- #1
fog37
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Hello everyone and happy new year. I have a quick dilemma on the concept of entanglement for a two-particle system and I am looking for some clarity:
Say we have particle ##A## and particle ##B##. Both particles can be in either of the two states that we call ##|\Phi>## and ## |\beta>## or in a superposition of them.
Particle ##A##'s most general state is the superposition $$\Psi_1 = a_1 |\Phi_1> +b_1 |\beta_1>$$
Is it possible for particle ##A## to be in a product state like ## |\Phi_1> |\beta_1>## which is a product of the two states ##|\Phi_1>## and ## |\Phi_2>## ? If so, what would that state physically mean?
Particle ##B##'s most general state is $$\Psi_2 = a_2 |\Phi_2> +b_2 |\beta_2>$$. Using the idea of tensor product, we can algebraically multiply ##\Psi_1## by ##\Psi_2##. What we get is the following: $$a_1 a_2 |\Phi_1> |\Phi_2> +a_1 b_2 |\Phi_1> |\beta_2> + b_1 a_2 |\beta_1> |\Phi_2> +b_1 b_2 |\beta_1> |\beta_2>$$
This summation is the most general joint state for the two particles together. Each one of the four addends is also a joint state, i.e. it is a product between the state of particle ##A## times the state of particle ##B##...
For no entanglement to exist, all terms except one must be zero. Is that correct? If so, why? I am still confused on this simple matter...
thank you
Say we have particle ##A## and particle ##B##. Both particles can be in either of the two states that we call ##|\Phi>## and ## |\beta>## or in a superposition of them.
Particle ##A##'s most general state is the superposition $$\Psi_1 = a_1 |\Phi_1> +b_1 |\beta_1>$$
Is it possible for particle ##A## to be in a product state like ## |\Phi_1> |\beta_1>## which is a product of the two states ##|\Phi_1>## and ## |\Phi_2>## ? If so, what would that state physically mean?
Particle ##B##'s most general state is $$\Psi_2 = a_2 |\Phi_2> +b_2 |\beta_2>$$. Using the idea of tensor product, we can algebraically multiply ##\Psi_1## by ##\Psi_2##. What we get is the following: $$a_1 a_2 |\Phi_1> |\Phi_2> +a_1 b_2 |\Phi_1> |\beta_2> + b_1 a_2 |\beta_1> |\Phi_2> +b_1 b_2 |\beta_1> |\beta_2>$$
This summation is the most general joint state for the two particles together. Each one of the four addends is also a joint state, i.e. it is a product between the state of particle ##A## times the state of particle ##B##...
For no entanglement to exist, all terms except one must be zero. Is that correct? If so, why? I am still confused on this simple matter...
thank you
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