![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg68.png\")
<\/p>\n\n\n\nBy \u201cstate\u201d of a system, we mean a collection of all possible knowledge we<\/p>\n\n\n\n
can gather about the system, which is practically achieved by studying the<\/p>\n\n\n\n
distribution of outcomes of measurements made on the system. In the case of<\/p>\n\n\n\n
pure states, these outcomes together are described by a ray in Hilbert space.<\/p>\n\n\n\n
Consider measuring an observable<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Q with N possible eigenvalues q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
with<\/p>\n\n\n\n
corresponding eigenstates |q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i. If we obtain a particular result q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
, then we can<\/p>\n\n\n\n
say that the projection operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
P<\/p>\n\n\n\n
i<\/p>\n\n\n\n
= |q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
ihq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
| has acted on the state of the<\/p>\n\n\n\n
system.<\/p>\n\n\n\n
If we know the state to be the pure state |\u03c8i, then the state is as well de-<\/p>\n\n\n\n
scribed by a projector |\u03c8ih\u03c8| along this direction. The probability of outcome<\/p>\n\n\n\n
q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
is given by<\/p>\n\n\n\n
P(q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
) = h\u03c8|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
P<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03c8i = h\u03c8|q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
ihq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
= hq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03c8ih\u03c8|q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i<\/p>\n\n\n\n
= hq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u02c6\u03c1|q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i (5.1)<\/p>\n\n\n\n
This de\ufb01nes the density operator \u02c6\u03c1 for a pure state described by a single state<\/p>\n\n\n\n
vector:<\/p>\n\n\n\n
\u02c6\u03c1<\/p>\n\n\n\n
pure<\/p>\n\n\n\n
= |\u03c8ih\u03c8|. (5.2)<\/p>\n\n\n\n
In general, the system could be composed of a number of (pure) states |\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i<\/p>\n\n\n\n
where n = 1, 2…d, with classical probability p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
: 0 \u2264 p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
\u2264 1,<\/p>\n\n\n\n
P<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
= 1. This<\/p>\n\n\n\n
mixture {p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
, |\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i} is referred to as an ensemble of pure states with associated<\/p>\n\n\n\n
probabilities. In this case, the probability of obtaining the outcome q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
on<\/p>\n\n\n\n
measuring<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Q is<\/p>\n\n\n\n
P(q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
) =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
h\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
P<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
h\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
ihq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
hq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i<\/p>\n\n\n\n
= hq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|<\/p>\n\n\n\n
!<\/p>\n\n\n\n
|q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i (5.3)<\/p>\n\n\n\n
where in the third equality, we have moved the term hq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i to the beginning<\/p>\n\n\n\n
<\/p>\n\n\n\n
Mixed States, Open Systems, and the Density Operator 79<\/p>\n\n\n\n
of the expression since it is a number. (This is an illustration of manipulating<\/p>\n\n\n\n
expressions using the Dirac notation.)<\/p>\n\n\n\n
The piece within parentheses in the middle is identi\ufb01ed as the density<\/p>\n\n\n\n
operator or the statistical operator \u02c6\u03c1 for that state. This operator is com-<\/p>\n\n\n\n
pletely given by the initial state.<\/p>\n\n\n\n
De\ufb01nition 5.1. The density operator for a system consisting of a mixed<\/p>\n\n\n\n
ensemble of states {p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
, |\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i} is<\/p>\n\n\n\n
\u02c6\u03c1 =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|. (5.4)<\/p>\n\n\n\n
The sum over states in this expression looks like a superposition of states:<\/p>\n\n\n\n
but this is an incoherent superposition, as opposed to coherent superposition<\/p>\n\n\n\n
of basis states that de\ufb01nes a pure state. The incoherence stems from the fact<\/p>\n\n\n\n
that the relative phases of the states |\u03c8<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i are not available to us.<\/p>\n\n\n\n
This operator uniquely prescribes the probabilities of outcomes on mea-<\/p>\n\n\n\n
surements on the system. Exactly as in Equation 5.1, we can then write the<\/p>\n\n\n\n
probability of obtaining the outcome q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
as<\/p>\n\n\n\n
P(q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
) = hq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u02c6\u03c1|q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i. (5.5)<\/p>\n\n\n\n
The expectation value of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Q in a state \u02c6\u03c1 is<\/p>\n\n\n\n
h<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Qi<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
P(q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
) =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
hq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03c1|q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i,j<\/p>\n\n\n\n
q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
hq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|q<\/p>\n\n\n\n
j<\/p>\n\n\n\n
ihq<\/p>\n\n\n\n
j<\/p>\n\n\n\n
|\u03c1|q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
j<\/p>\n\n\n\n
hq<\/p>\n\n\n\n
j<\/p>\n\n\n\n
|\u03c1<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
ihq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|<\/p>\n\n\n\n
!<\/p>\n\n\n\n
|q<\/p>\n\n\n\n
j<\/p>\n\n\n\n
i.<\/p>\n\n\n\n
Here we have introduced the resolution of identity =<\/p>\n\n\n\n
P<\/p>\n\n\n\n
j<\/p>\n\n\n\n
|q<\/p>\n\n\n\n
j<\/p>\n\n\n\n
ihq<\/p>\n\n\n\n
j<\/p>\n\n\n\n
| in the third<\/p>\n\n\n\n
line, and then moved the term hq<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|q<\/p>\n\n\n\n
j<\/p>\n\n\n\n
i to the end of the expression since it is<\/p>\n\n\n\n
a number. In the last line, we identify the term in the parentheses as the<\/p>\n\n\n\n
spectral representation of the operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Q.<\/p>\n\n\n\n
De\ufb01nition 5.2. The trace of an operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A is de\ufb01ned by<\/p>\n\n\n\n
Tr<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
j<\/p>\n\n\n\n
hj|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A|ji,<\/p>\n\n\n\n
a simple generalization of the trace of a matrix as the sum of its diagonal<\/p>\n\n\n\n
elements.<\/p>\n\n\n\n
The expectation value of the observable we are measuring is thus given by<\/p>\n\n\n\n
h<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Qi<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
= Tr(\u03c1<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Q). (5.6)<\/p>\n\n\n\n
(The trace here is apparently taken in the {|q<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i} basis, but trace is basis-<\/p>\n\n\n\n
independent, as you will prove.)<\/p>\n\n\n\n
<\/p>\n\n\n\n
80 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
This averaging of the physical property Q is twofold: \ufb01rst the quantum<\/p>\n\n\n\n
average hQi<\/p>\n\n\n\n
n<\/p>\n\n\n\n
= Tr(\u03c1<\/p>\n\n\n\n
n<\/p>\n\n\n\n
Q) over each of the (pure) states comprising the mixture,<\/p>\n\n\n\n
and the usual statistical average over the whole ensemble with each state<\/p>\n\n\n\n
average weighted by the probability p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
of its occurrence. We can make this<\/p>\n\n\n\n
explicit by writing<\/p>\n\n\n\n
h<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Qi<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
= h<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Qi =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
n<\/p>\n\n\n\n
p<\/p>\n\n\n\n
n<\/p>\n\n\n\n
Tr(\u03c1<\/p>\n\n\n\n
n<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Q). (5.7)<\/p>\n\n\n\n
Exercise 5.1. Show that for vectors |\u03c6<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i and |\u03c6<\/p>\n\n\n\n
j<\/p>\n\n\n\n
i, Tr(|\u03c6<\/p>\n\n\n\n
i<\/p>\n\n\n\n
ih\u03c6<\/p>\n\n\n\n
j<\/p>\n\n\n\n
|) = h\u03c6<\/p>\n\n\n\n
j<\/p>\n\n\n\n
|\u03c6<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i.<\/p>\n\n\n\n
Exercise 5.2. Show that the trace of an operator is independent of the basis<\/p>\n\n\n\n
chosen to evaluate it.<\/p>\n\n\n\n
Exercise 5.3. Show that trace as an operation is linear, i.e., Tr(A + B) = TrA +<\/p>\n\n\n\n
TrB and Tr(\u03bbA) = \u03bbTrA.<\/p>\n\n\n\n
Exercise 5.4. Show that the trace of products of operators is invariant under<\/p>\n\n\n\n
cyclic permutations of the operators. i.e., Tr(AB) = Tr(BA), Tr(ABC) =<\/p>\n\n\n\n
Tr(BCA) = Tr(CAB). etc.<\/p>\n\n\n\n
Exercise 5.5. Show that the Pauli matrices are traceless.<\/p>\n\n\n\n
The matrix representation of the density operator, called the density ma-<\/p>\n\n\n\n
trix of the system, is useful for computations. In the computational basis {|ii},<\/p>\n\n\n\n
we can represent the density operator (Equation 5.2) as a matrix:<\/p>\n\n\n\n
|\u03c8i =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
c<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|ii (5.8)<\/p>\n\n\n\n
=\u21d2 \u03c1<\/p>\n\n\n\n
pure<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i,j<\/p>\n\n\n\n
c<\/p>\n\n\n\n
i<\/p>\n\n\n\n
c<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
j<\/p>\n\n\n\n
|iihj|. (5.9)<\/p>\n\n\n\n
For a mixed state, in this basis we can represent the density matrix as<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
mixed<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i,j<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
|iihj|. (5.10)<\/p>\n\n\n\n
If a system consists of equal mixtures of all possible computational basis states<\/p>\n\n\n\n
it is said to be maximally mixed. In n dimensions, such a state is represented<\/p>\n\n\n\n
by a multiple of the identity matrix:<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
max<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
n<\/p>\n\n\n\n
n\u00d7n<\/p>\n\n\n\n
.<\/p>\n\n\n\n
<\/p>\n\n\n\n
Mixed States, Open Systems, and the Density Operator 81<\/p>\n\n\n\n
Example 5.1.1. The density matrix for the unpolarized electron beam dis-<\/p>\n\n\n\n
cussed above is the maximally mixed state<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
m<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|0ih0| +<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|1ih1| =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
0 1.<\/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
.<\/p>\n\n\n\n
In contrast, the density matrix for the pure state |\u2191<\/p>\n\n\n\n
x<\/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
(|0i + |1i) is<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
p<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0i + |1i)<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(h0| + h1|)<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0ih0| + |0ih1| + |1ih0| + |1ih1|)<\/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
1 1<\/p>\n\n\n\n
1 1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
.<\/p>\n\n\n\n
Note the di\ufb00erence, though when beams in either state are passed through an<\/p>\n\n\n\n
SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
machine, we get \u2191 and \u2193 outputs with equal probability! However, when<\/p>\n\n\n\n
passed through an SG<\/p>\n\n\n\n
x<\/p>\n\n\n\n
machine, the mixed state gives the same result, while<\/p>\n\n\n\n
the pure state |\u2191<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i gives only an \u2191 beam with probability 1.<\/p>\n\n\n\n
Example 5.1.2. We\u2019ll see how the usual results regarding experimental mea-<\/p>\n\n\n\n
surements follow using density matrices for pure states. Consider the state<\/p>\n\n\n\n
|\u2191<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i of the above example. The probability of obtaining +1 on measuring \u03c3<\/p>\n\n\n\n
z<\/p>\n\n\n\n
in this state is<\/p>\n\n\n\n
P<\/p>\n\n\n\n
+<\/p>\n\n\n\n
= h0|\u03c1|0i = [ 1 0 ]<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1<\/p>\n\n\n\n
0<\/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
.<\/p>\n\n\n\n
The matrix \u03c3<\/p>\n\n\n\n
z<\/p>\n\n\n\n
\u03c1 =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
0 \u22121<\/p>\n\n\n\n
#”<\/p>\n\n\n\n
1 1<\/p>\n\n\n\n
1 1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
=<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 1<\/p>\n\n\n\n
\u22121 \u22121<\/p>\n\n\n\n
#<\/p>\n\n\n\n
, and<\/p>\n\n\n\n
the expectation value of \u03c3<\/p>\n\n\n\n
z<\/p>\n\n\n\n
is the sum of its diagonal elements = 0.<\/p>\n\n\n\n
Example 5.1.3. A mixed state need not necessarily be composed of orthogonal<\/p>\n\n\n\n
states. For example, one could have a mixture containing 20% of the state<\/p>\n\n\n\n
|0iand 80% of the state |\u2191<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i, whose density matrix would be given by<\/p>\n\n\n\n
\u03c1 =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
5<\/p>\n\n\n\n
|0ih0| +<\/p>\n\n\n\n
4<\/p>\n\n\n\n
5<\/p>\n\n\n\n
|\u2191<\/p>\n\n\n\n
x<\/p>\n\n\n\n
ih\u2191<\/p>\n\n\n\n
x<\/p>\n\n\n\n
| =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
5<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
0 0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
+<\/p>\n\n\n\n
2<\/p>\n\n\n\n
5<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 1<\/p>\n\n\n\n
1 1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
5<\/p>\n\n\n\n
“<\/p>\n\n\n\n
3 2<\/p>\n\n\n\n
2 2<\/p>\n\n\n\n
#<\/p>\n","protected":false},"excerpt":{"rendered":"
By \u201cstate\u201d of a system, we mean a collection of all possible knowledge we can gather about the system, which is practically achieved by studying the distribution of outcomes of measurements made on the system. In the case of pure states, these outcomes together are described by a ray in Hilbert space. Consider measuring an […]<\/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-4023","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\/4023","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=4023"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4023\/revisions"}],"predecessor-version":[{"id":4565,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4023\/revisions\/4565"}],"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=4023"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4023"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4023"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}