Introduction to Quantum Information Theory

Quantum computation and information studies how information is encoded in nature according to the laws of quantum mechanics and what this means for its computational power. In this note, we present a short and rather schematic introduction to quantum information theory by drawing comparisons to classical probability theory. For more details on quantum information theory and computation we refer to [3]. A binary random variable X is a system with two possible states 0 and 1. Similarly, a quantum bit (qubit) is a quantum mechanical system, which can be in a state |0〉, |1〉 or any convex combination of these states. In other words, a quantum bit is a unit vector in a two dimensional Hilbert space a0|0〉 + a1|1〉, where a0, a1 ∈ C and |a0| + |a1| = 1. By tensoring such systems together we can define larger quantum states, for example over log n qubits as |φ〉 = ∑n−1 i=0 ai|i〉,with ∑n−1 i=0 |ai| = 1. A random variable X with probability distribution P = {p0, p1} evolves by multiplying the probability vector by a stochastic matrix S, i.e. a matrix that preserves the `1-norm. The new probability vector is P ′ = S · P . Moreover, a measurement of the random variable has Pr[X = b] = pb, pb ∈ [0, 1]. Let us see how a quantum bit evolves. A quantum bit |φ〉 = a0|0〉 + a1|1〉 can evolve by a unitary matrix U , i.e. a matrix that preserves the `2-norm, and the new state becomes |φ′〉 = U · |φ〉. In addition, we can perform a projective measurement of a state |φ〉 in an orthonormal basis {b1,b2, . . . ,bn} and have Pr[outcome is bi ] = |〈φ|bi〉|. More generally, we can define a mixed quantum state, i.e. a classical probability distribution over quantum states. For example, a mixed state ρ can be in an ensemble of states {|φi〉} with probabilities pi. We can rewrite a mixed state as a hermitian, positive, trace-one matrix, called density matrix ρ = ∑n i=1 pi|φi〉〈φi|. The density matrix contains all necessary information about a quantum state. More precisely, the quantum state ρ evolves by a unitary U to the state ρ′ = UρU† and a projective measurement has Pr[outcome bi ] = ∑ pi|〈φi|bk〉| = ∑ pi〈bk|φi〉〈φi|bk〉 = 〈bk| ( ∑ pi|φi〉〈φi| ) |bk〉 = 〈bk|ρ|bk〉. Let us note that two mixed states may look very different as an ensemble of quantum states, however they might correspond to the same density matrix. For example,