Density Matrices and Mixed States

Density Matrices and Mixed States#

Pure vs. mixed states, the density operator \(\rho\), partial trace, reduced density matrices, and decoherence.

So far, we have described quantum states as vectors \(|\psi\rangle\) in a Hilbert space. This is sufficient for isolated quantum systems in a known state. But in practice, quantum systems interact with their environment, and we often have incomplete knowledge about which state a system is in. The density matrix formalism extends quantum mechanics to handle both situations.

Pure States as Density Matrices#

Any pure state \(|\psi\rangle\) can be equivalently described by the density matrix:

\[\rho = |\psi\rangle\langle\psi|\]

For a qubit in state \(|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\):

\[\begin{split}\rho = \begin{pmatrix} |\alpha|^2 & \alpha\beta^* \\ \alpha^*\beta & |\beta|^2 \end{pmatrix}\end{split}\]

Mixed States#

A mixed state represents classical uncertainty about which quantum state the system is in. If a system is in state \(|\psi_k\rangle\) with probability \(p_k\), the density matrix is:

\[\rho = \sum_k p_k |\psi_k\rangle\langle\psi_k|\]

where \(p_k \geq 0\) and \(\sum_k p_k = 1\).

Example: Maximally Mixed State#

If a qubit is equally likely to be \(|0\rangle\) or \(|1\rangle\):

\[\begin{split}\rho = \frac{1}{2}|0\rangle\langle 0| + \frac{1}{2}|1\rangle\langle 1| = \frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \frac{I}{2}\end{split}\]

This state has no off-diagonal elements (no coherence) and is completely random. It sits at the center of the Bloch sphere.

Properties of Density Matrices#

A valid density matrix must satisfy:

Property	Mathematical condition
Hermitian	\(\rho = \rho^\dagger\)
Positive semidefinite	\(\langle\phi\|\rho\|\phi\rangle \geq 0\) for all \(\|\phi\rangle\)
Unit trace	\(\text{Tr}(\rho) = 1\)

Distinguishing Pure from Mixed#

Test	Pure state	Mixed state
\(\text{Tr}(\rho^2)\)	\(= 1\)	\(< 1\)
Rank of \(\rho\)	1	\(> 1\)
Eigenvalues	One eigenvalue is 1, rest are 0	Multiple non-zero eigenvalues
Bloch vector length	\(\|\vec{r}\| = 1\)	\(\|\vec{r}\| < 1\)

The quantity \(\text{Tr}(\rho^2)\) is called the purity. It ranges from \(1/d\) (maximally mixed in \(d\) dimensions) to \(1\) (pure).

Evolution and Measurement with Density Matrices#

Unitary Evolution#

\[\rho \mapsto U\rho U^\dagger\]

Measurement#

The probability of outcome \(k\) when measuring observable \(\{P_k\}\):

\[p(k) = \text{Tr}(P_k \rho)\]

Post-measurement state:

\[\rho_k = \frac{P_k \rho P_k}{\text{Tr}(P_k \rho)}\]

Expectation Values#

\[\langle A \rangle = \text{Tr}(A\rho)\]

This replaces \(\langle\psi|A|\psi\rangle\) from the pure-state formalism and works for both pure and mixed states.

The Bloch Sphere for Mixed States#

Any single-qubit density matrix can be written as:

\[\begin{split}\rho = \frac{I + \vec{r} \cdot \vec{\sigma}}{2} = \frac{1}{2}\begin{pmatrix} 1 + r_z & r_x - ir_y \\ r_x + ir_y & 1 - r_z \end{pmatrix}\end{split}\]

where \(\vec{\sigma} = (X, Y, Z)\) and \(\vec{r} = (r_x, r_y, r_z)\) is the Bloch vector with \(|\vec{r}| \leq 1\).

\(|\vec{r}| = 1\): pure state (surface of the sphere)
\(0 < |\vec{r}| < 1\): mixed state (interior of the sphere)
\(|\vec{r}| = 0\): maximally mixed state (center)

Partial Trace and Reduced Density Matrices#

For a composite system \(AB\) in state \(\rho_{AB}\), the reduced density matrix of subsystem \(A\) is obtained by tracing out \(B\):

\[\rho_A = \text{Tr}_B(\rho_{AB}) = \sum_k (I_A \otimes \langle k|_B) \rho_{AB} (I_A \otimes |k\rangle_B)\]

where \(\{|k\rangle_B\}\) is any orthonormal basis for \(B\).

Example: Bell State#

For \(|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\):

\[\begin{split}\rho_{AB} = |\Phi^+\rangle\langle\Phi^+| = \frac{1}{2}\begin{pmatrix} 1&0&0&1 \\ 0&0&0&0 \\ 0&0&0&0 \\ 1&0&0&1 \end{pmatrix}\end{split}\]

Tracing out \(B\):

\[\rho_A = \text{Tr}_B(\rho_{AB}) = \frac{1}{2}(|0\rangle\langle 0| + |1\rangle\langle 1|) = \frac{I}{2}\]

The reduced state of each qubit in a maximally entangled pair is the maximally mixed state. This is a defining property of maximal entanglement: all the information is in the correlations, none in the individual qubits.

Entanglement Test

A pure bipartite state \(|\psi_{AB}\rangle\) is entangled if and only if its reduced density matrix \(\rho_A\) is mixed (i.e., \(\text{Tr}(\rho_A^2) < 1\)). The more mixed the reduced state, the more entangled the composite state.

Decoherence#

Decoherence is the process by which a quantum system loses its coherence (off-diagonal elements) through interaction with its environment. It is the main enemy of quantum computing.

Dephasing (T2 Process)#

The off-diagonal elements decay while diagonal elements are preserved:

\[\begin{split}\rho(t) = \begin{pmatrix} |\alpha|^2 & \alpha\beta^* e^{-t/T_2} \\ \alpha^*\beta e^{-t/T_2} & |\beta|^2 \end{pmatrix}\end{split}\]

After time \(t \gg T_2\), the state becomes a classical mixture with no interference capability.

Relaxation (T1 Process)#

The excited state \(|1\rangle\) decays to the ground state \(|0\rangle\) with time constant \(T_1\):

\[P(|1\rangle \to |0\rangle) = 1 - e^{-t/T_1}\]

\(T_1\) sets the maximum useful computation time. \(T_2 \leq 2T_1\) always.

Decoherence Times by Platform#

Platform	\(T_1\)	\(T_2\)
Superconducting	\(\sim 100\;\mu\)s	\(\sim 100\;\mu\)s
Trapped ion	\(\sim 1\) s	\(\sim 1\) s
NV center	\(\sim 6\) ms	\(\sim 1\) ms
Spin qubit (Si)	\(\sim 1\) s	\(\sim 1\) ms

Quantum error correction is needed when the computation time exceeds \(T_2\). Current error correction schemes require physical qubit error rates below \(\sim 10^{-3}\) per gate.