Author: workhouse123
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Distinguishing pure and mixed states
A given density operator could represent a pure or a mixed state. If the system is pure, then the state is a ray in Hilbert space, and the density operator can be expressed as ρ = |ψihψ|, for some |ψi. Such a density matrix satisfies ρ 2 = |ψihψ|ψihψ| = ρ, (5.17) Tr(ρ 2 )…
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Properties of the density operator
The density operator on a Hilbert space, defined by Equation 5.4 satisfies the following properties: 1. ˆρ is Hermitian. Proof: ˆρ † = X n p ∗ n |ψ n i † hψ n | † = X n p n |ψ n ihψ n | = ˆρ. (5.11) 2. ˆρ is non-negative, that is,…
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The Density Operator
By “state” of a system, we mean a collection of all possible knowledge we can gather about the system, which is practically achieved by studying the distribution of outcomes of measurements made on the system. In the case of pure states, these outcomes together are described by a ray in Hilbert space. Consider measuring an…
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Introduction
The formalism for quantum systems developed so far applies to what are called pure states. A system in a pure state is completely specified by the state vector. A complete set of experimental tests will determine the system state fully: we have maximal knowledge of the system. For example, for a spin system, we can…
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Bell’s inequalities and non-locality
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…
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The EPR paradox
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…
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Quantum vs. classical correlations
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…
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Introduction to Quantum Physics and Information Processing
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…
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Entanglement
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…
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Distinguishability of Qubit States
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…