Entanglement(theoretically Explained)

In this article, we will discuss the following topics:

• QKD with Entangled States
• Entangled States
• Salient Features of Entanglement in QKD
• Verifying Entanglement

we will first learn the Basics of Entangled state / Bell State. It’s circuit basically consist of Hadamard gate and CNOT gate. following is the calculation to achive the entangled state.

So we have Four Maximally Entangled States, mentioned below

Following is the circuit of 4 Bell State:

with the variation of Z and X gate.

X and Z operation acts exactly oppsite on Standard basis and Hadamard basis.

X on standard basis acts as not gate.like X|0> -->|1>
                                          X|1> --> |0>

z on hadamard basis acts as not  gate. like Z|+> --> |->
                                            Z|-> --> |+>

X on Hadamard basis X|+> --> |+>
                    X|-> -->-|+>

Z on Standard basis Z|0> --> |0>
                    Z|1> --> -|1>

so if we make a measurement on |∅+> for the Alice and Bob entangled photon and if we get |0> on Alice side then its sure Bob will receive |1> OR Vice Versa. And in any of the above mentioned bell state we can see the same thing.

These bell state forms the basis set .

Salient features of Entanglement

1. Randomness
2. Whole vs Part
3. Monogamy of Entanglement

Randomness

will look Entanglement from perspective of Classical Way called Classical Correlation and then will look into the Quantum Correlation .

Classical Correlation:

suppose we have a pair of hand gloves, and what we do is select one randomly and put one on Earth and the other on Moon, then after we disclose whether its right hand or left hand glove be it on the Earth or moon , will be able to surely guess (100% surety) the other half.

this is what Classically correlation means. But not that here Randomness is because of Ignorance(Randomness → Ignorance).

Quantum Entanglement:

here we have two plain polarized light with basis as Hrizontal (purple)and vertical (yellow).It collapses into one of the two at the time of measurement But whatever it collapses into the other collapses into corresponding State.

Hence it is the totally random state we get as it was not predecided .as the state was nit already there but collapses at the time of measurement and decides which state is it going to.

Here, (Randomness → Intrinsic).

2. Whole vs Part of Entanglement States

classically knowing the whole ==> knowing the part

but in Entanglement State if we know the whole then the part is totally random, following is the one showing the maximally entangled state as whole.

looking at the part of it menas will take the density matrix and then take a trace over one of the state.

so looking at the state of one particle we cant tell what the original state was, as its clear from the above outcome equations.

Density matrix after measurement on density matrix

Density matrices for pure states and mixed states

3. Monogamy of Entanglement

ensure maximal entangle that means we have ensure the perfect security and there is no third connection(2 maximally entangled are totally disentangled from the third)

4. verifying entanglement

CHSH inequality :

we have four measurement basis X,Z W and V.

w=(X+Z)/2^(1/2)

V=(Z-X)/2^(1/2)

measurement outcome are +-1 for X,Z,W,V.

classical correlation(C)=Z1W2 + X1W2 + Z1V2 - X1V2

if we get the c>1/2 its says that CHSH inequality is violated means its successfully entangled