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M.E. Irizarry-Gelpí

Physics impostor. Mathematics interloper. Husband. Father.

Bell States and Tensor Products


The 1-qubit Bell states are given by

|+=12(|0+|1)|=12(|0|1)

These are ortho-normal eigen-states of X:

X|+=+1|+X|=1|

Taking tensor products gives a 2-qubit ortho-normal basis:

|++|+|+=12(|00+|01+|10+|11)|+|+|=12(|00|01+|10|11)|+||+=12(|00+|01|10|11)|||=12(|00|01|10+|11)

Note that, by construction, the four vectors in this 2-qubit basis do not describe 2-body entanglement since they are all tensor products. Linear combinations do, giving the 2-qubit Bell states:

|Φ+=12(|00+|11)=12(|+++|)|Φ=12(|00|11)=12(|++|+)|Ψ+=12(|01+|10)=12(|++|)|Ψ=12(|01|10)=12(|+|+)

These states cannot be factorized as tensor products.

Another 1-qubit basis is the two eigen-states of Y:

|=12(|0+i|1)|=12(|0i|1)

Taking tensor products gives another 2-qubit ortho-normal basis:

|||=12(|00+i|01+i|10|11)|||=12(|00i|01+i|10+|11)|||=12(|00+i|01i|10+|11)|||=12(|00i|01i|10|11)

I kind of like this notation: the star suggests complex conjugation.