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Stochastic Quantum Dynamics I. Born Rule
Robert B. Grifths
Version oF 25 January 2010
Contents
1 Introduction
1
2 Born Rule
1
2.1
Statement oF the Born Rule .
. . . . . . . . . . . . . . . . . . . . . . . . .......
1
2.2
Incompatible sample spaces .
. . . . . . . . . . . . . . . . . . . . . . .........
3
2.3
Born rule using preprobabilities .
. . . . . . . . . . . . . . . . . ...........
5
2.4
Generalizations oF the Born rule
. . . . . . . . . . . . . . . . . . . ..........
5
2.5
Limitations oF the Born rule .
7
3T
w
oq
u
b
i
t
s
8
3.1
Pure states at
t
1
......................................
8
3.2
Properties oF one qubit at
t
2
................................
9
3.3
Correlations and conditional probabilities .
. . . . . . . . . . . . . . . . . . . . . . .
10
ReFerences:
CQT =
Consistent Quantum Theory
by Grifths (Cambridge, 2002), Chs. 8, 9, 10, 11.
1
Introduction
±
Quantum dynamics, i.e., the time development oF quantum systems, is Fundamentally
stochas
tic
or probabilistic.
•
It is unFortunate that this basic Fact is not clearly stated in textbooks, including QCQI, which
present to the student a strange mishmash oF unitary dynamics Followed by probabilities introduced
by means oF
measurements
.
•
While the measurement talk is Fundamentally sound when it is properly interpreted it can be
quite conFusing.
◦
Measurements: topic For separate set oF notes
•
What we will do is to introduce probabilities by means oF some simple examples which
show what one must be careFul about when dealing with quantum systems, and the Fundamental
principles which allow the calculation oF certain key probabilities related to quantum dynamics.
2B
o
r
n
R
u
l
e
2.1
Statement of the Born Rule
±
The Born rule can be stated in the Following way. Let
t
0
and
t
1
be two times, and
T
(
t
1
,t
0
)
the unitary time development operator For the quantum system oF interest.
1
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This system must be isolated during the time period under discussion, or at least it must not
decohere, as otherwise the unitary
T
(
t
1
,t
0
) cannot be defned.
±
Assume a normalized state

ψ
0
±
at
t
0
and an orthonormal basis
{
φ
k
1
±}
,k
=0
,
1
,...
at
t
1
.
Then
Pr(
φ
k
1
)=Pr(
φ
k
1

ψ
0
)=
²
φ
k
1

T
(
t
1
0
)

ψ
0
±
2
=
²
ψ
0

T
(
t
0
1
)

φ
k
1
±²
φ
k
1

T
(
t
1
0
)

ψ
0
±
(1)
is the probability oF

φ
k
1
±
at
t
1
given the initial state

ψ
0
±
at
t
0
. In the argument oF Pr() and
sometimes in other places we omit the ket symbol; Pr(
φ
k
1
) means the probability that the system is
in the state

φ
k
1
±
, or in the ray (onedimensional subspace) generated by

φ
k
1
±
, or has the property

φ
k
1
±
corresponding to the projector

φ
k
1
±²
φ
k
1

=[
φ
k
1
]. The condition
ψ
0
to the right oF the vertical
bar

is oFten omitted when it is evident From the context. (±or more on conditional probabilities,
see separate notes.)
◦
The fnal equality in (1) is a consequence oF the general rule
²
χ

A

ω
±
*
=
²
ω

A
†

χ
±
, together
with the Fact that
T
(
t
1
0
)
†
=
T
(
t
0
1
).
◦
Mnemonic. In (1) subscripts indicate the time. In the matrix elements the time
t
j
as an
argument oF
T
always lies next to the ket/bra corresponding to this time.
±
It should be emphasized that the Born rule, (1), is on exactly the same level as the time
dependent Schr¨odinger equation in terms oF Fundamental quantum principles. That is, it is a
postulate or an axiom that does not emerge From anything else.
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