Classical and Quantum Information by Dan C. Marinescu

By Dan C. Marinescu

A brand new self-discipline, Quantum details technological know-how, has emerged within the final 20 years of the 20 th century on the intersection of Physics, arithmetic, and computing device technological know-how. Quantum details Processing is an program of Quantum details technological know-how which covers the transformation, garage, and transmission of quantum details; it represents a progressive method of details processing. This e-book covers themes in quantum computing, quantum details thought, and quantum blunders correction, 3 very important components of quantum info processing. Quantum details concept and quantum mistakes correction construct at the scope, recommendations, technique, and methods constructed within the context in their shut family members, classical info thought and classical mistakes correcting codes. provides fresh ends up in quantum computing, quantum details idea, and quantum mistakes correcting codes. Covers either classical and quantum details concept and mistake correcting codes. The final bankruptcy of the booklet covers actual implementation of quantum info processing units. Covers the mathematical formalism and the ideas in Quantum Mechanics severe for figuring out the homes and the alterations of quantum info.

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To check that U is unitary, observe that | Q | and | Q |−1 are self-adjoint hence | Q |† =| Q | . Then we have UU† =| Q |−1 QQ† (| Q |−1 )† =| Q |−1 QQ† | Q |−1 =| Q |−1 | Q || Q || Q |−1 = I. To prove that this decomposition is unique we assume that Q = V1 U1 and Q = V2 U2 . Then, we use the fact that U1 and U2 are unitary, U1 U†1 = U2 U†2 = I, to get: QQ† = [V1 U1 ] [V1 U1 ]† = V1 U1 U1† V1† = V1 V1† = V12 . and QQ† = [V2 U2 ] [V2 U2 ]† = V2 U2 U†2 V2† = V2 V2† = V22 It follows that V1 = V2 and immediately that U1 = U2 .

2 α∗  1 2 n   ρ =| ψ ψ |=  .. ..  .  . .  ∗ ∗ αn α1 αn α2 . . αn αn∗ The density matrix of a pure state is a Hermitian operator ρ† = (| ψ ψ |)† =| ψ ψ |= ρ. The trace of the density matrix of a pure state is equal to one n n αi αi∗ tr(ρ) = | αi |2 = 1. = i=1 i=q A mixed state is a statistical ensemble {(| ψ1 , p1 ), (| ψ2 , p2 ), . . , (| ψn , pn ), } with pi a n probability in the classical sense, i=1 pi = 1, rather than a probability amplitude. The n density matrix of a mixed state i=1 pi | ψi ∈ Hn is defined as n pi | ψi ψi | ρ= i=1 56 where more than one factor pi is greater than zero.

Indeed, there exits an orthonormal base in Hn consisting of eigenvectors of V; with respect to this basis V can be represented as a diagonal matrix, diag (p1 , p2 , . . pn ). Since V is positive then pi ≥ 0, hence they have unambiguously defined positive square roots. Define: Q= √ √ √ √ V to be represented by diag ( p1 , p2 , . . , pn ) . The modulus of any operator V is defined as √ | V |= V† V ≥ 0, √ it is always positive. One can also consider V† V ≥ 0 but, unless V is normal, that is not the same as (VV† ) ≥ 0.

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