What Are the Validity Conditions for Multi-Qubit Density Matrices?

[m~ray]

1 qubit can be expressed as 1/2(I + n.σ). where n = (n_x,n_y,n_z) is a 3D vector, with size <=1. Hence the condition is that the sum of squares of the variables must be 1 or less. In the general expression of multi-qubit systems, we tensor product these individual qubits and also add the correlator terms. For example a 3 qubit general state will have 63 variables. The question is what are the conditions on these variables in order for it to be a valid density matrix ?

 

[eljose79]

I can confirm that the expression 1/2(I + n.σ) is a valid representation of a single qubit state. The condition that the sum of squares of the variables must be 1 or less is known as the normalization condition, which ensures that the state has a probability of 1 when measured.

In the general expression of multi-qubit systems, the individual qubits are combined through tensor product and correlator terms are added. The total number of variables in a 3 qubit system would be 63, as mentioned in the forum post. In order for this system to be a valid density matrix, certain conditions must be met.

Firstly, the normalization condition must be satisfied for each individual qubit in the system. This means that the sum of squares of the variables in each qubit must be 1 or less.

Secondly, the overall state must be a valid quantum state, which means that it must be a superposition of basis states and the coefficients must be complex numbers with unit magnitude. This ensures that the state is normalized and can be used to calculate probabilities.

Lastly, the density matrix must be Hermitian, meaning that it must be equal to its own conjugate transpose. This ensures that the matrix is self-adjoint and can be used to calculate expectation values.

In summary, the conditions for a valid density matrix in a multi-qubit system are: normalization of individual qubits, valid quantum state, and Hermiticity. These conditions ensure that the density matrix accurately represents the quantum state of the system and can be used for calculations and predictions.