A short course in quantum information theory an approach by Lajos Diosi

By Lajos Diosi

This brief and concise primer takes the vantage element of theoretical physics and the cohesion of physics. It units out to strip the burgeoning box of quantum details technological know-how to its fundamentals through linking it to common options in physics. an in depth lecture instead of a complete textbook, this quantity is predicated on classes added over a number of years to complex undergraduate and starting graduate scholars, yet primarily it addresses an individual with a operating wisdom of uncomplicated quantum physics. Readers will locate those lectures a so much enough access aspect for theoretical reports during this box.

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Where the real α is called the vector of rotation. To interpret the name, let R(α) denote the orthogonal 3 × 3 matrix of spatial rotation along the direction α, by the angle α = |α|. 18) ˆ (α)σ ˆ † (α) = R(α)σ ˆU ˆ. 19) This feature of rotational covariance makes the quick proof of many structural properties possible in Pauli representation. 4) which corresponds to the pure state |n of a two-state qsystem takes this form: ˆ Iˆ + nσ |n n| = . 20) 2 In the distinguished case of z-up pure state we write: |↑ ↑| = Iˆ + σ ˆz 10 = .

Tr Π 2 =1− w . 24) had equal apriori probabilities. 27) 1+ 2 3 Cf. [16] by Ivanovic. 4 Applications of no-cloning and indistinguishability 47 ˆ ? , when the effect Π √ √ 2 2 2 ˆ ˆ ˆ? 28) where (1, 0, 1) n= √ . 29) 2 Now the ratio of conclusive cases is: 1 − p? = 1 w = 1 − √ ≈ 30% . 30) Hence the optimum non-projective measurement has increased the ratio of correct decisions by cca. 5% with respect to projective measurements. 3 of an unknown q-state seems a shortcoming literally. Yet it becomes a benefit if we take a new point of view, namely, the security of information.

39) while the probability distribution of the measurement outcome reads: pλ = |cλ |2 . 5 Composite systems The state space of the composite q-system, composed from the q-(sub)systems A and B, is the tensor product of the vector spaces of the q-subsystems, cf. 23): HAB = C dA ⊗ C dB = {cλl ; λ = 1, . . , dA ; l = 1, . . , dB } . 41) The pure state of the composite system is described by the normalized state vector of dimension dA dB : |ψAB = {cλl } ; |cλl |2 = 1 . 42) λ,l If the two subsystems are uncorrelated then the state vector of the composite system is a tensor product: 28 4 Foundations of q-physics |ψAB = |ψA ⊗ |ψB = {cAλ cBl } .

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