Survival Guide for Quantum Computing a Gentle Introduction VI: MultiQubit systems again
Measurement in multiQubit systems is just an extension of the Quantum Mechanical Axiom of what happens in a single Qubit system. Here it is again
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* > } form 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/ ).
Switch now to an n-Qubit system with its associated vector space (call it V) and its 2^n basis elements (which by definition are linearly independent of each other). If you’re foggy on linear independence have a look at https://luysii.wordpress.com/2010/01/11/linear-algebra-survival-guide-for-quantum-mechanics-iiiqrtq/
Each basis element is capable of forming a vector space of its own when it is multiplied by a scalar. Such a vector space is called a subspace (of the parent vector space V). So it is possible that there are as many as 2^n subspaces of V. But you can group several basis elements together to form different subspaces of V. By dividing all of the 2^n basis elements of V into groups, you can form k subspaces which completely cover V (k can range from 1 to 2^n.
Here comes the Quantum Mechanical Axiom for measurement of an n-Qubit system. To repeat Reiffel — “It is not derivable from other physical principles’ Rather it is derived from empirical observation of experiments with measuring devices.”
Any device measuring an n-Qubit system has an associated 2^n dimensional vector space associated with it. The same device has a decomposition into k subspaces called Si (Reiffel mixes up the device with the vector space). The mathematical term for this is direct sum decomposition and is usually notated by a plus sign with a circle around it — I’ll use + (bold plus).
V = S1 + . . . . + Sk.
The the number k corresponds to the maximum number of possible measurement outcomes for states of the 2^n system measured with that device. k can be different for different devices measuring the same 2^n system.
The internal workings of the measurement device are not given, just how it acts on the n-Qubit system. Measurement is one of central mysteries of quantum mechanics, but the following description of how measurement acts works perfectly and has never made an incorrect prediction since the axioms were laid down 100 years ago.
Given:
|Psi> the vector representing a quantum state.
|Psi_i> (read Psi sub i) is a unit vector in Si (the vector space decomposition mentioned above).
a_i (read a sub i) is a real number great than or equal to 0.
|Psi > = a_1 | Psi_1> + . . . + a_k| Psi_k >
First: When | Psi > is measured the state | Psi_i > is obtained with probability |a_i|^2 (although Reiffel doesn’t say, the a_i’s must be normalized so the sum of their squares equals 1 (as probabilities must vary between 0 and 1.
Second: After measurement, the state of | Psi > is now a unit vector in just one of the Si’s. There is no way to go back to the original | Psi >, it has been destroyed.
Paradoxical no? But this is the way the quantum world (and ultimately our world) works. Deal with it.
This will be the last post on Quantum Computation until I can figure out a way to get quantum circuit diagrams into the Word Press classic editor. Any ideas?
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