POVM, because it is a : probability measure on A that takes values in the E N →M positive cone + rather than in the real numbers. is normal if it has a pre-transpose We can constructB a sensible conditional state on in the same setting. It would be given by a quantumB # : # # . operation E M → N In general is unital if and only if the pre-transpose : # # # # (or theE transpose #) is trace-preserving. The F B → B⊗ A E E category vN is defined as the category of pre-duals such that the composition TrB (in which Alice ◦F of von Neumann algebras with TPCP maps as its applies and then destroys Bob) is a POVM F morphisms. Equivalently it is the category of von : # # . Neumann algebras with normal UCP (unital and E B → A CP) maps as contravariant morphisms (Exercise ??). The category vN is the most satisfactory model of We further suppose that is initial among all such factors of , in the sense thatF Bob retains as much in- infinite quantum probability, and its morphisms can E be called quantum operations. An interesting also- formation as possible about his previous state. Then (Exercise ??) one form for is ran is the category C∗ of C∗ algebras with arbitrary F UCP maps as contravariant morphisms. Although = [a] ρ√Ea , vN can be viewed as a subcategory of C∗, C∗ can F ⊗ a A also be viewed as a subcategory of vN, by taking ∈ ## M each C∗-algebra to its enveloping algebra . A A where in general the state ρX is defined as

X ρ (Y )= ρ(X∗YX); 1.9. Classical and quantum coexistence also if X 0, √X is its unique positive square root. Because the category vN includes commutative In light of≥ this possible structure for , the condi- algebras, which are the corresponding models of clas- tional state of the POVM applied toF ρ with out- sical probability, they can model coexistence of clas- come a is E sical and quantum objects and interactions between √Ea them. We can then use these categories to study ρ =a = ρ . measurements of quantum systems by classical ob- |E servers, and classical behavior of quantum systems. Finally the POVM has mutually exclusive out- (As discussed in Section 1.6.2, the converse of the comes if and only ifE its non-destructive lift is a latter is impossible.) projection, i.e., 2 = . In this case (ExerciseF ??) F F As a first case, suppose that Alice is classical and the Eas are projections such that finite, so that her von Neumann algebra is EaEa′ =0 = CA A when a = a′. Thus the projective measurements de- 6 for some finite set A. Suppose that Bob has some fined in Sections ?? and ?? are exactly those POVMs other von Neumann algebra . Then we can ax- with mutually exclusive outcomes. iomatically define a destructiveB measurement to be The above discussion can be extended to the case an arbitrary quantum operation where Alice is classical but infinite. In this case her von Neumann algebra is L∞(A) for some σ-field A, : # # . and a POVM is a measure on A that takes values in E B → A +. This structure is closely related to a normalized Then (Exercise ??) the general form of is Bmeasure on + itself. P We turn toB a result from the theory of operator (ρ)= ρ(E ), algebras that shows how decoherence can reduce a E a a A quantum object to a classical one. Suppose that M∈ A 23

is Alice’s algebra of observables and that she evolves The vector (λ1, λ2,...,λn) in Theorem 1.9.2 can according to a quantum operation be called the shape of the algebra . A : Examples 1.9.2. A joint system consisting of a fi- P A → A nite classical Alice with algebra Cn and a finite quan- in a unit period of time. Very often we can suppose tum Bob with algebra has the rectangular shape that is an idempotent, i.e., 2 = (Exercise ??). Mk P P P (k,k,...,k). For example, Bob could be a quantum Intuitively this means that Alice stabilizes in a short computer with qubit memory and Alice could be a period of time and, once stable, her temporal evolu- classical controllers with classical bit memory. The tion does not further affect her state. first finite quantum system which is not of this form Theorem 1.9.1 (Choi-Effros). If is a C -algebra is the hybrid trit 2 C. The author [? ] has ∗ M ⊕ and is an SUCP idempotent on A, then the image analyzed the storage properties of finite quantum P A of is a C∗-algebra with a modified product systems such as the hybrid trit. P B X Y = (XY ). ◦ P Exercises

Theorem 1.9.1 says that if Alice’s evolution is an 1.10. Appendix: A classical review idempotent , then she can be modelled by a smaller P effective C∗-algebra . In particular, can be com- B B Consider a classical probabilistic system with a fi- mutative even when is not. The algebra is a vec- nite set A of configurations. The general probability tor subspace of , butA not in general a subalgebra.B A distribution or measure µ on A can be written in the Rather, the modified product defined by Choi and form Effros matches the process of applying between external interactions with Alice. Note alsoP that if A µ = pa[a], is a von Neumann algebra and if is normal, then a A is also a von Neumann algebra.P X∈ B where each probability p 0 and Example 1.9.1. The blind projective measurement a ≥ operation defined in Equation (4) is an idempotent. In thisP case im is closed under multiplication pa =1. P a A and is the algebra X∈ ( ). The symbol [a] represents both the configuration a M Hs and the probabilistic state in which a is certain. s S M∈ (This state is also called an atom or a Kronecker Finally, the classification of finite-dimensional C∗- delta function.) The set of all normalized states algebras, or von Neumann algebras, also fits the forms a simplex ∆A whose vertices are the set A. theme of classical and quantum coexistence. Say These simplices are the objects of a reasonable if that a -algebra is positive if slightly non-standard definition of finite, classical ∗ A probability theory. The other elements of this defi- X∗X =0 = X =0. ⇒ nition are as follows. If is positive and finite-dimensional, then it is au- An event is a subset E A. A state µ assigns a A ⊆ tomatically a C∗-algebra, indeed a von Neumann al- probability to E between 0 and 1 by the formula gebra. This can be seen from the following classification theorem. P [E]= Pµ[E]= pa. a A Theorem 1.9.2 (Artin-Schreier). If a positive, X∈ A finite-dimensional -algebra, then it is a direct sum If P [E] > 0, then E and µ induce a conditional state ∗ of matrix algebras: on E given by the formula n 1 = λ µ = p [a]. A ∼ M k |E P [E] a k=1 a E M X∈ d for some integers A random variable or measurement is a function X : A B for some set B. It is interpreted as λ1 λ2 λn. ≥ ≥···≥ a partition→ of A into disjoint events: the equation X = b, as an event, is the set of all a such that 24

X(a)= b. If X takes values in R or C, then it also Example 1.10.1. Two fair dice are rolled. If we has an expectation, defined as take the dice as subsystems A and B with a joint state, then A = B = 1,..., 6 and the state µ is { } E[X]= paX(a). the uniform state a X 1 If A and B are two finite configuration sets, then µ = [(s,t)]. 36 a function 1 s,t 6 ≤X≤ M : ∆A ∆B The event of rolling 7 is the set E = (s,t) s+t =7 . → The chance of E is P [E] = 1 . The{ state| µ is an} is a stochastic map (or a Markov map) if it comes 6 independent state, but the conditional state µ is from a linear map E correlated. A protocol such as “if you rolled 7,| pick M : RA RB. one die at random and roll it again” is modelled by → a stochastic map. The linearity of M is the classical superposition principle. Another way to describe a stochastic map is The rules of classical probability theory are the to start with a linear map M and impose the posi- basis of a (Bayesian) statistician’s model of ordinary tivity condition M(∆A) ∆B. If M is viewed as a human experience, as well as most scientific experi- matrix, the positivity condition⊆ says that the entries ments, according to the following interpretation: of M are non-negative and each column sums to 1. 1. State: Each observer can model external re- In this case M is called a stochastic matrix. Stochas- ality as a measure space, such as a finite set tic maps are the natural empirical class of maps in A, that carries has probability distribution at probability theory. each point in time. A non-negative vector µ in RA is a subnormalized state if the total probability is at most 1, and a linear 2. Independence: Reality decomposes into ap- map proximately disjoint subsystems modelled by Cartesian products of measure spaces. An ob- M : RA RB → server is an approximately independent sub- that preserves subnormalized states is called subsystem whose residual non-independence is de- stochastic. A substochastic map is also called an scribed by witnessed information. extinction process. It can be converted to a stochas- 3. Evolution: After an observer witnesses an tic map by adding an extinction state to the target event, the new state of reality is given by set B. conditional expectation. More generally the If A and B are the configuration sets of two proba- state of reality evolves by stochastic and sub- bilistic systems, then the two systems together have stochastic maps. a joint configuration set A B. The joint simplex × ∆A B lies in a tensor product: × 4. Statistics: An observer’s experiences are ( A B viewed as independently repeatable experi- ∆A B R A B)= R R . × ⊂ × ⊗ ments. The probability of an event is the frac- A joint state µ ∆A B is independent or a product tion of times that it occurs in repeated trials state if it factors∈ as a× tensor product: of the experiment.

µ = µA µB. These rules can be accepted in one of two ways: ⊗ (1) they hold empirically; (2) they can be mimicked More typically µ does not have this form, in which by deterministic systems with hidden information. case it is correlated. A stochastic map that affects The first reason, but not the second, applies to the only one of A and B has the form M I or I M; quantum analogue of these rules. likewise an event that depends on only⊗ one of⊗A or B has the form E B or A E. Whether or not µ is independent, it× induces states× on A and B called Exercises marginals. They are defined as: 1.11. Appendix: Categories and tensors µA = p(a,b) [a] µB = p(a,b) [b]. a A b B b B a A X∈ X∈ X∈ X∈ Much of the presence of category theory in math- Evidently, if µ is a product, then it is the product of ematics is in the spirit of Moliere: You can use its marginals. categories without knowing it. In many areas of 25

mathematics there is a distinguished class of objects tensor products of morphisms is more complicated (e.g., sets, groups, topological spaces, vector spaces) because the axioms include compatibility relations and a distinguished class of functions between them between the operations of composing and tensoring (e.g.all functions, group homomorphisms, continu- morphisms. The tensor categories of interest in this ous functions, linear maps). In some cases two ob- article are all symmetric, meaning that “ ” is com- jects are related by a transformation which isn’t mutative as well as associative, both for objects⊗ and strictly a function but behaves a lot like one; the his- morphisms. torically important example was a function between The most important example of a tensor category two topological spaces considered up to homotopy. is Vect, the category of vector spaces. A category is a possibly abstract class of set- like objects andC a class of function-like relations, or morphisms. Every morphism f has a domain A and a target B (both of them objects in ) and is then denoted C f : A B. → There is a partial composition law: If the target of f is the same object as the domain of g, or f : A B g : B C, → → then there is a composition f g : A C. ◦ → The composition law for is required to be asso- ciated, and for every objectC A there should be an identity morphism i : A A → which satisfies the obvious identity axiom with respect to composition. These axioms are not all that restrictive and there are a wide variety of different kinds of categories. Examples 1.11.1. The category of sets, with all functions as the morphisms, is called Set. The category of vector spaces over a field F, with linear transformations as the morphisms, can be called VectF. The category of topological spaces with continuous functions as the morphisms is called Top. Two different categories can have the same objects but different morphisms. For example, the obejcts of the category iSet are sets, but the morphisms are just the injective functions. As an example of a more abstract category, if G is a group, it yields a category with one object whose morphisms are the elements of G. A tensor category is a category with a multiplication law, denoted “ ” and called a “tensor product”, for both objects and⊗ morphisms. The multiplication law should be associative: A (B C) should be the which A ⊗ (B ⊗ C) and (A ⊗ B) ⊗ C are exactly the same same object as (A B) C4.⊗ The⊗ axioms for taking object, and a lax tensor category, in which they are iso- ⊗ ⊗ morphic and the isomorphisms are meta-associative; this meta-associativity is called “coherence”. We don’t have to worry about lax tensor categories here, but note that they arise in physics in the guise of spin calculus; the main co- 4 There is a distinction between a strict tensor category, in herence axiom is known as the Biedenharn-Elliott identity. 26

2. MECHANICS 2.1. Wave mechanics

Basic quantum mechanics assumes the rules of Wave mechanics is a model of a single particle quantum probability plus a single additional rule. moving in Rd with 1 d 3. To give the particle If a physical system is independent and autonomous something to do, it interacts≤ ≤ with a potential V (~x). (or closed), then quantum mechanics postulates that Then wave mechanics posits the specific Hamilto- it has a pure state space and its state evolves ac- nian cording to a one-parameterH group U(t) of unitary ~p ~p operators. The group has the form H = · + V (~x), 2 U(t) = exp( itH) − where ~x and ~p are two vector-valued operators. The for some self-adjoint (but often unbounded) operator component xk of ~x is interpreted as multiplication by the coordinate x , while ~p = i∂/∂~x. The H, which is called the Hamiltonian of the system. If k − the state at time t is ψ = ψ(t) , it evolves by the Schrödinger equation then becomes the linear par- Schrödinger equation,| i | i tial differential equation ∂ ∂ψ ∆ψ i ψ = H ψ . i = + V (~x)ψ, ∂t| i | i ∂t − 2 Besides the system’s autonomous behavior, it can where also be measured. In particular, H itself is a mea- ∂ ∂ surement and its value is called the energy of the ∆= ∂~x · ∂~x system. A state ψ with definite energy E satisfies the eigenvalue equation| i is the Laplacian. Since this is a wave equation, it illustrates half of particle-wave duality. The state E ψ = H ψ , of a particle travels through space as a wave, which | i | i is why it is called a wave function (but see exercise which is also called the time-independent 2.1.3). Schrödinger equation. The operator ~x is interpreted as position and has The operator ∂ has physical units of inverse time, ∂t physical units of distance. The potential V (~x) has while the Hamiltonian H has units of energy. There- units of energy, the same as H. With a factor of ~, fore quantum mechanics requires a conversion factor ~, known as Planck’s constant, to relate the two op- ∂ ~p = i~ , erators. Written with units, the Schrödinger equa- − ∂~x tion is the operator ~p is interpreted as the particle’s linear ∂ i~ ψ = H ψ . momentum. The kinetic energy of a particle with ∂t| i | i momentum ~p is (~p ~p)/2m, where m is its mass, so · If ~ is small compared to the scale of a physical sys- with units the Hamiltonian becomes tem, its quantum phase will oscillate wildly, which ~p ~p H = · + V (~x). leads to classical probabilistic behavior. In metric 2m units, The mass m, an energy scale E, and Planck’s con- 34 ~ 6.6262 10− Js, stant ~ fully determine the dimensional scale of ≈ × Schrödinger’s equation. For example, the length where J is joules and s is seconds. So Planck’s con- scale of the particle’s quantum behavior is d = stant is extremely small on the human scale. ~/√Em, which decreases as m increases. Thus if The fact that the Hamiltonian H often has a dis- several particles interact on a common length and crete spectrum points to the origin of quantum me- energy scale, heavier particles behave more classi- chanics and its name. The word “quantum” now of- cally than lighter ones. ten means simply “non-commutative”, but the original meaning is “discrete”. The original purpose of Example 2.1.1. A particle in the potential V (~x)= quantum mechanics was to explain why many physi- 0 is called free. In this case the Hamiltonian cal measurements unexpectedly take discrete values. has no point spectrum and the time-independent For example, the energy spectrum of the quantum Schrödinger equation has no solutions with finite Z 1 harmonic oscillator is 0 + 2 . The energy of a Hilbert norm. But the plane wave classical harmonic oscillator≥ can of course be any i~k ~x non-negative real number. ψ(~x)= e · 27

is an interesting infinite-norm solution. It has defi- so the state ~ nite momentum ~p = k as well as definite energy. It n (a∗) 0 can be approximated by a wave packet (or wavelet) n = | i that, according to the Schrödinger equation with | i √n! physical units, travels with velocity ~k/m. 1 is normalized and is an eigenstate with energy n+ 2 . The behavior of a free particle illustrates an inter- Thus the harmonic oscillator has a sequence of pretation of the continuous spectrum of an arbitrary eigenstates 0 , 1 ,.... To show that they span 2 | i | i Hamiltonian. A state ψ is unbound if it wanders L (R), observe that H 0 as an operator, so its ≥ under unitary evolution:| i spectrum is non-negative. If the spectrum of a state ψ lies in the interval [0,n], then lim ψ U(t) ψ =0. | i t n+1 →∞h | | i a ψ =0. | i It is bound if it recurs: It follows that lim sup ψ U(t) ψ =1. t |h | | i| ψ = P (a∗) ψ →∞ | i | i These are the quantum analogues of the notions of for some polynomial P of degree at most n. Thus wandering and non-wandering points in (reversible) ψ is a linear combination of 0 ,..., n . classical dynamical systems. The spectral theorem | i | i | i implies that point-spectrum states are bound and Wave mechanics also postulates a multiparticle continuous-spectrum states are unbound. In partic- Schrödinger equation. The state space of n parti- ular, every bound state is a quantum superposition cles in d dimensions is of fixed-energy or stationary states. 2 d n 2 dn L (R )⊗ = L (R ). Example 2.1.2. The simple harmonic oscillator is ∼ a 1-dimensional system with Hamiltonian The dn coordinates of this Hilbert space are divided into n vector-valued position operators ~x ,...,~x , p2 x2 1 n H = + . and there are n corresponding momentum operators 2 2 ~p1,...,~pn. The Hamiltonian has the general form

The coherent state n 2 ~pk ~pk e x /2 H = · + V (~x1,...,~xn). ψ(x)= − √dx 2 (2π)1/4 Xk=1

1 The corresponding Schr¨odinger equation is a PDE in is an eigenstate with energy 2 . The rest of the spec- dn dimensions. It is usually intractable, even with trum can be found with ladder operators. This is a the aid of computers (but see Chapter 3). Almost Lie-algebraic method that begins with the commu- the only case with a satisfactory solution is the one tation relations in which the potential factors,

[x, p]= i [H, x]= ip [H,p]= ix. n − V (~x ,...,~x )= V (~x ), Deﬁne the lowering and raising operators 1 n k k kY=1 x + ip x ip a = a∗ = − . which means that the particles do not interact with √2 √2 each other.

Then H, a, and a∗ satisfy Exercises [a,a∗]=1 [H,a]= a [H,a∗]= a. − These commutation relations imply that if a state 1. Show that ψ has energy E, then a ψ has energy E 1 and | i | i − sin k ~x a∗ ψ has energy E + 1, provided that either vector ψ(~x)= | | is| non-zero.i The coherent state spans the kernel of ~x a and we rename it 0 . The commutation relations | | imply, by induction,| thati is an inﬁnite-norm solution to the free-particle Schr¨odinger equation. It represents a spheri- n n 0 a (a∗) 0 = n!, cally radiating particle. h | | i 28

2. The function 2.2. A classical limit 1 ψ(~x)= ~x In this section we will reformulate wave mechanics | | as a version of Hamiltonian mechanics. In classical is not considered an infinite norm solution to the free-particle Schrödinger equation. For ex- physics, Hamiltonian mechanics produces a dynamical system (Hamilton’s equations) on the configu- ample, f(~x) is a smooth bump function, then f(~x)ψ(~x) is not an approximate eigenstate of rations of a physical object for every smooth func- H. Why not? tion H (the Hamiltonian) on the configuration space. The configuration space must have a symplectic or 3. A nonlinear Schrödinger equation is a partial Poisson structure and is also called phase space. The differential equation quantum version is an important generalization of 2 wave mechanics, and it also limits to classical Hamil- ∂ψ 1 ∂ ψ ~ i = 2 + V (x, ψ) tonian mechanics as 0. ∂t −2 ∂x We first restate quantum→ mechanics so that mea- for some function V which is not linear in ψ. surement operators evolve and states does not. Similarly, given a pair of wave functions Given a general Hamiltonian H acting on an arbi- 2 2 (ψ1, ψ2) L (R) L (R), trary Hilbert space , let ψ(t) be the state at time ∈ ⊕ t and let H | i we can write coupled Schrödinger equations (either linear or nonlinear): ψ = ψ(0) . | i | i ∂ψ 1 ∂2ψ 1 1 If A is a measurement operator to be applied at time i = 2 + V1(x, ψ1, ψ2) ∂t −2m1 ∂x t, define ∂ψ 1 ∂2ψ i 2 = 2 + V (x, ψ , ψ ). 2 2 1 2 A(t)= U( t)AU(t). ∂t −2m2 ∂x − Explain why a nonlinear Schrödinger equation Then evolving the operator A and fixing the state ψ cannot model a quantum particle and coupled is statistically equivalent to fixing the operator and Schrödinger equations cannot model coupled evolving the state: quantum particles. ψ A(t) ψ = ψ(t) A ψ(t) . 4. Prove that if a state lies in the point spectrum h | | i h | | i of H, then it recurs, while if it lies in the con- This equivalence is a special case of the conjuga- tinuous spectrum, then it wanders. Prove that tion principle in group theory: If a group G acts in the Schrödinger wave equation, if the po- 1 on a set S, then ghg− does to g(s) what h does tential V (x) is non-negative, then an unbound to s. As applied to quantum mechanics, it is called particle must escape to infinity. the Heisenberg picture. The differential form of the 5. Prove that the harmonic oscillator state n Heisenberg picture is an operator-valued differential has the form | i equation called the Heisenberg equation: x2/2 Hn(x)e− ∂A i = [A, H]. for some polynomial Hn(x) of degree n. Since ∂t the polynomials are orthogonal, they are Her- We can now rename variables to match Hamilton’s mite polynomials up to rescaling x and multi- equations in symplectic R2n. Substitute ~q for ~x and plying by a constant factor. n for d and drop the restriction d 3; the state 2 n ≤ 6. Let ρ be the standard quasifree state with pa- space becomes L (R ). The operator ~q consists of Rn rameter a, defined in Example 1.7.1. Show the coordinates q1,...,qn on , while ~p is defined that the harmonic oscillator Hamiltonian H by has a discrete exponential distribution with re- ∂ spect to the state ρ with parameter pk = i . − ∂qk a 1 t = − . a +1 These operators satisfy the commutation relations I.e., show that [qj ,pk]= iδj, k. ∞ ρ = (t 1)tn n n . Now assume that the Hamiltonian operator H is a − | ih | n=0 X “function” H(~p, ~q) of ~p and ~q. For example it might 29

be a non-commutative polynomial in the coordinates group SO(3). By Noether’s theorem for quantum of ~p and ~q or a suitably convergent power series. systems, if the system’s Hamiltonian H is invariant Then formally under rotation, the Lie generators, if multiplied by i, are Hermitian operators with conserved values. ∂H ∂H [qk, H]= i [pk, H]= i . The angular momentum operators in the x, y, and z 3 ∂pk − ∂qk directions in R are written Jx, Jy, and Jz; the angular momentum in the direction of a general unit Combining this with the Heisenberg equation yields vector ~v is ~v J~. These operators do not commute an operator form of Hamilton’s equations for conju- · gate variables: with each other, so they are not simultaneously definite. There is another twist as well. Since global ∂q ∂H ∂p ∂H phase is not statistically meaningful, the state space k = k = . ∂t ∂pk ∂t −∂qk might only be a projective representation of any givenH symmetry group. This can happen with spa- To see the classical limit, assume for simplicity tial rotations, so the true quantum rotation group is that n = 1 and the conjugate variables are just p and the double cover q. In units of Planck’s constant, their commutator ^ is SO(3) ∼= SU(2). [q,p]= i~. Whether is a projective representation or a linear one, theH analysis will show that angular momen- Now take the limit ~ 0. The Heisenberg uncer- → tum takes discrete values that differ by multiples of tainty relation (Exercise 1.2.8) yields the inequality ~, even though there is a continuous family of di- ~2 rections in which to measure it. Moreover, the ro- V [q]V [p] . tational state space of a physical system is typically ≥ 4 finite. In fact the inequality is sharp: Gaussian wave pack- The operators Jx, Jy, and Jz satisfy the commu- ets (i.e., coherent states) achieve equality. If ~ is tation relations small, then p and q can both be nearly definite in the initial state ψ(0) . A Hamiltonian expressed [Jx, Jy]= iJz [Jy, Jz]= iJx [Jz, Jx]= iJy. | i in terms of p and q is also nearly definite and the Angular momentum has the same physical units as quantum dynamics of the system preserves near def- Planck’s constant, so for example [Jx, Jy]= i~Jz in initeness at least for a while. In conclusion, classical units. Since SU(2) is compact, every unitary repre- Hamiltonian dynamics is a valid short-term approx- sentation decomposes as an orthogonal direct sum imation to quantum Hamiltonian dynamics. of finite-dimensional irreducible representations (irreps). By Cartan-Weyl theory, it has a unique irrep of dimension n +1 for every integer n 0, which we Exercises ≥ will call Vj with j = n/2. It is also called a spin-j system. The standard basis is 1. Let ψ be a state with respect to which | i j , 1 j ,..., j , ~2 |− i | − i | i V [q]V [p]= . 4 where m is an eigenstate of Jz with eigenvalue m. The proof| i that the irreps of SU(2) have this struc- Show that ψ is pure and has the form | i ture uses the same ladder operator method as in the 2 harmonic oscillator system. In this case the ladder ea(x b+ic) /2 ψ(x)= − operators are (πa)1/4 + J = Jx + iJy J − = Jx iJy. for some real constants a, b, and c. (The con- − verse of this exercise is also worthwhile and is By convention their action on the standard basis is much easier.) J + m = (j m)(j + m + 1) m +1 | i − | i J − m = p(j + m)(j m + 1) m 1 . 2.3. Symmetry and spin | i − | − i Another importantp operator is If a physical system can be rotated in space, its state space becomes a representation of the Lie J 2 = J 2 + J 2 + J 2, H x y z 30

which in representation theory is called a Casimir of a Lie algebra on a tensor product. I.e., the action operator. Its sole eigenvalue on the spin-j system is of J~ on j(j + 1). V V V The term “spin” often refers to the intrinsic an- j1 ⊗ j2 ⊗···⊗ jn gular momentum of a particle. Electrons, protons, is given by 1 neutrons, and neutrinos all have spin 2 , while the n n (k) k 1 N k cobalt-60 nucleus, for example, has spin 5. The spin- J~ = J~ = I⊗ − J I⊗ − . 1 ⊗ ~v ⊗ 2 spin states are also written and , and they k=1 k=1 are also called right-handed spin|↑iand left-handed|↓i spin X X when measured in the direction of the particle’s motion. A photon is a spin-1 particle, but the spin state Exercises 0 never occurs in the direction of motion. (This is |onlyi possible because of special relativity.) 2.4. Identical particles Although some of the most interesting behavior of a particle is due to its spin, it is worth remembering Following Section 1.1, two coherent trajectories that most of its state is in its position. In general if of a physical system obey quantum superposition if a particle has spin-j, its total state space (in a flat they arrive at the same state and classical super- universe) is V L2(R3). j ⊗ position if they arrive at different states. Thus in There are two ways that Vj arises extrinsically. quantum mechanics there is a difference between a 2 First, if a particle lies on the unit 2-sphere S , or if pair of particles (or any other two physical systems) R3 it exists in but its radial state is an independent that are logically identical and a pair that merely factor, then it possesses orbital angular momentum. appear the same. If they are identical, then a co- (Orbital angular momentum operators are often de- herent beam apparatus that conditionally switches + noted Lz,L ,..., but we will stay with J.) The them, rotation group SO(3) acts on S2 and by extension L2(S2). It decomposes as (picture) exhibits quantum interference. By this test and its L2(S2) = V V V ..., ∼ 0 ⊕ 1 ⊕ 2 ⊕ consequences, many classes of particles are known to be identical. All electrons are identical, all helium where a vector in the summand Vj is a spherical harmonic of degree j. If the particle moves freely on nuclei are identical, etc. S2, then its Hamiltonian is More explicitly, suppose that two particles have the same state space . Then by the independence H 2 ∆ rule, their joint state space is ⊗ . This state space H = . H − 2 has an operator X that switches the two factors: X( ψ ψ )= ψ ψ . The Laplacian ∆ equals the action of the Casimir | 1i ⊗ | 2i | 2i ⊗ | 1i operator J 2 on the representation L2(S2). Thus the Since X is a Hermitian involution, it has two j(j+1) 2 spectrum of H is 2 , and the jth eigenvalue eigenspaces. The space S of symmetric tensors (numbered from 0){ has multiplicity} 2j+1. Especially has eigenvalue 1, while theH space of antisymmetric in atomic physics, the first several harmonic spaces tensors Λ2 has eigenvalue 1. If the two particles are often denoted by letters: s, p, d, f, g, . . . are identical,H then X must fix− the density operator Second, when two or more spin systems are com- ψ ψ , but not necessarily the state vector ψ it- bined, their joint state space decomposes as a direct self.| ih Thus| ψ can lie in either eigenspace of|Xi, so sum of spin systems. The Clebsch-Gordan rule gives particles can| i be identical in two different ways. If the decomposition of two spin systems: X ψ = ψ , | i | i Vj Vk = Vj+k Vj+k 1 V j k . then the particles are called bosons and obey Bose- ⊗ ∼ ⊕ − ⊕···⊕ | − | Einstein statistics. If Thus a spin-j particle and a spin-k particle can to- X ψ = ψ , gether be in a spin-ℓ state with | i −| i then the particles are called fermions and obey ℓ j k ,...,j + k . ∈ {| − | } Fermi-Dirac statistics. More generally, given n particles with the same state space , if they are iden- Higher tensor products also decompose, of course, tical bosons, then their joint stateH space is the sym- but in a more complicated way. The main rule to metric space remember is that the total angular momentum oper- n n ators match the Leibniz rule for the diagonal action S ⊗ . H ⊆ H 31

If they are identical fermions, then their joint state 3. COMPUTATION space is the antisymmetric space

n n Quantum computation is a computational model Λ ⊗ . H ⊆ H based on quantum probability as described in Chap- ter 1. It was first proposed by Feynman [? ] that Here is a completely explicit description in terms artificial quantum systems — quantum computers of amplitudes. If n particles each have state space — could be used to simulate natural quantum sys- with a basis A, then a joint pure state can be writtenH tems (Chapter 2). A sequence of further ideas [? ? ? ] culminated in Shor’s discovery of polynomial- ψ = α a ,a ,...,a . | i a1,a2,...,an | 1 2 ni time quantum to factor integers [? ]. It is now X reasonable to conjecture that quantum computation If the particles are identical bosons, then is sometimes exponentially faster than determinis- α = α tic (or classically randomized) computation. This a1,a2,...,an aσ(1),aσ(2) ,...,aσ(n) possibility is as almost as surprising as quantum probability itself, and it would give a new reason for any permutation σ Sn. If they are identical fermions, then ∈ that quantum probability does not reduce to classical probability (Section 1.6). σ Quantum algorithms to simulate many natural αa1,a2,...,an = ( 1) αaσ(1) ,aσ(2),...,aσ(n) . − Hamiltonians [? ] close the circle with Feynman’s In this case, any amplitude α with a repeated index proposal. They indicate that quantum probability, must vanish, so identical fermions cannot appear in and not any further features of quantum mechanics, the same state. This conclusion is called the Pauli provides the likely acceleration of quantum comput- exclusion principle. ers. Indeed, since experimental quantum computation is very difficult in practice, we can say that the Example 2.4.1. Let = L2(S1) and let H realistic physics that we might exploit for quantum computation is both a friend and a foe. sin(θ1 θ2) ψ(θ1,θ2)= − Since quantum computation depends on continu- √2π ous state, it is fair ask whether it unrealistically ex- ploits analog precision. This is related to the prob- be a joint wave function for two particles on the lem that quantum computation is more fragile than circle S1. It is antisymmetric and can describe iden- classical computation and must be protected from tical fermions, by the natural continuous analogue decoherence. These issues are addressed by the the- of amplitude antisymmetry. At the same time, the ory of quantum error correction and fault-tolerant state is invariant under rotation of the circle, so the computation Section 3.6. Although theory leans to measured value of θ has the uniform distribution. 1 the conclusion that quantum computation is possible, useful quantum computers have not yet been 2.5. Atomic structure built. There could conceivably be some unknown practical barrier to quantum computation. Quantum secrecy is a parallel development [? ] In this section we will draw together the ideas of which, unlike quantum computation, has been con- Sections 2.1, 2.3, and 2.4 to analyze the hydrogen vincingly demonstrated [? ]. The idea is to use atom. We will also qualitatively predict some of the the violation of Bell’s inequalities and related phe- features of other atoms and molecules. nomena to detect eavesdropping in communication, rather than to defy the eavesdropper’s computa- 2.6. Quantum field theory tional power. Since quantum secrecy assumes that eavesdroppers have unlimited computational power, it is more trustworthy than classical cryptography, but it is also more limited. Regardless, the first practical use of quantum computers could be as repeaters for quantum secrecy protocols.

3.1. Computational models

To put quantum probability into a computational form, we begin with the qubit, which has a 2- 32

dimensional state space C2 with standard basis 0 program answers a question in NP if at least and 1 . The memory of a quantum computer could| i one leaf of the computation tree says YES n consist| i of n qubits with state space = C2 . The when the true answer is YES, but all leaves general pure state is given by a unitH vector in , say NO when the true answer is NO, and if while the general state is a matrix in the Bloch re-H the tree has polynomial depth. gion 2n 2n . A computation could consist of a sequenceB ⊂ of M2-qubit, unitary gates applied to this 3. BPP is probabilistic polynomial time with n 2n-dimensional state space C2 , followed by a mea- bounded error. It can be defined using the surement. By definition such a gate is a unitary same non-deterministic model as NP, except operator U : C4 C4 applied to some pair of the that each conditional is assigned a computed qubits. More generally,→ a (j, k)-qubit gate is a quan- probability. The program answers a question in BPP if a random computation path provides tum operation 2 the true answer with probability at least 3 . : j k . E M2 →M2 4. BQP is the quantum analogue of BPP. A problem is in BQP if it is solved by a polyno- The general quantum circuit is a sequence of such mial quantum circuit. As in other circuit quantum operations applied to j-tuples of qubits in models, the circuit must be efficiently precom- a memory with n qubits. There is usually a uniform puted (say with a classical computer) using bound on the gates such as j, k 2. If j = k, the length of the input. The final stage is a then n changes by k j as the operation≤ is applied.6 boolean-valued measurement, which must be As in classical Boolean− circuitry, a quantum circuit correct with probability 2 as in the definition is equivalent to an acyclic digraph with each node 3 of BPP. labelled by a quantum operation. 5. PSPACE is deterministic polynomial space Example 3.1.1. with no restriction on computation time. It is Before considering quantum algorithms in this equivalent to a non-deterministic polynomial- model, we can compare qubits to randomized bits. time computation model in which each node of Since a state of n qubits is a quantum superposition the computation tree is assigned an arbitrary of the 2n basis states, or a mixture of these, quan- binary function of its child nodes, and the fi- tum computation is a form of parallel computation. nal answer is the boolean value assigned to the Since the coefficients are complex numbers, it is also root. It is also equivalent to polynomial-time a form of analog computation. However, randomized parallel computation with exponentially many classical computation shares both of these features. networked processors. Indeed, the simplex ∆4n of states of 2n classical bits A complexity class can also be modified by an or- has the same dimension as the Bloch region n of B2 acle, which is a black-box function that a program n qubits. Qubits are more useful than randomized can invoke in one step, or that can be used as a large bits because they allow more operations, not because gate in a circuit. they carry more state. (But see Section 3.5.) The inclusions We can also place quantum computation in the context of standard complexity classes. To review, BPP BQP PSPACE. ⊆ ⊆ a complexity class is a set of computational deci- are elementary. BPP can also be defined by cir- sion problems (YES-NO-valued functions on the set of finite bit strings) that can be answered with spe- cuits of stochastic maps, which places it inside BQP. For the other inclusion, any quantum or stochastic cific computational resources. The computational resources defining a particular complexity class may circuit can be evaluated by path summation (Sec- tion ??), which requires only as much memory as or may not be realistic. Here are some standard complexity classes with the most natural quantum the number of nodes in the circuit. In fact quantum amplitudes can be estimated in a space-efficient class included: way with stochastic sampling. Thus quantum ran- 1. P is the set of problems that can be solved in domness only saves time and not space over classical deterministic polynomial time. randomness. It is reasonable to conjecture that P = BPP, 2. NP is problems that can be solved in non- that BQP is somewhat larger than BPP, and that deterministic polynomial time. A non- PSPACE is vastly larger than BQP. It is also rea- deterministic computer program is one with sonable to conjecture that BQP does not contain blank conditionals. Its execution history is a NP. For one reason, it is easy to find an oracle A tree rather a sequence. A non-deterministic such that BQPA does not contain NPA. 33

3.2. Low-level operations If and ′ are two quantum operations with the F F same domain and target, their distance d( , ′) is F F A first step in constructing quantum algorithms defined as the Lipschitz constant of their difference ′ with respect to trace distance on states. is to identify a convenient set of universal quantum F − F gates. At the unitary level, it is convenient to in- This distance behaves predictably with respect to clude multiplication by a global phase even though composition and tensor products. Its relevance is it is irrelevant. The group of unitary operators on a that each step in a quantum algorithm only needs single qubit is then U(2). Some important examples to be approximated to within a tolerance. Among of single-qubit (or unary) gates are the NOT gate other uses, approximate computation is needed for X, the Hadamard gate H, and the phase rotation the fault-tolerance problem in Section 3.6. Z(θ):

0 1 1 1 1 3.3. Dilations and Grover’s algorithm X = H = 1 0! √2 1 1! − It is useful to explicitly construct unitary dilations 1 0 of classical algorithms. Suppose for simplicity that Z(θ)= iθ f is boolean-valued with classical input s and that it 0 e ! is computed by a sequence of classical gates. Then The gate Z = Z(π) is a phase flip. The phase sub- we can dilate the computation one gate at a time. group Z(θ) and the Hadamard gate H generate For example, the Toffoli gate is a (3, 3)-qubit gate U(2). { } defined by If U U(2) is unitary, then there is a correspond- ∈ T a,b,c = a,b,c + ab ing coherently controlled extension C(U) of U to 2 | i | i qubits. It applies U to the right qubit when the left with a,b,c Z/2. Here and below we assume the qubit is in the state 1 . As a matrix, ∈ | i abbreviations

n n I 0 a,b = a b a = a ⊗ . C(U)= . | i | i ⊗ | i | i | i 0 U! The Toﬀoli is a unitary dilation of the classical AND Coherently controlled unitaries applied to both gate applied to a and b. In general a dilation Df of f qubits generate the 2-qubit group U(4), and 2-qubit takes as input s , n scratch qubits initialized to 0 , unitaries applied to arbitrary pairs of qubits in an n- and a receiving| qubiti also initialized to 0 . (Extra| i qubit memory generate its full unitary group U(2n). qubits used in dilation are also called ancillas| i .) The Strictly speaking the unitary group on n qubits output can be written is not full quantum computation, because the lat- n+1 ter also includes decoherent quantum operations. Df s, 0 = s, x(s),f(s) , By Theorem 1.5.1, unitary operators together with | i | i qubit erasure (taking the marginal from n qubits to where x(s) is the scratch work. | i n 1) and qubit creation in the state 0 generate Many quantum algorithms require a more conve- the− entire category of quantum operations.| i nient dilation of f that we can call minimal unitary Furthermore, any quantum computation can be form. It is a unitary operator Uf that preserves the dilated to a unitary computation. If the output is input and adds the output (in Z/2-arithmetic) to an classical, the unitary part is followed by the mea- extra qubit: surement of output qubits. Such a dilation changes computation time by only a constant factor, but it Uf s,b = s,b + f(b) . | i | i can be expensive in space. Nonetheless, many quantum algorithms are unitary with classical pre- and It is implemented as the conjugation postprocessing. Decoherent operations often (but 1 U = D− N D not always) squander the quantum acceleration. f f ◦ ◦ f Quantum operations also need not be exact in or- where trivial tensor factors are suppressed and der to work. Given two states ρ and ρ′, the probability that two measurements might give a diﬀerent N f(s),b = f(s),f(s)+ b . answer is bounded by the trace distance | i | i 1 is a controlled NOT. This trick is also called uncom- d(ρ,ρ′)= ρ′ ρ . 2|| − ||1 puting the function f. 34

A related construction is a phase rotation con- When N is large, trolled by f, denoted Z (θ). It is defined as f 4 1 θ Zf (θ)= D− Z(θ) Df , ≈ π√N f ◦ ◦ where the phase rotation Z(θ) acts on f(s) . It is and +n and s are nearly orthogonal. Thus the | i | i also given by the formula | i full course of Grover’s algorithm brings ψ close to the state s . | i Z (θ) s = eif(s)θ s . | i f | i | i One significant use of these unitary tools is 3.4. Shor’s algorithm Grover’s general quantum search algorithm. The input to this algorithm is a black-box function, or Grover’s algorithm and its variations represent oracle, one of two main families of existing quantum al- f : S 0, 1 , gorithms. The other family consists of algorithms →{ } related to Shor’s algorithm for finding the period of where S is some finite set. The function f comes a periodic function. with the promise that f(s) = 1 for a single s S and The input to Shor’s algorithm is a black-box func- ∈ the task is to find the solution s. Clearly any clas- tion sical algorithm must evaluate f at least N/2 times on average in order to find s. But if the oracle is f : Z S, → available in minimal unitary form Uf , then Grover’s where S is some target set. It comes with the algorithm can find s using O(√N) quantum queries. promise that For simplicity assume that S = [0, 2n) for some n. Grover’s algorithm is a loop that alternates between f(x)= f(x + p) the standard qubit basis, 0 , 1 , and the basis + , . We can either implement{| i | i} this alternation for some period p, and that otherwise f takes dis- {| i |−i} with Hadamard gates, or assume that quantum gates tinct values. The cost of computing f(x) is polyno- are available in both bases. The algorithm requires mial in log x, and the task is to find the period p n qubits and is initialized in the state in polynomial time in log p. Since f is given as an oracle, this is classically impossible just from count- ψ = +n . | i | i ing necessary queries (exercise ??). But if the oracle Relative to the standard basis, the qubits are then is available in minimal unitary form, then O(log p) in the constant pure state quantum queries suffice. (Technically O(log p) classical queries suffices, but they must be very large 1 queries.) S = s . | i S | i The main computational step of Shor’s algorithm s S | | X∈ is the quantum Fourier transform on a cyclic group Note that S can be defined for any finite orthonor- Z/N. This is a unitary operator FN defined by mal set of| statesi of any quantum system. π√N 1 xy Grover’s algorithm then consists of itera- FN x = ω y 4 | i √ | i tions of the following two steps, with ψ⌊ as the⌋ state N | i X of the qubits at each step. with x, y Z/N and ∈ 1. In the standard basis, apply the phase flip ω = e2πi/N . Zf = Zf (π) controlled by f. This reflects the vector ψ in the hyperplane perpendicular to | i This map has the same formula as the discrete s . Fourier transform and the algorithm for it when | i n 2. By a similar computation in the + , ba- N = 2 is very similar to the classical FFT. But {| i |−i} the interpretation is very different, because the FFT sis, reflect φ through the hyperplane perpen- n dicular to |+in and formally negate ψ . transforms a list of 2 stored numbers, while the | i | i QFT transforms the amplitudes of n stored qubits. The effect of both reflections together is to rotate (See Exercise ??.) ψ from +n to s by an angle θ given by To compute F2n , first express the residue x as a | i | i | i high bit plus a low remainder, n 1 sin θ = + s = . n 1 h | i √ x =2 − xn 1 + x′, N − 35

n 1 with 0 xn 1 1 and 0 x′ < 2 − . Second, and ≤ − ≤ ≤ use the remainder x′ to control a unitary operator applied to the high qubit xq : p 1 | i 1 − kN kN FN (ρ)= . ′ p | p ih p | 1 xn−1 x k=0 U(x′) xn 1 = 0 + ( 1) ω 1 X | − i √2 | i − | i 0 + ωx 1 The measurement of a few copies of this state in the = | i | i √2 standard basis reveals N/p and therefore p. To use Shor’s algorithm to factor a number M, This operator can be expressed as deﬁne the periodic function xn 2π xn 3π x0π − − U(x′)= Z( )Z( ) ...Z( n 1 )H, 2 4 2 − f(n)= an

where xk is the kth bit of x. In words, U(x′) is a Hadamard gate followed by a sequence of phase rota- for some prime residue a Z/M. It can be com- tions controlled by each bit of x′ separately. To ob- puted quickly for any fixed∈n by repeated squaring. tain Fn, recursively apply the QFT operator Fn 1 to Z − Its period divides the exponent of ( /M)× and once x′ and rechristen the high qubit with input xn 1 | i | − i this exponent is known it is easy to factor M. as the low qubit with output y0 . 3.5. Feasibility Shor’s algorithm is as follows.| i First, choose a number N ... Prepare an integer in the constant pure state [0,N) , where Like a bit, a qubit can in principle be any 2-state | i physical system. The operational difference is that [0,N)= 0,...,N 1 . a randomized bit only needs to be suitably indepen- { − } dent from other memory bits, while a qubit state de- Second, supply this state to the minimal unitary coheres if it is leaked to any physical observer, even form Uf and discard the output. This step partially an accidental one. A randomized computation can measures the input according to the value of f(x), proceed as intended even if its entire state is moni- which only depends on the residue of x mod p. The tored by the programmer; a quantum computation result ρ is a mixture of the corresponding residue can be ruined if one qubit is witnessed by a stray classes within [0,N). atom. The final step of Shor’s algorithm is to reveal the residual coherence of ρ with a quantum Fourier transform on Z/N = [0,N). To understand its effect, first fictitiously suppose that p divides N. (This is of course infeasible given that p is not known.) In 3.6. Error correction this case

p 1 3.7. Quantum secrecy 1 − ρ = k + pZ/(N/p) k + pZ/(N/p) p | ih | Xk=0

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