What is density matrix of one on two entangled qubits?

[Dims]

Let we have two qubits A and B. First qubit has eigenstates |A0> and |A1>, and second has |B0> and |B1>.

Let them be in the entangled state, described with vector

c1 * |A0> * |B0> + c2 * |B0> * |B1>|

where c1 and c2 are complex numbers with |c1|^2 + |c2|^2 = 1.

Then what is density matrix, describing only first qubit separately? Is this matrix 1x1 or 2x2?

 

[Dims]

Is the correct answer is 1x1 matrix

|c1|^2 * |A0> * <A0| + |c2|^2 * |A1> * <A1|

?

 

[denian]

The density matrix of one of two entangled qubits is a 2x2 matrix that represents the state of the first qubit, also known as subsystem A. It is calculated by taking the partial trace of the density matrix of the entire system, which in this case includes both qubits A and B. The resulting matrix will have elements that correspond to the probabilities of the first qubit being in its two eigenstates, |A0> and |A1>. The diagonal elements will represent the probabilities of measuring the first qubit in the states |A0> and |A1>, while the off-diagonal elements will represent the coherences between these two states. The density matrix for the first qubit will depend on the values of c1 and c2, as well as the entangled state of the two qubits. It is important to note that the density matrix for the first qubit will not be a pure state, as it will have non-zero off-diagonal elements indicating the presence of quantum entanglement with the second qubit.