3.4 Measurements
The measurement of one or more particles in a quantum system results in a projection of the state of the system prior to measurement onto the subspace of the state space compatible with the measured values. The amplitude of the projection is rescaled to make sure that the resulting state vector has length one. The probability that the result of the measurement is a given value is the sum of the squares of the absolute values of the amplitudes of all components compatible with that value of the measurement.
A simple example of measurement in a two qubit system will illustrate the above points. Let’s fix the basis 
, 
, and assume that all measurements of individual qubits will be done with respect to this basis. An arbitrary state of a two qubit system can be written as
where 
and 
are complex numbers such that
When the first qubit is measured, then the probability that the result is 
is 
. Assume now that the measurement gives the first qubit exactly that value, that is, . Consequently, the state is projected onto the subspace compatible with the measurement which is the subspace spanned by 
and 
and the result of this projection is 
. Renormalizing we get:
In general, consider a system containing 
qubits 
. Any state 
of the system can be expressed as
where
When the first qubit is measured with respect to the basis 
, then the result is obtained with probability
After rescaling, the new state obtained after the measurement is
Similarly, the measurement gives the outcome 
with the probability
and the state changes correspondingly.
There are many speculations about the “collapse of the wave function (state)" due to an irreversible interaction of the microphysical quantum system with the macroscopic measurement apparatus. Some authors (see, for example, Greenberg and YaSin _[
65
] or Herzog, Kwiat, Weinfuter and Zeilinger _[
70
]) have argued that it is, in fact, possible to reconstruct the state of the physical system before the measurement, that is, to “reverse the collapse of the wave function" if the process of measurement is reversible. After “reconstruction" no information about the measurement is left.
The act of measurement gives another perspective about entangled particles. Particles are not entangled if the measurement of one has no effect on the other. For instance, the state
is entangled since the probability that the first bit is measured to be is 
if _the second bit has not been measured. However, if the second bit had been measured, then the probability that the first bit is measured as is different from , it is either 1 or 0, depending on whether the second bit was measured as or , respectively. Hence, the probability of measuring the first bit has been changed by the measurement of the second bit.
In contrast, the state
is not entangled. Reason: any measure of the first qubit will produce the result independently whether a measurement is performed or not on the second qubit, and the second qubit has probability 
to be measured to regardless of whether the first qubit was measured or not.
In a sense, entangled states can be equivalently presented in mathematical terms (they cannot be represented as a tensor product of two states) or in physical terms (the measurement on one affects the other); however, the physical meaning is richer than the mathematical formalism.
An important consequence of the existence of entangled states is the fact that if a quantum memory register exists in an entangled state, one can change the state of one part of the register simply by measuring another part of it. This is a unique feature of quantum physics
which has no parallel in classical physics. Entanglement is one of the most important features which distinguishes quantum from classical computing.
Which is crucial in many quantum algorithms, teleportation, information transmission, etc.
