![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg78.png\")
<\/p>\n\n\n\nThere is another sense in which density operators are a useful way to<\/p>\n\n\n\n
describe nature. In general, it is impossible to isolate the system of interest<\/p>\n\n\n\n
from some parts of its environment. We then have to regard our system as a<\/p>\n\n\n\n
subsystem of a larger system: \u201csystem + environment\u201d. If the large system in a<\/p>\n\n\n\n
pure quantum state consists of subsystems, then the state of any subsystem is<\/p>\n\n\n\n
essentially described by a density operator. The way to get there is to perform<\/p>\n\n\n\n
a reduction of the density matrix of the larger system, by a procedure called<\/p>\n\n\n\n
the partial trace over all subsystems except the one of interest.<\/p>\n\n\n\n
5.3.1 Reduced density operator<\/p>\n\n\n\n
Consider a composite of two systems A and B, described by a pure state<\/p>\n\n\n\n
density operator \u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
.<\/p>\n\n\n\n
FIGURE 5.5: Illustrating a bipartite composite system.<\/p>\n\n\n\n
For the purposes of this book, we will only concentrate on systems consist-<\/p>\n\n\n\n
<\/p>\n\n\n\n
Mixed States, Open Systems, and the Density Operator 93<\/p>\n\n\n\n
ing of two subsystems, the so-called bipartite systems (Figure 5.5). We can<\/p>\n\n\n\n
perform a partial trace over the system B alone to obtain the state of system<\/p>\n\n\n\n
A. If the set {|k<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i} forms a basis for system B then<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
= Tr<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
k<\/p>\n\n\n\n
hk<\/p>\n\n\n\n
B<\/p>\n\n\n\n
|\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
|k<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i. (5.34)<\/p>\n\n\n\n
Trace operation is linear, and if we demand that partial trace is also linear in<\/p>\n\n\n\n
its inputs, we can compute partial traces in practice.<\/p>\n\n\n\n
De\ufb01nition 5.4. If subsystems A and B are given by Hilbert spaces spanned<\/p>\n\n\n\n
by the bases {|i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i} and {|j<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i} respectively, we de\ufb01ne the partial trace of \u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
with respect to subsystem A as<\/p>\n\n\n\n
Tr<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
hi<\/p>\n\n\n\n
A<\/p>\n\n\n\n
|\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
|i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i (5.35)<\/p>\n\n\n\n
which will be an operator on the Hilbert space of subsystem B alone.<\/p>\n\n\n\n
Example 5.3.1. Consider a simple example where the system state can be<\/p>\n\n\n\n
written in separable form:<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
= \u03c3<\/p>\n\n\n\n
A<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u2297 \u03c3<\/p>\n\n\n\n
B<\/p>\n\n\n\n
2<\/p>\n\n\n\n
.<\/p>\n\n\n\n
Then quite trivially,<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
= Tr<\/p>\n\n\n\n
B<\/p>\n\n\n\n
(\u03c3<\/p>\n\n\n\n
A<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u2297 \u03c3<\/p>\n\n\n\n
B<\/p>\n\n\n\n
2<\/p>\n\n\n\n
) = \u03c3<\/p>\n\n\n\n
A<\/p>\n\n\n\n
1<\/p>\n\n\n\n
Tr<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\u03c3<\/p>\n\n\n\n
B<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= \u03c3<\/p>\n\n\n\n
A<\/p>\n\n\n\n
1<\/p>\n\n\n\n
.<\/p>\n\n\n\n
Example 5.3.2. A less trivial case where the two subsystems are entangled,<\/p>\n\n\n\n
so that the state of the system is a Bell state:<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\ue000<\/p>\n\n\n\n
|0<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i|0<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i + |1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i|1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\ue001<\/p>\n\n\n\n
.<\/p>\n\n\n\n
=\u21d2 \u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
= |\u03c8<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
|<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|00ih00| + |00ih11| + |11ih00| + |11ih11|) .<\/p>\n\n\n\n
To obtain \u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
by a partial trace over B, we sandwich each term between the<\/p>\n\n\n\n
basis states of B and add up:<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
= Tr<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
= h0<\/p>\n\n\n\n
B<\/p>\n\n\n\n
|\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
|0<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i + h1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
|\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
|1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i<\/p>\n\n\n\n
<\/p>\n\n\n\n
94 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
We illustrate the calculation of this by \ufb01rst evaluating the contribution by the<\/p>\n\n\n\n
\ufb01rst term in \u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
:<\/p>\n\n\n\n
h0<\/p>\n\n\n\n
B<\/p>\n\n\n\n
|(|00ih00|) |0<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i + h1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
|(|00ih00|) |1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i<\/p>\n\n\n\n
= |0ih0<\/p>\n\n\n\n
B<\/p>\n\n\n\n
|0ih0|h0|0<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i + |0ih1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
|0ih0|h0|1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i<\/p>\n\n\n\n
= |0ih0| + 0.<\/p>\n\n\n\n
Evaluating the other terms similarly, we \ufb01nd that<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0ih0| + |1ih1|) =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
.<\/p>\n\n\n\n
Thus the subsystem A is in a maximally mixed state! Similarly,<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
= Tr<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
.<\/p>\n\n\n\n
This result is a hallmark of entanglement: though the composite system is<\/p>\n\n\n\n
in a well-de\ufb01ned state, i.e., its density operator contains maximal informa-<\/p>\n\n\n\n
tion about all measurement outcomes in the state, we can say nothing about<\/p>\n\n\n\n
measurement outcomes on either of the component subsystems: they are in<\/p>\n\n\n\n
maximally mixed states.<\/p>\n\n\n\n
Exercise 5.9. Calculate the density matrices for both subsystems for the other<\/p>\n\n\n\n
three Bell states.<\/p>\n\n\n\n
Exercise 5.10. Consider a 2-qubit system AB with the density matrix \u03c1 =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
00<\/p>\n\n\n\n
ih\u03b2<\/p>\n\n\n\n
00<\/p>\n\n\n\n
| +<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|10ih10|. Compute the reduced density matrices \u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
and<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
.<\/p>\n\n\n\n
The fact that the reduced density matrices for entangled systems represent<\/p>\n\n\n\n
mixed states is generic, and can be used to characterize entanglement. As we<\/p>\n\n\n\n
have already seen, the reduced density matrices for separable systems will<\/p>\n\n\n\n
always be pure.<\/p>\n\n\n\n
Box 5.1: POVM from Projective Measurements on a Composite System<\/p>\n\n\n\n
POVM measurements on quantum systems can be realized as projective<\/p>\n\n\n\n
measurements on an extended \u201csystem+ancilla\u201d Hilbert space. Let\u2019s consider<\/p>\n\n\n\n
a system A that is not interacting with the independent ancilla B. The com-<\/p>\n\n\n\n
bined AB system is in a product state that can be represented by the density<\/p>\n\n\n\n
operator<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
= \u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297 \u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
.<\/p>\n\n\n\n
A projective measurement on this state is the action of projection operators<\/p>\n\n\n\n
<\/p>\n\n\n\n
Mixed States, Open Systems, and the Density Operator 95<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
m<\/p>\n\n\n\n
on this state. The probability of outcome m is then<\/p>\n\n\n\n
P(m) = Tr<\/p>\n\n\n\n
h<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
m<\/p>\n\n\n\n
(\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297 \u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
)<\/p>\n\n\n\n
i<\/p>\n\n\n\n
= Tr<\/p>\n\n\n\n
A<\/p>\n\n\n\n
h<\/p>\n\n\n\n
Tr<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\ue010<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297 \u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\ue011i<\/p>\n\n\n\n
= Tr<\/p>\n\n\n\n
A<\/p>\n\n\n\n
(<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
) (5.36)<\/p>\n\n\n\n
where the<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
s are operators on the system A. We can identify the matrix<\/p>\n\n\n\n
elements of these operators by expressing the above equation in components:<\/p>\n\n\n\n
using orthonormal bases {|ii} for the system A and {|\u00b5i} for the ancilla B,<\/p>\n\n\n\n
Tr<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\ue010<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297 \u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\ue011<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
ij\u00b5\u03bd<\/p>\n\n\n\n
(<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
m<\/p>\n\n\n\n
)<\/p>\n\n\n\n
j\u03bdi\u00b5<\/p>\n\n\n\n
(\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
)<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
(\u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
)<\/p>\n\n\n\n
\u00b5\u03bd<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
(E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
)<\/p>\n\n\n\n
ji<\/p>\n\n\n\n
(\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
)<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
=\u21d2 (E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
)<\/p>\n\n\n\n
ji<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
\u00b5\u03bd<\/p>\n\n\n\n
(<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
m<\/p>\n\n\n\n
)<\/p>\n\n\n\n
j\u03bdi\u00b5<\/p>\n\n\n\n
(\u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
)<\/p>\n\n\n\n
\u00b5\u03bd<\/p>\n\n\n\n
.<\/p>\n\n\n\n
It is easy to see that the<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
s de\ufb01ned this way are complete. Suppose \u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
is<\/p>\n\n\n\n
diagonal in the basis {|\u00b5i}:<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
\u00b5<\/p>\n\n\n\n
p<\/p>\n\n\n\n
\u00b5<\/p>\n\n\n\n
|\u00b5ih\u00b5|,<\/p>\n\n\n\n
X<\/p>\n\n\n\n
m<\/p>\n\n\n\n
E<\/p>\n\n\n\n
m<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
\u00b5<\/p>\n\n\n\n
p<\/p>\n\n\n\n
\u00b5<\/p>\n\n\n\n
h\u00b5|<\/p>\n\n\n\n
X<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
m<\/p>\n\n\n\n
|\u00b5i = .<\/p>\n\n\n\n
5.3.2 Schmidt decomposition<\/p>\n\n\n\n
Another useful way of dealing with composite systems, the Schmidt de-<\/p>\n\n\n\n
composition is about expressing the state of a bipartite system in terms of<\/p>\n\n\n\n
orthonormal states of the two subsystems.<\/p>\n\n\n\n
Theorem 5.2. If {|u<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i} and {|v<\/p>\n\n\n\n
B<\/p>\n\n\n\n
j<\/p>\n\n\n\n
i} are orthonormal sets of vectors in the<\/p>\n\n\n\n
Hilbert spaces of subsystems A and B, respectively, the state of the combined<\/p>\n\n\n\n
system can be expressed as<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u03bb<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|u<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i|v<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i. (5.37)<\/p>\n\n\n\n
The constants \u03bb<\/p>\n\n\n\n
i<\/p>\n\n\n\n
are called Schmidt coe\ufb03cients, and are non-negative real<\/p>\n\n\n\n
numbers satisfying<\/p>\n\n\n\n
P<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u03bb<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i<\/p>\n\n\n\n
= 1. The number of terms in the expansion is known<\/p>\n\n\n\n
<\/p>\n\n\n\n
96 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
as the Schmidt number. While such an expansion may not in general be<\/p>\n\n\n\n
unique, the Schmidt number is unique for a given state.<\/p>\n\n\n\n
Proof. The Schmidt decomposition theorem (5.37) can be proved by simple<\/p>\n\n\n\n
results in linear algebra.<\/p>\n\n\n\n
Consider a general pure state in the computational basis {|i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i|j<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i}:<\/p>\n\n\n\n
|\u03c8i =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
C<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
|ii|ji.<\/p>\n\n\n\n
Now the matrix C of complex numbers is a square matrix, and therefore (from<\/p>\n\n\n\n
results in linear algebra) has a singular value decomposition (SVD) of the form<\/p>\n\n\n\n
C = UDV where D is a diagonal matrix and U and V are unitaries. So we<\/p>\n\n\n\n
can write<\/p>\n\n\n\n
|\u03c8i =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
X<\/p>\n\n\n\n
k<\/p>\n\n\n\n
U<\/p>\n\n\n\n
ik<\/p>\n\n\n\n
D<\/p>\n\n\n\n
kk<\/p>\n\n\n\n
V<\/p>\n\n\n\n
kj<\/p>\n\n\n\n
|ii|ji.<\/p>\n\n\n\n
By de\ufb01ning D<\/p>\n\n\n\n
kk<\/p>\n\n\n\n
= \u03bb<\/p>\n\n\n\n
k<\/p>\n\n\n\n
,<\/p>\n\n\n\n
P<\/p>\n\n\n\n
i<\/p>\n\n\n\n
U<\/p>\n\n\n\n
ik<\/p>\n\n\n\n
|ii = |u<\/p>\n\n\n\n
k<\/p>\n\n\n\n
i,<\/p>\n\n\n\n
P<\/p>\n\n\n\n
j<\/p>\n\n\n\n
V<\/p>\n\n\n\n
kj<\/p>\n\n\n\n
|ji = |v<\/p>\n\n\n\n
k<\/p>\n\n\n\n
i we get the form of<\/p>\n\n\n\n
Equation 5.37 for |\u03c8i.<\/p>\n\n\n\n
In terms of density matrices,<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u03bb<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|u<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i<\/p>\n\n\n\n
ihu<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i<\/p>\n\n\n\n
| \u2297 |v<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i<\/p>\n\n\n\n
ihv<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|. (5.38)<\/p>\n\n\n\n
If we perform partial traces on this, we will get<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
A<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u03bb<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|u<\/p>\n\n\n\n
i<\/p>\n\n\n\n
ihu<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|, \u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u03bb<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|v<\/p>\n\n\n\n
i<\/p>\n\n\n\n
ihv<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|, (5.39)<\/p>\n\n\n\n
There are some important take-home points to note here:<\/p>\n\n\n\n
\u2022 Both the reduced density matrices have the same eigenvalues.<\/p>\n\n\n\n
\u2022 \u03c1 could have zero eigenvalues and those terms are not present in the<\/p>\n\n\n\n
expansion above. So the sets {|u<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i} and {|v<\/p>\n\n\n\n
B<\/p>\n\n\n\n
j<\/p>\n\n\n\n
i} are not bases for H<\/p>\n\n\n\n
A<\/p>\n\n\n\n
and H<\/p>\n\n\n\n
B<\/p>\n\n\n\n
, but can be extended to bases by including eigenvectors for the<\/p>\n\n\n\n
zero eigenvalues.<\/p>\n\n\n\n
If the composite system is in a product state, then there is obviously only<\/p>\n\n\n\n
one term in the Schmidt decomposition. Thus the Schmidt number for product<\/p>\n\n\n\n
states is always 1. Therefore an entangled state has Schmidt number > 1. This<\/p>\n\n\n\n
is one of the \ufb01rst ways of quantifying entanglement.<\/p>\n\n\n\n
Example 5.3.3. Let\u2019s \ufb01nd the Schmidt form of some simple states:<\/p>\n","protected":false},"excerpt":{"rendered":"
There is another sense in which density operators are a useful way to describe nature. In general, it is impossible to isolate the system of interest from some parts of its environment. We then have to regard our system as a subsystem of a larger system: \u201csystem + environment\u201d. If the large system in a […]<\/p>\n","protected":false},"author":1,"featured_media":4020,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[491],"tags":[],"class_list":["post-4035","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-5-grand-unification"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/quantum-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4035","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4035"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4035\/revisions"}],"predecessor-version":[{"id":4570,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4035\/revisions\/4570"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4020"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4035"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4035"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4035"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}