From density matrix, how can I know what state it belongs to

[munirah]

Homework Statement

Given a density matrix of three qubit pure state, how can I know after do some transformation, this state belong to what class?. Class I mean here, either separable state, biseparable, GHZ state or W state?

I mean here what is the indicator to me classify it?
It is the indicator must has been done by measuring the state with certain measure?
If yes (mean have to measure it) , what the range that must be satisfy of each class (class-separable,biseparable, GHZ and W).

Please help me.

thank you

 

[jambaugh]

Keep in mind that the definitions are relative to a particular factorization of the original "state" into factor "states".

(I enquote the word "state" because of it's philosophical baggage, and prefer the term "mode" indicating it represents a class of physical systems rather that representing the "state of the system")

Assuming a sharp mode initially then the total system will have 0 entropy. ($S=-\kappa trace(\rho ln(\rho)$) If it is representative of a single tensor product of three sharp modes then you will note that the partial entropies for each factor system obtained from the reduced density matrices (partial traces over other factor systems of the composite density matrix) should also be zero. In general, even if the composite system is not in a sharp mode, if the composite entropy is the sum of the partial entropies then this indicates no information is lost by considering the system as three separate parts, i.e. there is no entanglement.

You can also check this in combinations. For system with factors A, B, and C tracing over C gives a reduced AB composite density operator and you can then compare the entropy of that composite to the further reduced factors of A and B each traced over C and the other.

I'm not sure about actually quantifying entanglement but this entropy comparison can allow qualifying its existence or absence. This may be helpful in answering your specific questions.

 

[munirah]

Keep in mind that the definitions are relative to a particular factorization of the original "state" into factor "states".

(I enquote the word "state" because of it's philosophical baggage, and prefer the term "mode" indicating it represents a class of physical systems rather that representing the "state of the system")

Assuming a sharp mode initially then the total system will have 0 entropy. ($S=-\kappa trace(\rho ln(\rho)$) If it is representative of a single tensor product of three sharp modes then you will note that the partial entropies for each factor system obtained from the reduced density matrices (partial traces over other factor systems of the composite density matrix) should also be zero. In general, even if the composite system is not in a sharp mode, if the composite entropy is the sum of the partial entropies then this indicates no information is lost by considering the system as three separate parts, i.e. there is no entanglement.

You can also check this in combinations. For system with factors A, B, and C tracing over C gives a reduced AB composite density operator and you can then compare the entropy of that composite to the further reduced factors of A and B each traced over C and the other.

I'm not sure about actually quantifying entanglement but this entropy comparison can allow qualifying its existence or absence. This may be helpful in answering your specific questions.

thank you so much .