Collapsing an Entangled Quantum State Through Projection

Explore quantum entanglement using ideas from linear algebra and quantum computation

A quantum bit (qubit) is a simple quantum system that has only two basic states:

|0〉

and

|1〉

. For example, an electron may be found in one of the two following states:

The state of the qubit at any time is a linear superposition of basis states:

v=a|0〉+b|1〉

where

and

satisfy

a

b

. Typically,

and

are complex numbers. The following are examples of coordinate representations of such linear superposition:

In[1]:=

={1,0};(*

=|0〉*)

In[2]:=

={0,1};(*

=|1〉*)

In[3]:=

={1/Sqrt[2],1/Sqrt[2]};(*

|0〉+

|1〉*)

Use ResourceFunction["BlochSpherePlot"] to visualize these example states:

In[4]:=

[◼]	BlochSpherePlot

[{

}]

Out[111]=

Out[106]=

The evolution of a closed quantum system is described by a unitary matrix

: if at time

the state of the quantum system is

, then a later time

, the state is

U.v

. Matrix

is unitary if its inverse is its conjugate transpose:

†

I

. The evolution of a quantum system is reversible since

is an invertible matrix.

Verify that

1 2	1 2
1 2	-1 2

is a unitary matrix:

In[5]:=

U=1

,1

,1

,-1

;U.ConjugateTranspose[U]IdentityMatrix[2]

Out[5]=

True

Alternatively, use the built-in test function

UnitaryMatrixQ

In[6]:=

UnitaryMatrixQ[U]

Out[6]=

True

acts on the basis states as follow. On

In[7]:=

U.{1,0}//MatrixForm

Out[7]//MatrixForm=

In[8]:=

U.{0,1}//MatrixForm

Out[8]//MatrixForm=

These states are now a linear combination of the two basis states.

Quantum measurements are described by a set of projection operators onto the various basis states of the system.

In the case of a single qubit, one can define the following pair of projection operators:

In[9]:=

(

={{1,0},{0,0}})//MatrixForm

Out[9]//MatrixForm=



1	0
0	0



In[10]:=

(

={{0,0},{0,1}})//MatrixForm

Out[10]//MatrixForm=



0	0
0	1



The results of measurement using these operators:

In[11]:=

.{a,b}//MatrixForm

Out[11]//MatrixForm=





Out[12]//MatrixForm=





The probabilities for the measurements are given by

a

and

b

, respectively, assuming that

a

b

. Note that neither projection operator is invertible because their determinants are zero:

In[13]:=

{Det[

],Det[

]}

Out[13]=

{0,0}

Hence we see that application of a projection operator is not reversible.

Consider a pair of qubits, each of which has the basis states

|0〉

and

|1〉

. Suppose that these qubits are allowed to interact for some time. After the interaction, the quantum state of this system of qubits will be a linear superposition of the following basis states:

|00〉,|01〉,|10〉,|11〉

, where the first entry is a label for the first qubit and the second entry is a label for the second qubit.

The following is a linear superposition of the first and fourth states in coordinates:

In[14]:=

(

={1/Sqrt[2],0,0,1/Sqrt[2]})//MatrixForm(*

|00〉+

|11〉*)

Out[14]//MatrixForm=

State

cannot be "factored" into a tensor product of states because nothing in common is shared. This is an EPR pair, the simplest example of quantum entanglement. Since the first and third entries of the vector represent

|00〉

and

|10〉

, respectively, both with the

|0〉

state for the second qubit, the projection operator for the

|0〉

state of the second qubit is:

In[15]:=

(

={{1,0,0,0},{0,0,0,0},{0,0,1,0},{0,0,0,0}})//MatrixForm

Out[15]//MatrixForm=

1	0	0	0
0	0	0	0
0	0	1	0
0	0	0	0

Apply the projection operator

to the EPR state:

In[16]:=

//MatrixForm

Out[16]//MatrixForm=

Thus, the measurement collapses the EPR pair into the

|00〉

state and the first qubit will be in the

|0〉

state after measurement.

External Links

Wolfram U Introduction to Linear Algebra

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Contributed by: Juan Ortiz

Wolfram Language Example Repository

Collapsing an Entangled Quantum State Through Projection

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See Also

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