General 3-Qubit Mixed State: 63 Variables & Bloch Vectors

[m~ray]

How do I write the general form of a 3 qubit mixed state ? It has 63 variables. Well one way is to use bloch vectors. So basically we take 3 single qubit mixed states (3 variables each) and tensor product them. Then we add the 2 qubit correlators and 3 qubit correlator terms. Finally we can divide it by its trace (to make its trace 1). It is hence unit trace and hermitian. However positive semidifinite condition is violated.

 

[eljose79]

Thank you for your question! Writing the general form of a 3 qubit mixed state can be done using the Bloch vector representation, as you have mentioned. However, there are other ways to write the general form as well.

One approach is to use the density matrix formalism, where the mixed state is represented by a 8x8 matrix. This matrix can be written as a linear combination of the 64 possible basis states, each of which corresponds to a specific combination of qubit states (e.g. |000>, |001>, |010>, etc.). The coefficients of this linear combination are the 63 variables that you have mentioned.

Another approach is to use the Pauli matrices, which are a set of 4x4 matrices that represent the possible spin states of a single qubit. By taking the tensor product of three Pauli matrices, we can construct a 64x64 matrix which represents the mixed state. This matrix can also be written as a linear combination of the 64 possible basis states, with the coefficients being the 63 variables.

In both of these approaches, the positive semidefinite condition is automatically satisfied, as it is a necessary condition for a valid density matrix. However, the trace condition must still be enforced by dividing the matrix by its trace.

I hope this helps in writing the general form of a 3 qubit mixed state. it is important to explore different approaches and find the one that best suits your needs and research goals. Good luck!