Information Theory

Dan C. Marinescu's Classical and quantum information PDF

By Dan C. Marinescu

ISBN-10: 0123838746

ISBN-13: 9780123838742

A new self-discipline, Quantum details technology, has emerged within the final 20 years of the 20th century on the intersection of Physics, arithmetic, and desktop technology. Quantum info Processing is an software of Quantum details technological know-how which covers the transformation, garage, and transmission of quantum info; it represents a innovative method of info processing.

This ebook covers themes in quantum computing, quantum details conception, and quantum mistakes correction, 3 very important components of quantum details processing.

Quantum info thought and quantum errors correction construct at the scope, techniques, technique, and strategies built within the context in their shut relations, classical info idea and classical blunders correcting codes.

Presents contemporary leads to quantum computing, quantum details idea, and quantum blunders correcting codes.

Covers either classical and quantum details conception and mistake correcting codes.

The final bankruptcy of the publication covers actual implementation of quantum details processing devices.

Covers the mathematical formalism and the ideas in Quantum Mechanics severe for knowing the houses and the changes of quantum information.

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Extra info for Classical and quantum information

Example text

A unitary transformation U is linear and preserves the inner product. The spontaneous evolution of an unobserved quantum system with the density matrix, ρ, is ρ → UρU† , with U† , the adjoint of U. 4. Given two independently prepared quantum systems, Q described in Hn and S described in Hm , the bipartite system consisting of both Q and S is described, in a Hilbert space, Hn ⊗ Hm , the tensor product of the two Hilbert spaces. 24 CHAPTER 1 Preliminaries 5. A measurement of the quantum system Q in the state | ψ described in Hn corresponds to a resolution of Hn to orthogonal subspaces, {Hj }, and a projection of the system’s state to these subspaces, {Pj }, such that the sum of the projections is Pj = 1.

Rotate it, or let it evolve for some time t), the system evolves to a different state, | ϕ . , the interaction of an atom with a magnetic field) is represented in our formalism by an operator. Consider a canonical base, {| 0 , . . , | j , . . , | k , . . , | (n − 1) } ∈ Hn . A Hermitian operator, A = A† , applied to the state | ψ = k αk | k , with αk = k | ψ , produces a new state | ϕ : |ϕ =A|ψ or ϕ | = ψ | A† = ψ | A. It follows that j|ϕ = j|A|ψ = j|A αk | k = k j | A | k αk = k j|A|k k|ψ .

Proof. Let | ψi , 1 ≤ i ≤ n, be an eigenvector of A and λi the corresponding eigenvalue. If the eigenvalues are distinct, λi = λj for i = j, 1 ≤ i, j ≤ n, we construct the orthonormal basis as follows: we select an eigenvector | ψj from a subspace that is orthogonal to the subspace spanned by (| ψ1 , | ψ2 , . . , | ψi ). Then:   λ1 0 0 · · · 0 λ  0 ··· 2      . 0 0 λ · · · 3 A=  .. .  ..  . .  . λn If the eigenvalues, λj , are degenerate (not distinct), then there are many bases of eigenvectors that diagonalize matrix A.

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Classical and quantum information by Dan C. Marinescu

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