Entanglement Flashcards
Composite system
|q1>=a0|0>+a1|1>
|q2>=b0|0>+b1|1>
|phi>=a0b0|00>+a0b1|01>+a1b0|10>+a1b1|11>
This is a superposition. The sum of the amplitudes squared is 1.
|01> tensor of |0>|1>=[1 0]’[0 1]’=[0 1 0 0]’
Can we derive the qbits of a composite system ?
Take the general form:
|q1>=a0|0>+a1|1>
|q2>=b0|0>+b1|1>
|phi>=a0b0|00>+a0b1|01>+a1b0|10>+a1b1|11>
Then replace the amplitudes of the composite system and see what a1,a2,b1,b2 should equal to.
If there is a contradiction: the 2 states are entangled and they cannot be extracted !
if a0b0.a1b1/=a0b1.a1b0 then entangled ! if equal, unentangled
if unentangled:
put common factor of 1stqbit and amplitudes (do length parenthesis).
x0|0>(y0|0>+y1|1>)+x1|1>(y0|0>+y1|1>)
the 2 parenthesis should be the same, otherwise would mean entangled.
then we have (x0|0>+x1|1>)(y0|0>+y1|1>)
The 2 qbits are unentangled
Entangled
State where qbits cannot be described individually
If we measure the 1st qbit, it collapse and we know what the second qbit will be.
This does not imply an exchange of information !
Bell’s State
Entangled states
1/sqrt(2)|00> + 1/sqrt(2)|11>
Changing base will always give the same !
(rewrite +/- in terms of 0/1 : |+> = 1/sqrt(2) |0> + 1/sqrt(2) |1> and |-> = 1/sqrt(2) |0> - 1/sqrt(2) |1> then 1/sqrt(2)|++> + 1/sqrt(2)|- -> with this new notation and cancel out terms until finding 1/sqrt(2)|00> + 1/sqrt(2)|11>)
1/sqrt(2)|00> + 1/sqrt(2)|11>=1/sqrt(2)|++> + 1/sqrt(2)|–>
state will collapse accordingly no matter which base we measure in
This equality is the base of the EPR paradox
EPR paradox
measuring 1 entangled qbit means that the other collapse: info exchange ?
This would break the uncertainty principle: closer you are to a base the more uncertain you are to the other.
This paradox is rejected
CNOT operator
Unitary matrix
[1 0 0 0; 0 1 0 0; 0 0 0 1; 0 0 1 0]
Second qbit flips if 1st qbit is 1
This output an entangled state
Flip operator
Unitary matrix
[0 0 0 1; 0 0 1 0; 0 1 0 0; 1 0 0 0]
both qbits flip
Partial measurement rule
if have
|phi>=a0b0|00>+a0b1|01>+a1b0|10>+a1b1|11>
and we measure 1st qbit, eg |0>
then second qbit is narrowed to: b0|00>+ b1|01>
the superposition is |0>(x)(b0|0>+ b1|1>)/(sqrt(b0^2+b1^2)
NOT operator
NOT is [0 1;1 0]
flips the qbit
Operations on Qbit
tensor product of the matrices
if no transformation : I
if transformation: U
eg: no transformation on 1st qbit and transformation on 2nd
-> I (x) U
This allows to use tensor product to build larger unitary matrices
does not work for CNOT !
