# How can I write the maximally mixed state on m qubits as a linear combination of basis vectors?

The maximally mixed state on m qubits is defined to be the quantum state with associated density operator $$\rho_m = \frac{1}{2^m} I$$. Examples are

• On one qubit this is $$\rho_1 = \frac{1}{2}(|0\rangle\langle0|+|1\rangle\langle1|) = \frac{1}{2}I$$
• On two qubits we have $$\rho_2 = \frac{1}{4} (|00\rangle\langle00| +|01\rangle\langle01|+|10\rangle\langle10|+|11\rangle\langle11|$$.

My question is the following: how can the corresponding state vector $$|\phi_m\rangle$$ be expressed in terms of the standard basis elements, eg. $$|\phi_2\rangle = \sum_{i,j} a_{i,j}|ij\rangle$$? What are the values of $$a_{i,j}$$?

If by "corresponding state vector" you mean a pure state $$\lvert\psi\rangle$$ such that $$\lvert\psi\rangle\!\langle\psi\rvert$$ is maximally mixed, then the answer is that there isn't one.
A density matrix $$\rho$$ can be written as $$\rho=\lvert\psi\rangle\!\langle\psi\rvert$$ for some ket state $$\lvert\psi\rangle$$ if and only if it is pure. One easy way to check for this is to compute $$\mathrm{Tr}(\rho^2)$$, which is the so-called purity of the state, and equals $$1$$ if and only if the state is pure.
• Well what I meant by corresponding is this: a one qubit system can be described either via it's density matrix or just as an element of the Hilbert space in question ie as $|\phi\rangle = \sum_i |\phi_i\rangle$ for basis vectors $|\phi_i\rangle$. Hmm, on a second look I think I might be confusing mixed states with superposition... So I am guessing when I say $|\phi\rangle$ can be written as linear combination of basis vectors, that's only true for pure states, right? That would explain my confusion. – gen Jun 9 '19 at 14:53
• @gen yes, when you say that a state can be described either via its density matrix $\rho$ or its ket state $\lvert\psi\rangle$, the correspondence between the two is that $\rho=\lvert\psi\rangle\!\langle\psi\rvert$. Not all density matrices correspond to a ket (pure) state. They only do when the state is pure. Otherwise, the density matrices are mixtures of DMs corresponding to pure states, that is, objects of the form $\rho=\sum_k p_k\lvert\psi_k\rangle\!\langle\psi_k\rvert$ for some ensemble of pure states $\{\lvert\psi_k\rangle\}$ – glS Jun 9 '19 at 14:55