Category: 4. Properties of Qubits

What are the principles of quantum computing?

Bell’s original work, and many subsequent variants show how quantum correlations in an entangled state are essentially different from classical ones. One of the inequalities of Bell applies to a physical system consisting of two subsystems, obeying the principle of local realism. He shows that the quantum statistics for such a system involving entangled subsystems…

A famous 1936 paper by Einstein, Podolsky, and Rosen [31] brought the whole matter to a head. Popularly known as EPR, they examined a thought experiment with entangled particles 1 and concluded that the quantum me- chanical description of nature is incomplete, or else a paradox arises. Niels Bohr countered their claim in a paper…

In what way are the quantum correlations in an entangled quantum state different from correlations in a classical system? If a measurement of a quan- tum state yields a probabilistic outcome, could we not assume that the ob- servable measured has a definite value that was merely uncovered by the measurement? Then the probabilities encoded…

possible to construct higher-dimensional states by taking direct products of lower-dimensional states. However not all higher-dimensional states can be constructed this way. There will always exist states that cannot be expressed as a direct product. Such states are called entangled states. This nomenclature is due to Erwin Schr¨odinger who first discovered the implication of such…

We now discuss in detail one of the most startling and yet most useful aspects of superposition. The most general n-qubit state would be a superposition |ψi n = 2 n −1 X x=0 α x |xi n , X x |α x | 2 = 1, (4.5) where the subscript n is to emphasize…

Classically, the outcomes of decision processes are always distinguishable: it is taken for granted that a tossed coin will land either on heads or on tails and upon looking at it, we can distinguish the different outcomes with cer- tainty. In applications to quantum information processing too, we will usually measure the output state after…

The full specification of a superposition state |ψi = α|0i+ β|1i is given by the complex numbers α and β. The meaning of these numbers is physically derived by making measurements on this state, in the computational basis. This process would randomly “collapse” the state to either |0i or |1i. The probability of obtaining |0i…

A generic qubit could have a non-definite state expressed as a superposition |ψi = α|0i + β|1i, |α| 2 + |β| 2 = 1. How do we picture a qubit? As a vector in Hilbert space, the description is abstract. The 2-d Hilbert space is a space with 4 dimensions. To get a better feel…