xii Notation

A∗

set of all finite words in the alphabet A

E∗

group of characters on the Abelian group E,

i.e., Hom(E, U(1))

z∗

complex conjugate of z

M∗

space of linear functionals on the space M

ξ| bra-vector (p. 56)

|ξ ket-vector (p. 56)

ξ|η inner product

A†

Hermitian adjoint operator

G unitary operator corresponding to the

permutation G (p. 61)

IL identity operator on the space L

ΠM projection (the operator of projecting

onto the subspace M)

TrF A partial trace of the operator A over the

space (tensor factor) F (p. 96)

A · B superoperator ρ → AρB (p. 108)

M⊗n

n-th tensor degree of M

C(a,b,... ) space generated by the vectors a, b, . . .

Λ(U) operator U with quantum control (p. 65)

U[A] application of the operator U to a quantum

register (set of qubits) A (p. 58)

E[A], E(n, k) error spaces (p. 156)

σ (α1,β1,...,αn,βn) basis operators on the space

B⊗n

(p. 162)

SympCode(F, μ) symplectic code (p. 168)

| · | cardinality of a set or modulus of a number

· norm of a vector (p. 71)

or operator norm (p. 71)

·

tr

trace norm (p. 98)

·

♦

superoperator norm (p. 110)

Pr[A] probability of the event A

P (·|·) conditional probability (in various contexts)

P (ρ, M) quantum probability (p. 95)

f(n) = O(g(n)) there exist numbers C and n0

such that f(n) ≤ Cg(n) for all n ≥ n0

f(n) = Ω(g(n)) there exist numbers C and n0

such that f(n) ≥ Cg(n) for all n ≥ n0

f(n) = Θ(g(n)) f(n) = O(g(n)) and f(n) = Ω(g(n)) at the

same time

f(n) = poly(n) means the same as f(n) =

nO(1)

poly(n, m) abbreviation for poly(n + m)