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Cryptography in a quantum world
Wehner, S.D.C.
Publication date 2008
Link to publication
Citation for published version (APA):
Wehner, S. D. C. (2008). Cryptography in a quantum world.
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Symbols
Symbol Page
log binary logarithm
ln natural logarithm
a∗ complex conjugate of a
|a| absolute value of a
|S| number of elements of the set S N set of natural numbers 1, 2, 3, . . .
R set of real numbers
C set of complex numbers
[n] set of numbers {1, . . . , n}
x|S string x restricted to the indices in S
δij δij = 1 if i = j and δij = 0 otherwise
I identity matrix
Id d × d identity matrix
A[1] matrix A acting on subsystem 1
A−1 inverse of the matrix A
AT transpose of the matrix A
A∗ conjugate of the matrix A
A† conjugate transpose of the matrix A
[aij] matrix whose entry in the i-th row and j-th column is aij
Aij the entry in the i-th row and j-th column of the matrix A
Tr(A) trace of A = [aij] given by
jajj
rank(A) rank of the matrix A
A > 0 A is positive definite 189
A ≥ 0 A is positive semidefinite 189
||A||1 trace norm of A, given by Tr
√ A†A a real vector a = (a1, . . . , ad)
|Ψ complex vector|Ψ = (α1, . . . , αd)
Ψ| conjugate transpose of the vector |Ψ 249
250 Symbols
|xb string x encoded in basis b
Ψ|Φ inner product of|Ψ and |Φ
x · y standard inner product of real vectors x and y
|ΨΦ| outer product of|Ψ and |Φ
|ΨΨ| projector onto the vector |Ψ
|||Ψ|| 2-norm given byΨ|Ψ
H a Hilbert space
B(H) set of all bounded operators onH
S(H) set of states on H [A, B] commutator AB − BA
{A, B} anti-commutator AB + BA
Comm(A ) commutant of the algebra A 198
ZA center of the algebra A 198
A1, . . . , An algebra generated by A1, . . . , An 194
S algebra generated by operators from the setS 194
D(ρ, σ) trace distance of ρ and σ 32
F (ρ, σ) fidelity of ρ and σ 35
d(X|ρ) distance from uniform of r.v. X given state ρ 165
h(p) binary entropy 36
H(X, Y ) joint entropy of X and Y 36
H(X|Y ) conditional entropy of X given Y 36
I(X, Y ) mutual information of X and Y 36
Ic(ρAB) classical mutual information of ρAB 36
Iacc(E) accessible information of an ensemble E 38
S(ρ) von Neumann entropy of the state ρ 37
χ(ρ) Holevo quantity 38
H∞(X) min-entropy 37
H2(X) collision entropy 37
