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A Short Course in Quantum Information Theory: An Approach by Ben Q. Li

By Ben Q. Li

This brief and concise primer takes the vantage element of theoretical physics and the team spirit of physics. It units out to strip the burgeoning box of quantum info technological know-how to its fundamentals by means of linking it to common techniques in physics. an intensive lecture instead of a accomplished textbook, this quantity is predicated on classes introduced over a number of years to complicated undergraduate and starting graduate scholars, yet primarily it addresses a person with a operating wisdom of simple quantum physics. Readers will locate those lectures a so much enough access aspect for theoretical stories 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 = .

6, where the computational basis becomes indispensable. To explore the theoretical structure of a qubit, however, the well known physical representation is more convenient, cf. next Section. Lajos Diósi: A Short Course in Quantum Information Theory, Lect. Notes Phys. 2 Pauli representation The mathematical models of all two-state q-systems are isomorphic to each other, regarding the state space and the physical quantities. Also the equations of motion are isomorphic for all closed two-state systems.

4) An obvious parametrization takes normalization and the free choice of the complex phase into the account: θ θ cos |↑ + eiϕ sin |↓ . 5) 2 2 The angular parameters θ, ϕ can be identified with the standard directional angles of a 3-dimensional real unit vector n. This way the above q-state can be parametrized by that unit vector itself: |n = cos θ θ |↑ + eiϕ sin |↓ . 6) All pure states of a 2-dimensional q-system have thus been brought to a one-toone correspondence with the surface points of the 3-dimensional unit-sphere called the Bloch sphere.

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