Matrix Representation of Quantum Entangled States:
Understanding Bell's Inequality and Teleportation
Frank Wang, fwang@lagcc.cuny.edu
In 1935, Einstein, Podolsky and Rosen published a paper revealing a counterintuitive situation in quantum mechanics which was later known as the EPR paradox. The phenomenon involved an entangled state., which Schrodinger called "not one but the characteristic trait of quantum mechanics." In textbooks, entanglement is often presented in abstract notations. In popular accounts of quantum mechanics, entanglement is sometimes portrayed as a mystery or even distorted in a nearly pseudoscientific fashion. In this worksheet, we use Maple's LinearAlgebra package to represent quantum states and measurements in matrix form. The famous Bell's inequality and teleportation can be understood using elementary matrix operations.

Spin1/2 System


In 1922, Otto Stern and Walther Gerlach performed experiments to measure the deflection of beams of silver atoms as they were sent through inhomogeneous magnetic fields. Rather than finding continuous range of deflections, the beam was split into two beams. This result served as evidence of quantization of angular momentum, and was later explained by the spin1/2 nature of electrons. (Protons, neutrons, neutrinos, and quarks are also spin1/2 particles). If we ignore other attributes of a spin1/2 particle, the spin quantum state of such a particle is represented by a twodimensional complex vector, and the SternGerlach measurements are Hermitian (linear and selfadjoint) operators acting on this vector space. We use the Dirac notation to write u> for the spin up state (along the zaxis) and d> for the spin down state. They form two orthonormal basis vectors, and can be represented by column vectors.
> 
u := <1,0>; d := <0,1>;


(1.1) 
The general vector in this twodimensional complex vector space takes the form (with a, b 2C)

(1.2) 
For computer scientists, this is what they refer to as a "qubit." When we represent the quantum states as column vectors, the operators that act on the states are represented by twobytwo matrices. If we measure the spin in units , the spin operators are the Pauli matrices:
> 
s1 := <<01>,<10>>; s2 := <<0I>,<I0>>; s3 := <<10>,<01>>;


(1.3) 
They measure the spin component in x, y, and z direction, respectively. Physically, one can rotate the SternGerlach magnets to select the direction of a measurement.
We can check that u> and d> are eigenvectors of s3 with eigenvalues and , respectively:

(1.4) 
In quantum mechanics, the expected value of an observation A in the state described by a normalized vector v> is <vAv>. For example, the expected value of s3 in u> and d> are
> 
DotProduct(u, s3.u); DotProduct(d, s3.d);


(1.5) 
It is evident that u> and d> are not eigenvectors of s1:
> 
v, e := Eigenvectors(s1);


(1.6) 
The eigenvalues of s1 are also and . We define the normalized eigenvectors of s1 as r> and l> (right and left states).
> 
Normalize(e[1..1, 1], 2); Normalize(e[1..1, 2], 2);


(1.7) 
> 
r := <1/sqrt(2), 1/sqrt(2)>; l := <1/sqrt(2), 1/sqrt(2)>;


(1.8) 
(We manually enter the r> and l> vectors because the order of eigenvectors from the Maple output might vary. The negative sign for l> made the first element positive; it is just a convention and does not affect the calculation of expected values.)
Let's verify that r> and l> are eigenvectors of s1:

(1.9) 
While the state u> (which is the state of spin up in the z direction) is not an eigenvector of s1, it can be written as a linear combination of r> and l> state. Let u> = ar>+bl>, the coefficients are
> 
a := DotProduct(r, u); b := DotProduct(l, u); u = a*r + b*l;


(1.10) 
If we measure the spin of u> in the x direction, the probability of getting is , or 1/2, and the probability of getting is , also 1/2. The expected value is <us1u>:

(1.11) 
This result is not surprising, as :

(1.12) 
Let us consider a measurement made at an angle θ with respect to the z axis on the xz plane. Such an operator is :
> 
s := theta > cos(theta)*s3 + sin(theta)*s1; s(theta);


(1.13) 
The eigenvalues and vectors are
> 
v, e := Eigenvectors(s(theta));


(1.14) 
With elementary trigonometric identities, the reader can verify that the normalized eigenvectors are
> 
<cos(theta/2), sin(theta/2)>; <sin(theta/2), cos(theta/2)>;


(1.15) 
for eigenvalues and , respectively. Consider a specific example of .

(1.16) 
> 
v, e := Eigenvectors(s(2*Pi/3));


(1.17) 
The normalized eigenvector for eigenvalue is denoted by one>, and that for eigenvalue is denoted by two>.
> 
Normalize(e[1..1,2], 2); Normalize(e[1..1,1], 2);


(1.18) 
> 
one := <1/2, sqrt(3)/2>; two := <sqrt(3)/2, 1/2>;


(1.19) 
Consider the spindown state d>: it is not the eigenvector of the operator when the SternGerlach magnets are rotated by 120 degrees, but it can be written as the linear combination of the eigenvectors one> and two>, with coefficients <oned> and <twod>:
> 
a := DotProduct(one, d); b := DotProduct(two, d); d = a*one + b*two;


(1.20) 
The probabilities of getting or are (3/4) and (1/4), respectively. The expected value <dsd> is
> 
DotProduct(d, s(2*Pi/3).d);


(1.21) 
This is explained by the fact

(1.22) 


Composite Systems and Entanglement


Consider a system composed of two spin1/2 particles A and B. The quantum state of the system is a vector in a new vector spacethe tensor product (or Kronecker product) space. Let particle A be described by v=<a1, a2> and particle B by w=<b1, b2>. The composite system is described by
> 
v := <a1, a2>; w := <b1, b2>; T := KroneckerProduct(v, w);


(2.1) 
As an example, consider a system with particle A in spin up state (along the z axis) and particle B in spin down state, denoted by u>d> or simply ud>. It is represented by
> 
phi := KroneckerProduct(u, d);


(2.2) 
To measure the z component of particle A, the operator is
> 
A3 := KroneckerProduct(s3, IdentityMatrix(2));


(2.3) 
for particle B,
> 
B3 := KroneckerProduct(IdentityMatrix(2), s3);


(2.4) 
The calculations below show that φ is an eigenvector of A3 and B3, as expected.
> 
A3.phi = phi; B3.phi =  phi;


(2.5) 
Consider another system with particle A in state u> and particle B in r> state: ur>:
> 
phi2 := KroneckerProduct(u, r);


(2.6) 
> 
B1 := KroneckerProduct(IdentityMatrix(2), s1);


(2.7) 
> 
A3.phi2 = phi2; B1.phi2 = phi2;


(2.8) 
A general state of the composite system can be described by
> 
psi := <delta1, delta2, delta3, delta4>;


(2.9) 
We say that Ψ is an entangled state of the two particle if it cannot be written as the tensor product of two subsystems.

(2.10) 
We notice that
> 
T[1,1]*T[4,1]  T[2,1]*T[3,1];


(2.11) 
This means that the condition for entanglement is . An obvious entangled state is
> 
psi := Normalize(<0, 1, 1, 0>, 2);


(2.12) 
We can verify that ψ> = 1/sqrt(2)*(ud>  du>):
> 
1/sqrt(2)*(KroneckerProduct(u, d)  KroneckerProduct(d, u));


(2.13) 
Physically, it is possible to obtain such a state from the decay of a particle with 0 angular momentum into two particles, each with spin 1/2.


Bell's Inequality


In 1981, the Cornell physicist N. David Mermin published a paper in the American Journal of Physics; he provided a gedanken demonstration designed to convey the essential character of John Bell's analysis of the EPR experiment. Richard P. Feynman wrote to Mermin saying "one of the most beautiful papers in physics that I know of is yours in the American Journal of Physics." Mermin's demonstration consisted of two widely separated detectors, each triggered by one of a pair of particles that originated earlier from a common source. Each detector had a switch that could be set to one of three positions before it was triggered. When triggered it flashed either a red light or green one.
Source: Physics Today.
The device is operated repeatedly in the following way: the switch on either detector is set at random to one of its three positions; this gives nine equally likely settings for the pair of detectors: 11, 12, 13, 21, 22, 23, 31, 32, and 33. The setting of the switches and the colors that flashed are recorded using the following notation: 31GR indicates that detector A was set to 3 and flashed green, while B was set to 1 and flashed red. A typical fragment from a record of many runs is shown below.
Source: American Journal of Physics.
There are three relevant features:
• 
If one examines only those runs in which the switches have the same setting, then one finds that the lights always flash the same colors.

• 
In those runs in which the switches end up with different settings both detectors flash the same color only a quarter of the time; the other three quarters of the time the detectors flash different colors.

• 
If one examines all runs, without any regard to how the switches are set, the one finds that the pattern of flashing is completely random. In particular, half the time the lights flash the same colors, and half the time different colors.

Under the framework of "local realism" (roughly speaking, each particle can be divided into eight types: RRR, RRG, RGR, RGG, GRR, GRG, GGR, and GGG) it is impossible to construct a theory that satisfies all three features. A particle whose state is of type RGG, for example, will always cause its detector to flash red for setting 1 of the switch, green for setting 2, and green for setting 3; a particle in a state of type GGG will cause its detector to flash green for any setting of the switch; and so on. Consider a particular instruction set, for example, RRG. Should both particles be issued the instruction set RRG, then the detectors will flash the same colors when the switches are set to 11, 22, 33, 12, or 21; they will flash different colors for 13, 31, 23, or 32. Because the switches at each detector are set randomly and independently, each of these nine cases is equally likely, so the instruction set RRG will result in the same colors flashing 5/9 of the time.
Evidently the same conclusion holds for the sets RGR, GRR, GGR, GRG and RGG: all six such instructions sets also result in the same colors flashing 5/9 of the time. But the only instruction sets left are RRR and GGG, and these each result in the same colors flashing all of the time. Therefore if instruction sets exist, the same colors will flash in at least 5/9 of all the runs (2/3 is the precise ratio). But this is in conflict with the demonstration: the same colors flash only 1/2 the time.
There is no problem in obtaining the result using quantum mechanics. Let the source produce two particles of spin 1/2 in entangled state, ψ>=(1/sqrt(2))(ud>  du>), flying apart toward the two detectors. Each detector contains a SternGerlach magnet, oriented along one of three directions, perpendicular to the line of flight of the particles, and separated by 120 degrees, as indicated in the figure below.
Source: Physics Today.
> 
psi := 1/sqrt(2)*(KroneckerProduct(u, d)  KroneckerProduct(d, u));


(3.1) 
When the switches have the same setting, the spins of both particles are or , with equal probability. We can calculate the mean value of the product of the spins, or the correlation, to be
> 
DotProduct(psi, KroneckerProduct(s(0), s(0)).psi);


(3.2) 
> 
DotProduct(psi, KroneckerProduct(s(2*Pi/3), s(2*Pi/3)).psi);


(3.3) 
> 
DotProduct(psi, KroneckerProduct(s(4*Pi/3), s(4*Pi/3)).psi);


(3.4) 
When the switches have different settings (say 0 and 120 degrees), the correlation is
> 
DotProduct(psi, KroneckerProduct(s(0), s(2*Pi/3)).psi);


(3.5) 
This correlation can be understood based on our discussion in the first section. If detector A record spin up, then the particle arriving at detector B must be in d> state. But d> can be written as d> = a one> + b two>, where one> and two> are the eigenvectors of detector B when the magnets are rotated by 120 degrees.
> 
a; b; a^2*(+1) + b^2*(1);


(3.6) 
The probability of having the spins of both particles is , and having is . The correlation is thus .
To establish the third feature of the data, we must show that the mean value of the product of the spins vanishes when averaged over all the nine distinct pairs of orientations:
> 
sum(sum(DotProduct(psi, KroneckerProduct(s(i*2*Pi/3), s(j*2*Pi/3)).psi), i=0..2), j=0..2);


(3.7) 
This completely accounts for the data.
More generally, if we rotate the first SternGerlach magnets about the particle's line of flight by an angle , and the second by an angle , the correlation is
> 
DotProduct(psi, KroneckerProduct(s(theta), s(theta1)).psi);


(3.8) 

(3.9) 
We plot this correlation as a function of the difference of the two angles, and compare it with the correlation under the local realism theory.
> 
plot([abs(theta)/(Pi/2)  1, cos(theta)], theta = Pi..Pi, linestyle = [3, 1], title = "correlation", legend = ["local realism", "quantum mechanics"]);

Experimental results by Alain Aspect and other scientists show excellent agreement with the quantum mechanical predictions.


Teleportation


In 1993, Charles H. Bennett and his collaborators published a paper in Physical Review Letters demonstrating that teleportation is possible in quantum mechanics through dual classical and EPS channels.
Suppose Alice has been given a spin1/2 particle prepared in a state au> + bd>:

(4.1) 
To provide Bob with all the information in ψ>, one can prepare an entangled state
> 
Phi0 := 1/sqrt(2)*(KroneckerProduct(u, u) + KroneckerProduct(d, d));


(4.2) 
This is one of the four Bell basis states. The other three are
> 
Phi1 := KroneckerProduct(IdentityMatrix(2), s1).Phi0;


(4.3) 
> 
Phi2 := KroneckerProduct(IdentityMatrix(2), s2).Phi0;


(4.4) 
> 
Phi3 := KroneckerProduct(IdentityMatrix(2), s3).Phi0;


(4.5) 
One particle in Φ0> (called particle 2) is given to Alice, and the other (particle 3) is given to Bob. The complete state of the three particles is ψ123>
> 
psi123 := KroneckerProduct(psi, Phi0);


(4.6) 
We can express direct product of particles 1 and 2 in terms of the Bell states. Let particle 3 be the following unknown vectors

(4.7) 

(4.8) 

(4.9) 

(4.10) 
We solve for these unknowns:
> 
P := KroneckerProduct(Phi0, B0) + KroneckerProduct(Phi1, B1) + KroneckerProduct(Phi2, B2) + KroneckerProduct(Phi3, B3);


(4.11) 

(4.12) 
> 
soln := solve({seq(epr1[i,1] = 0, i=1..8)}, {alpha0, alpha1, alpha2, alpha3, beta0, beta1, beta2, beta3});


(4.13) 
Observe that each of these four states is related to the original ψ> in a simple way.

(4.14) 

(4.15) 

(4.16) 

(4.17) 
In summary, we have written the threeparticle system ψ123> in terms of the following
> 
psi123 = 1/2*(KroneckerProduct(Phi0, psi) + KroneckerProduct(Phi1, s1.psi)  KroneckerProduct(Phi2, s2.psi) + KroneckerProduct(Phi3, s3.psi));


(4.18) 
When Alice measures the state of particles 1 and 2 in the Bell basis, the four outcomes are equally likely: . If Alice gets Φ0>, then Bob's particle is just ψ>. If Alice gets Φ1>, then Bob's particle goes into the state s1ψ>. After Alice informs Bob of her result, Bob can apply s1 to his particle to recover ψ>:

(4.19) 
Similarly, if Alice gets Φ2>, Bob will apply s2 to his particle:

(4.20) 
And if Alice gets Φ3>, Bob should apply s3 to his particle:

(4.21) 
We have found the method to recover ψ> for the particle in Bob's possession in all 4 cases of Alice's measurement result.


References


1. Alain Aspect, Philippe Grangier, and Gerard Roger, "Experimental realization of EinsteinPodolskyRosenBohm Gedankenexperiment: A new violation of Bell's inequalities," Physical Review Letter, 49, 9194 (1982).
2. J. S. Bell, Speakable and unspeakable in quantum mechanics, 2nd ed. (Cambridge University Press, 2004).
3. Charles H. Bennett, Gilles Brassard, Claude Crepeau, Richard Jozsa, Asher Peres, and William K. Wootters, "Telporting an unknown quantum state via dual classical and EinsteinPodolskyRosen channels," Physical Review Letters, 70, 18951899 (1993).
4. N. David Mermin, "Bringing home the atomic world: Quantum mysteries for anybody," American Journal of Physics, 49, 940943 (1981).
5. N. David Mermin, "Is the moon there when nobody looks? Reality and the quantum theory." Physics Today, (April 1985), 3847.
6. Barton Zweibach, MIT 8.05 Quantum Physics II Lecture Notes, http://ocw.mit.edu/courses/physics/805quantumphysicsiifall2013/lecturenotes/.

