Download Observables and Measurements

Observables and Measurements
AQM Lecture 16
(Dated: 15 May 2006)
A.
Measuring observables
In “traditional” quantum mechanics, a property of a system that we can measure is referred to as an observable,
and is represented by a Hermitean operator. Thus, if a system is in a given state (a pure state |φi or a mixed state
ρ), one can determine expectation values and uncertainties in this observable. These properties refer to the expected
outcomes of an ensemble of measurements (i.e., measuring many identically prepared systems).
Sometimes we wish to determine the probabilities of a particular measurement, rather than an ensemble. Such
probabilities are determined using observables as follows. First, recall that every Hermitean operator  has a spectral
decomposition, that is, a set of eigenstates {|ii} and corresponding real eigenvalues ai . (In the following, we consider
only finite-dimensional quantum systems, for simplicity.) Assume that the eigenstates are all non-degenerate (i.e.,
each has a unique eigenvalue). Then the set of eigenstates {|ii} forms a basis for the Hilbert space of the system.
If a system in in a state |ψi and we measure the observable Â, we get the measurement result ai with probability
Pi = |hi|ψi|2 . (If the eigenstates are degenerate, things become a little more complicated, but we won’t concern
ourselves with that situation.)
B.
Measurements in a basis
John von Neumann, in his axiomatization of quantum mechanics, introduced a slightly different way to describe
measurements than using observables. His measurement axiom states that a measurement is described by a basis {|ii}
of the Hilbert space being measured. One can then speak of “measuring in a basis {|ii}.” If a system in in a state
|ψi and we measure in a basis {|ii}, we get the measurement result i with probability Pi = |hi|ψi|2 . Measurements of
this form are commonly referred to as projective measurements or von Neumann measurements.
Note that this way of describing measurements is equivalent to that of observables. To “measure in a basis {|ii}” is
the same as measuring an observable (Hermitean operator) with non-degenerate eigenstates {|ii}. Note that the choice
of the eigenvalues of this observable are arbitrary (as long as they are unique, because it must be non-degenerate).
Thus, for projective measurements, we usually label the measurement outcome with the same label as the basis, i,
rather than using the eigenvalue ai of a corresponding Hermitean operator Â.
C.
Post-measurement state of the system
Sometimes we will be interested in the state of the system after the measurement. Von Neumann defined the
post-measurement state to be as follows. If we measure in a basis {|ii} and obtain the measurement outcome i, the
system is left in (“collapsed to”) the state |ii. This postulate ensures the repeatability of measurements. If a system
is measured in a basis {|ii} and the outcome i is obtained, and then the system is measured again immediately in the
same basis, the outcome i is obtained again with certainty (probability equal to 1).
Measurement postulate (von Neumann): A (projective) measurement is described by a basis for the Hilbert
space. If the system is prepared in a state |ψi, and a measurement is performed in the basis {|ii}, the probability of
obtaining the measurement outcome i is Pi = |hi|ψi|2 . After the measurement, the state of the system is |ii.
D.
Measuring part of a system
A quantum system can be composed of several “parts;” the Hilbert space for the whole system is given by the
tensor product of the Hilbert spaces for all of its parts.
What happens if we measure just part of the system? Consider a system composed of two parts, subsystem 1 and
subsystem 2. The Hilbert spaces for these subsystems are H1 and H2 , respectively, and the Hilbert space of the whole
system is given by the tensor product of the two, H = H1 ⊗ H2 . Let {|ii1 } be an arbitrary basis for H1 and let {|ji2 }
be a basis for H2 . We want to consider what happens when we measure, say, just subsystem 1.
2
1.
Product states
The simplest situation is if the system is in a product state, i.e., the state of the whole system is of the form
|Ψprod i12 = |φi1 |χi2 .
(1)
Because the subsystems are uncorrelated, we intuitively expect that a measurement on the first system will not effect
the state of the second. This intuition is correct; now let’s see how it’s described.
Consider a projective measurement on system 1 in the basis {|ii1 }. First, we calculate the probability of obtaining
the outcome i. This probability is given by the following rule:
°
°2
°
°
Pi = °1 hi|Ψi12 ° .
(2)
Note that the bra vector in only a vector for system 1, where as the ket vector is a vector for systems 1 and 2. We
can use the expression (5) to work out this probability:
°
°2
°
°
Piprod = °1 hi|φi1 |χi2 °
°
°2
°
°
= |1 hi|φi1 |2 °|χi2 °
= |1 hi|φi1 |2 .
(3)
The last line follows because the state |χi2 is normalized. Thus, the probability of getting the measurement outcome
i is exactly the same as if the second subsystem wasn’t there. If the outcome i is obtained, the post-measurement
state is
|Ψprod0 i12 = |ii1 |χi2 .
(4)
That is, the measurement collapses the first system into the state |ii1 , and does not affect the second system.
2.
General (entangled) states
Things get more interesting if the system is in an entangled state, and we measure just the one of the subsystems.
Consider a general entangled state, which is a superposition of product states,
X
|Ψi12 =
cij |ii1 |ji2 ,
(5)
ij
P
where the complex coefficients cij have to give a normalized state, and thus satisfy ij |cij |2 = 1.
Consider a projective measurement on system 1 in the basis {|ii1 }. Again, we first calculate the probability of
obtaining the outcome i. This probability is given by the same rule as above:
°
°2
°
°
Pi = °1 hi|Ψi12 ° .
(6)
Because the system is not in a product state, the calculation is a little trickier:
°
°2
X
°
°
Pi = °1 hi|
ci0 j |i0 i1 |ji2 °
i0 j
°X
°2
°
°
=°
ci0 j (1 hi|i0 i1 )|ji2 °
ij
°X
°2
°
°
=°
cij |ji2 °
j
=
X
j
|cij |2 .
(7)
3
If the outcome i is obtained, the post-measurement state is proportional to
X
|Ψ0 i12 ∝ |ii1 hi|Ψi12 = |ii1 (
cij |ji2 ) .
(8)
j
The post-measurement state |Ψ0 i12 is clearly a product state. However, the state of the second system is not normalized. We normalize it by dividing by the square root of the probability of the measurement outcome i. Define
P
P
j cij |ji2
j cij |ji2
0
√
= qP
|χi i2 =
.
(9)
Pi
2
j |cij |
Show that this state is normalized.
Note that, with the definition of this state |χ0i i2 for every i, we could rewrite the original entangled state of the
two systems as
Xp
|Ψi12 =
Pi |ii1 |χ0i i2 .
(10)
i
Then, by measuring the first system in the basis {|ii1 } and obtaining the measurement result i, the post-measurement
state of both systems is
|Ψ0 i12 = |ii1 |χ0i i2 .
(11)
This result can be interpreted as follows: the system 1, which is measured, is collapsed to the state |ii1 , and the
second system 2, even though it is not measured, is also collapsed to the component of the original entangled state
|Ψi12 that was associated with the state |ii1 of system 1.