Survival Guide for Quantum Computing a Gentle Introduction: II Measurement

I certainly don’t intend to repeat the book but certain topics are so crucial to understanding the book, that some elaboration of what Reiffel says are indicated.

Measurement of a quantum state is discussed starting on p. 16. The following is a postulate or axiom of Quantum Mechanics (QM). As Reiffel notes “It is not derivable from other physical principles’ Rather it is derived from empirical observation of experiments with measuring devices.”

Any device that measures a two state quantum system (such as a qubit) must have two preferred states whose representative vectors {| u >, | u* > } for an orthonormal basis for the associated vector space (see the links to survival guide for Linear Algebra in QM — https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/ ).

Unfortunately, Reiffel doesn’t say that these preferred states are fixed in the measuring device, which you’ll need to know when she discusses relative and global phases later on. She describes the wierdness of quantum measurement very well in the example of polaroids given on pp. 9 –> 13. The pieces of polaroid she uses as measuring devices have fixed directions; they let light through or do not. Spoiler alert, your polaroid sunglasses won’t really show this unless you are looking at light which has become polarized by reflection from a smooth surface (such as water).

Now suppose you have a quantum state vector in the standard basis of {| 0 >, | 1 > } e.g. | v > = a | 0 > + b |1>. This must be transformed into the basis of the measuring device (done easily with a 2 x 2 invertible matrix — see the linear algebra survival guide link above).

This gives | v’ > = a’ | u > + b’ | u* >

Now for the rest of the bizarre postulate. (1) Measurement gives you | u > with probability | a’ |^2 or | u* > with probability | b’|^2.

Note that | a’ |^2 + | b’|^2 = 1 (by orthogonality of {| u >, | u* > }

Not only that, but (2) the act of measurement changes | v’ > into | u > or into | u* > meaning that you can never go back and measure the original |v’> again — you’ve destroyed it, e.g. “you can’t go home again”. It’s an experimental fact, but nobody understands just how this happens, although happen it does.

A second measurement of | v’ > will gives you the result you got the first time with 100% probability (probability 1).

So whether a measurement in QM is probabilistic as it was for | v > = a | 0 > + b | 1 > or deterministic as it was for | v’ > = | u > or |v ‘ > = | u* > depends on the basis it was measured in.

So states are superpositions with respect to some bases {| 0 >, | 1 > } and not with respect to others {| u >, | u* > }.

Reiffel closes with the excellent ” Some people are tempted to think of the state | v > = a | 0 > + b |1> as being a probabilistic mixture of |0 > and | 1 >. It is not. In particular, it is not true that the state is really either | 0 > or | 1 > and that we just do not happen to know which. Rather | v > is a definite state, which when measured in certain bases gives deterministic results while in others it gives random results.”

Notice that getting | u > or |u* > back by a measurement just gives you only 1 bit of information, not the infinite amount which could be stored in the complex numbers a and b.

How in the world this leads to efficient parallel computation seems miraculous and something I hope to find out from plowing through the book.

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