Pure vs Mixed States
| A pure state is a single, definite quantum state $ | \psi\rangle \in \mathcal{H}$, described by the rank-1 projector: |
| A mixed state arises from classical uncertainty about which pure state the system is in. If the system is in state $ | \psi_i\rangle$ with probability $p_i$ (a classical ensemble), the density matrix is: |
where $p_i \geq 0$ and $\sum_i p_i = 1$. The ensemble is not unique: many different ensembles give the same $\rho$, and $\rho$ contains all physically observable information.
Characterising properties of valid density matrices:
| Property | Statement | Meaning |
|---|---|---|
| Hermitian | $\rho = \rho^\dagger$ | Observable (real eigenvalues) |
| Positive semidefinite | $\rho \geq 0$ | Probabilities are non-negative |
| Unit trace | $\text{tr}(\rho) = 1$ | Probabilities sum to 1 |
| Purity | $\text{tr}(\rho^2) \leq 1$ | Equality iff pure state |
| Example: A qubit with equal classical probability of being $ | 0\rangle$ or $ | 1\rangle$ (maximally mixed): |
| Compare with the pure state $ | +\rangle\langle + | $: |
Both have $\text{tr}(\rho) = 1$, but $\text{tr}(\rho^2) = 1/2$ for the mixed state and $= 1$ for the pure state.
Bloch Sphere Representation
Any single-qubit density matrix can be written uniquely as:
\[\rho = \frac{1}{2}(I + \mathbf{r}\cdot\boldsymbol{\sigma}) = \frac{1}{2}(I + r_x X + r_y Y + r_z Z)\]| where $\mathbf{r} = (r_x, r_y, r_z) \in \mathbb{R}^3$ is the Bloch vector with $ | \mathbf{r} | \leq 1$. |
The Bloch vector components are expectation values of the Pauli operators:
\[r_k = \text{tr}(\sigma_k \rho) = \langle\sigma_k\rangle, \quad k \in \{x, y, z\}\]Interpretation:
-
$ \mathbf{r} = 1$: Pure state (surface of Bloch sphere). $\rho^2 = \rho$. -
$ \mathbf{r} < 1$: Mixed state (interior of Bloch sphere). $\rho^2 \neq \rho$. -
$ \mathbf{r} = 0$: Maximally mixed state $\rho = I/2$ (centre of sphere).
| Purity: $\text{tr}(\rho^2) = \frac{1}{2}(1 + | \mathbf{r} | ^2)$. |
Under a unitary gate $U$, the density matrix transforms as $\rho \to U\rho U^\dagger$, which rotates the Bloch vector: $\mathbf{r} \to R_U\,\mathbf{r}$ where $R_U \in SO(3)$ is the corresponding rotation matrix.
Partial Trace and Reduced Density Matrices
For a bipartite system $AB$ in state $\rho_{AB} \in \mathcal{H}_A \otimes \mathcal{H}_B$, the partial trace over subsystem $B$ gives the reduced density matrix of $A$:
\[\rho_A = \text{tr}_B(\rho_{AB}) = \sum_j \langle j|_B \,\rho_{AB}\, |j\rangle_B\]| where ${ | j\rangle_B}$ is any orthonormal basis for $\mathcal{H}_B$. The result is independent of the chosen basis. |
Physical meaning: $\rho_A$ correctly predicts the statistics of any measurement performed on subsystem $A$ alone:
\[\langle A \otimes I \rangle = \text{tr}_{AB}[(A \otimes I)\rho_{AB}] = \text{tr}_A[A\,\rho_A]\]| Example — Bell state: For $ | \Phi^+\rangle = \frac{1}{\sqrt{2}}( | 00\rangle + | 11\rangle)$: |
The reduced state of Alice is maximally mixed, reflecting complete entanglement — Alice’s local state is maximally uncertain, yet the joint state is pure.
Von Neumann Entropy
The von Neumann entropy generalises the Shannon entropy to quantum states:
\[S(\rho) = -\text{tr}(\rho \log_2 \rho) = -\sum_k \lambda_k \log_2 \lambda_k\]where $\lambda_k$ are the eigenvalues of $\rho$ (with $0 \log 0 \equiv 0$).
Properties:
| Property | Statement |
|---|---|
| Non-negativity | $S(\rho) \geq 0$ |
| Pure states | $S(\rho) = 0$ iff $\rho$ is pure |
| Maximum | $S(\rho) \leq \log_2 d$ where $d = \dim\mathcal{H}$; achieved by $\rho = I/d$ |
| Unitary invariance | $S(U\rho U^\dagger) = S(\rho)$ |
| Concavity | $S(\sum_i p_i\rho_i) \geq \sum_i p_i S(\rho_i)$ |
| Subadditivity | $S(\rho_{AB}) \leq S(\rho_A) + S(\rho_B)$ |
| Strong subadditivity | $S(\rho_{ABC}) + S(\rho_B) \leq S(\rho_{AB}) + S(\rho_{BC})$ |
Quantum mutual information:
\[I(A:B) = S(\rho_A) + S(\rho_B) - S(\rho_{AB}) \geq 0\]For a pure bipartite state, $S(\rho_A) = S(\rho_B)$ (Schmidt symmetry), and $I(A:B) = 2S(\rho_A)$ equals twice the entanglement entropy.
Quantum Channels as CPTP Maps
Evolution of open quantum systems is described by quantum channels — completely positive, trace-preserving (CPTP) maps $\mathcal{E}: \mathcal{L}(\mathcal{H}{\text{in}}) \to \mathcal{L}(\mathcal{H}{\text{out}})$.
Kraus representation: Every CPTP map can be written as:
\[\mathcal{E}(\rho) = \sum_k K_k \rho K_k^\dagger, \quad \sum_k K_k^\dagger K_k = I\]where the Kraus operators $K_k$ are not unique (related by unitary freedom). The minimum number of Kraus operators is the Kraus rank.
Physical interpretation: $K_k$ corresponds to the environment “jumping” to outcome $k$ (Kraus operators arise from tracing out the environment after a joint unitary $U_{SE}$).
Common channels and their Kraus operators:
| Channel | Kraus operators | Bloch effect |
|---|---|---|
| Unitary | $K_0 = U$ | Rotation |
| Depolarising ($p$) | $\sqrt{1-p}I, \sqrt{p/3}X, \sqrt{p/3}Y, \sqrt{p/3}Z$ | Shrinks $\mathbf{r} \to (1-4p/3)\mathbf{r}$ |
| Dephasing ($p$) | $\sqrt{1-p}I, \sqrt{p}Z$ | Shrinks $r_x, r_y \to (1-2p)r_x, (1-2p)r_y$ |
| Amplitude damping ($\gamma$) | $K_0, K_1$ as above | Pulls Bloch vector toward $\mathbf{r} = \hat{z}$ |
Complete positivity (vs merely positivity) is required because $\mathcal{E}$ must remain positive when applied to one part of an entangled state: $(\mathcal{E} \otimes I)(\rho_{AB}) \geq 0$. The Choi-Jamiołkowski isomorphism maps each channel to a positive semidefinite state (the Choi matrix), enabling efficient characterisation.
Lindblad Master Equation
For Markovian open systems (short environment memory), the time evolution of $\rho$ obeys the Lindblad master equation:
\[\frac{d\rho}{dt} = -\frac{i}{\hbar}[H, \rho] + \sum_k \gamma_k\left(L_k \rho L_k^\dagger - \frac{1}{2}L_k^\dagger L_k \rho - \frac{1}{2}\rho L_k^\dagger L_k\right)\]where $H$ is the system Hamiltonian, $L_k$ are jump operators (Lindblad operators) describing dissipation channels, and $\gamma_k \geq 0$ are decay rates.
Interpretation of terms:
- $-\frac{i}{\hbar}[H, \rho]$: Unitary evolution (Hamiltonian part).
- $L_k \rho L_k^\dagger$: Quantum jump — system jumps into a new state.
- $-\frac{1}{2}{L_k^\dagger L_k, \rho}$: Anti-commutator term ensures trace preservation.
Example — single-qubit amplitude damping ($T_1$ relaxation):
\[L_1 = \sqrt{\gamma}\, \sigma^- = \sqrt{\gamma}\begin{pmatrix}0&1\\0&0\end{pmatrix}\] \[\frac{d\rho_{11}}{dt} = -\gamma\rho_{11}, \quad \frac{d\rho_{01}}{dt} = -\frac{\gamma}{2}\rho_{01}\]Solution: $\rho_{11}(t) = \rho_{11}(0)e^{-\gamma t}$, $\rho_{01}(t) = \rho_{01}(0)e^{-\gamma t/2}$. The off-diagonal (coherence) decays at half the rate of the population — the $T_2 = 2T_1$ relation in the absence of pure dephasing.
Adding a pure dephasing channel $L_2 = \sqrt{\gamma_\phi}Z$:
\[\frac{d\rho_{01}}{dt} = -\left(\frac{\gamma}{2} + \gamma_\phi\right)\rho_{01} \equiv -\frac{1}{T_2}\rho_{01}\]giving $\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\phi}$.