A Mathematical elements The language of quantum mechanics is rooted in linear algebra and functional analysis. It often expands when the foundations of the theory are reassessed, which they periodically have been. As a result, additional mathematical struc- tures not ordinarily covered in physics textbooks, such as the logic of linear subspaces and Liouville space, have come to play a role in quantum mechan- ics. Information theory makes use of statistical quantities, some of which are covered in the main text, as well as discrete mathematics, which are also not included in standard treatments of quantum theory. Here, before describ- ing the fundamental mathematical structure of quantum mechanics, Hilbert space, some basic mathematics of binary arithmetic, finite fields and random variables is given. After a presentation of the basic elements of Hilbert space theory, the Dirac notation typically used in the literature of quantum informa- tion science, and operators and transformations related to it, some elements of quantum probability and quantum logic are also described. A.1 Boolean algebra and Galois fields n A Boolean algebra Bn is an algebraic structure given by the collection of 2 subsets of the set I = {1, 2,...,n} and three operations under which it is closed: the two binary operations of union (∨) and intersection (∧), and a unary operation, complementation (¬). In addition to there being comple- ments (and hence the null set ∅ being an element), the following axioms hold. (i) Commutativity: S ∨ T = T ∨ S and S ∧ T = T ∧ S; (ii) Associativity: S ∨(T ∨U)=(S ∨T )∨U and S ∧(T ∧U)=(S ∧T )∧U; (iii) Distributivity: S ∧ (T ∨ U)=(S ∧ T ) ∨ (S ∧ U)andS ∨ (T ∧ U)= (S ∨ T ) ∧ (S ∨ U); (iv) ¬∅ = I, ¬I = ∅,S∧¬S = ∅,S∨¬S = I, ¬(¬S)=S , for all its elements S, T, U. The algebra B1 is the propositional calculus arising from the set I = {1}, which is also used in digital circuit theory, where ∅ 232 A Mathematical elements corresponds to FALSE, I to TRUE, ∨ to OR, ∧ to AND, and ¬ to NOT. The “basic” logic operation XOR corresponds to (S ∨ T ) ∧ (¬S ∨¬T ). A field is a set of elements, with two operations (called multiplication and addition) under which it is closed, that satisfies the axioms of associativity, commutativity and distributivity and in which there exist additive and mul- tiplicative identity elements and inverses. The order of a field is the number of its elements. Theorem (Galois): There exists a field of order q if and only if q is a prime power, that is, q = pn where p is prime and n is a positive integer; furthermore, if q is a prime power, then there is (up to a relabeling), only one field of that order. A field of order q is known as a Galois field and is denoted GF (q). If p is prime, then GF (p) is simply the set {0, 1, ..., p−1} with n mod-p arithmetic. GF (p ) is a linear space of dimension n over Zp (the set of integers 0 to p − 1) containing a copy of it as a subfield. For Z2, GF (2) is such a field over the set {0, 1} with XOR as addition and AND as multiplication. In computing, one is primarily interested in functions f : {0, 1}m →{0, 1}n; for example, two-bit logic gates g : GF (2) → GF (2). A.2 Random variables A random variable is a deterministic function from a given sample space S, that is, the set of all possible outcomes of a given experiment, the subsets of which are known as events, those containing only a single element being the elementary events, to the real numbers. The expectation value, E[Y (X)], of a function Y (X) of a random variable X is given by a linear operator such that n E[Y (X)] = Y (xi)P (X = xi) , (A.1) i=1 +∞ E[Y (X)] = Y (x)f(x)dx , (A.2) −∞ in the cases of discrete and continuous variables, respectively, where in the former P (X = xi)istheprobability andinthelatterf(x)istheprobability density (p.d.f.), that is, a collection of numbers between 0 and 1 summing to unity. For a random variable with a (differentiable) cumulative distribution function F (X), f(x) ≡ dF/dx; a random variable has the cumulative distri- bution function F (X) if the probability of an experiment with sample space S to yield x<X as an outcome is x F (x)=P (X<x)= f(x)dx . (A.3) −∞ A Markov chain is a sequence of random variables X1,X2,... such that a given random variable Xi is independent of all the variables X1,X2,...Xi−1, that is, is memoryless. A.3 Vector Spaces and Hilbert space 233 A.3 Vector Spaces and Hilbert space The central mathematical structure of quantum mechanics is Hilbert space, which is a specific sort of vector space. The Dirac notation commonly used to describe Hilbert-space calculations in the physics literature is introduced below, after this structure is described in standard mathematical notation. A vector space V over a field F is a set of vectors together with operations of scalar-multiplication and addition satisfying the following. For all elements a, b ∈ F and u, v, w ∈ V (with the zero vector denoted 0), scalar multiples and vector sums are elements of V such that: (i) There is a unique scalar zero element 0 ∈ F such that 0u = 0 for all u; (ii) 0 + u = u; (iii) a(u + v)=au + av and (a + b)u = au + bu;and (iv) u + v = v + u, and u +(v + w)=(u + v)+w. A (scalar) inner product (u, v) is assumed for all pairs u, v ∈ V such that: (i) (u, v)=(v, u)∗,where∗ indicates complex conjugation; (ii) (u, u) ≥ 0, with (u, u) = 0 if and only if u = 0; (iii) (u, v + w)=(u, v)+(u, w); and (iv) (u, av)=a(u, v). The norm of a vector, ||u|| ≡ (u, u), satisfies: (i) ||u|| ≥ 0 for all u ∈ V ,with||u|| = 0 if and only if u = 0; (ii) ||u + v||≤||u|| + ||v||, for all u, v ∈ V ;and (iii) ||au|| = |a|||u||, for all a ∈ F and all u ∈ V . The field F used in quantum mechanics is usually taken to be that of the complex numbers, C. The vectors for which ||u|| =1aretheunit vectors. Vectors u and v are orthogonal (u ⊥ v) if and only if (u, v)=0. A function of two vector arguments that is more general than the inner product, which serves similar purposes but is applicable in broader contexts, is the Hermitian form, h(u, v), which satisfies: (i) h(u, v)=h(v, u)∗; (ii) h(u, av)=ah(u, v), for all a ∈ F , u, v ∈ V ;and (iii) h(u + v, w)=h(u, w)+h(v, w), for all u, v, w ∈ V . A positive Hermitian form is an Hermitian form that also satisfies the con- dition h(u, u) ≥ 0 for every u ∈ V . An Hermitian form is positive-definite if h(u, u) = 0 implies u = 0;apositive-definite Hermitian form is an inner product. A basis for a vector space is a set of mutually orthogonal vectors such that every u ∈ V can be written as a linear combination of its elements; a basis is orthonormal if it is composed entirely of unit vectors. A set S of vectors in V is a subspace of V if S is a vector space in the same sense as V itself is. The one-dimensional subspaces of V are called rays. 234 A Mathematical elements A Hilbert space, H, is a complete complex vector space with an inner product for which (u, u) ≥ 0 for all u ∈H.AmapA : H→H, u → Au is a linear operator if, for all u, v ∈Hand scalars a ∈ F : (i) A(u + v)=Au + Av;and (ii) A(au)=a(Au). If, instead of (ii), one has A(au)=a∗(Au), then A is an anti-linear operator. (Together, the linear and anti-linear operators are fundamental to quantum mechanics; see Wigner’s theorem, below.) A vector space with an inner product, such as a Hilbert space, can be attributed a norm ||·||=(v, v)1/2, which provides a distance via d(v, w)= ||v − w||. Such a vector space is separable if there exists a countable subset in the space that is everywhere dense, that is, for every vector there is an element of the space within a distance of it for every positive real ; the space is complete if every Cauchy sequence—namely, every sequence such that for every >0thereisanumberN( ) such that ||vm − vn|| < if m, n > N( )— has a limit in the space. (For finite-dimensional spaces, one usually considers the norm topology, although weak topologies may be required to define needed limits and to give proper definitions of continuity.) A subspace of a Hilbert space H is a closed linear manifold, that is, a linear manifold containing its limit points; a linear manifold in H is a collection of vectors such that the scalar multiples and sums of all its vectors are in it. A bounded linear operator is a linear transformation L between normed vector spaces V and W such that the ratio of the norm of L(v) to the norm of v is bounded by the same number, for all non-zero vectors v ∈ V .The set of bounded linear operators on a Hilbert space H is designated B(H). The sum of two operators A and B, A + B, is another operator defined by (A + B)v = Av + Bv, for all v ∈H; multiplication of an operator by a scalar a is defined by (aA)v = a(Av), for all v ∈H; multiplication of two operators is defined by (AB)v = A(Bv), for all v ∈H.Thezero operator, O,and the unit operator, I, are defined by Ov = 0 and Iv = v, respectively.