# Characterization of Quantum Entangled States [thesis] - download pdf or read online

By J. Batle-Vallespir

The current Thesis covers the topic of the characterization of entangled states through recourse to entropic measures, in addition to the outline of entanglement regarding a number of concerns in quantum mechanics, resembling the rate of a quantum evolution or the connections present among quantum entanglement and quantum section transitions.

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10) is Fourier transformed producing the state 1 |ψ3 = √ q q−1 q−1 ei 2πam q m=0 a=0 |m, xa modN . 11) Now a measurement is performed on the arguments, obtaining m = c, xa = xk for 0 < k < r. The probability of this particular outcome is given by 1 P (c, x ) = √ q q−1 ei k 2πac q . 12) a=0,xa =xk modN This probability is periodic in c with period q/r, being sharply peaked at c = pq/r for some integer p. After few trials, one obtains the period r probabilistically. The classical algorithm for checking whether a given number is a factor of N is a faster one, so it is not a big deal to multiply large integers.

Computation as such is then understood as a sequence of repeated unitary transformations, and the time of computation of each one of those is a multiple of the finite time T that is necessary to perform a logical action. For instance, let us consider the evaluation of a function f (x) at N values. To do so, we encode the numbers into states 1 As f: x → f(x) 0 (|0 ) 1 (|1 ) → → ... → f (0) (|f (0) ) f (1) (|f (1) ) N − 1 (|N − 1 ) f (N − 1) (|f (N − 1) ). 1) we have seen, it was also Einstein’s (with Podolsky and Rosen) insight into the possible incompleteness of quantum mechanics who triggered Schr¨ odinger’s fundamental response about entanglement, starting the whole thing out.

This algorithm is discussed in detail in Chapter 13 in connection with entanglement. A more drastic improvement over a classical algorithm (from exponential to shortened to polynomial time) due to quantum mechanics is given by Shor’s algorithm for factorizing large integers. A strong incentive for attempts to develop practical quantum computers arises from their possible use in the speed-up of factoring very large numbers for cryptographic purposes (see Chapter 3). While the best classical algorithm known to date requires of the order of 2/3 1/3 e(lnL) L steps to factorize a L-digit number [72], Shor’s requires only of the order of L3 steps.

### Characterization of Quantum Entangled States [thesis] by J. Batle-Vallespir

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