4.1 Quantum states and density matrices

Quantum states

Quantum states describe the complete physical situation of a system at the atomic and subatomic level. In statistical mechanics, they're the starting point for understanding how large collections of quantum particles produce the thermodynamic behavior you observe macroscopically. This section covers the mathematical machinery you need: state vectors, density matrices, ensembles, time evolution, and how measurement works in the quantum framework.

Pure states vs mixed states

A pure state describes a system where you have complete quantum information. It's represented by a single state vector $|\psi\rangle$ living in a Hilbert space, and it exhibits maximum quantum coherence.

A mixed state arises when you don't fully know the system's quantum state. Instead of a single vector, you describe it as a statistical ensemble of pure states, weighted by classical probabilities. Mixed states require the density operator $\rho$ to capture their probabilistic nature.

The key distinction matters for statistical mechanics: thermal systems are almost always in mixed states, because you lack complete information about every microscopic degree of freedom.

• Pure states satisfy $\rho^2 = \rho$ and $\text{Tr}(\rho^2) = 1$
• Mixed states have $\text{Tr}(\rho^2) < 1$, reflecting reduced coherence
• A pure state is a special case of a density matrix: $\rho = |\psi\rangle\langle\psi|$

Superposition principle

Superposition means a quantum system can exist in a linear combination of basis states simultaneously:

$|\psi\rangle = \sum_i c_i |\phi_i\rangle$

The coefficients $c_i$ are complex probability amplitudes. The probability of finding the system in basis state $|\phi_i\rangle$ is $|c_i|^2$, and normalization requires $\sum_i |c_i|^2 = 1$.

Superposition leads directly to interference effects (as in the double-slit experiment) and underpins quantum parallelism in quantum computing, where a register of qubits can represent many states at once.

Measurement and collapse

When you measure a quantum observable, the superposition collapses into a definite eigenstate of that observable. The probability of getting outcome $a$ is given by Born's rule:

$P(a) = |\langle a|\psi\rangle|^2$

This collapse is instantaneous and irreversible, which is fundamentally different from classical measurement where you're just revealing a pre-existing value. The measurement problem concerns exactly how and why this transition from superposition to a definite outcome occurs.

The quantum Zeno effect is a striking consequence: measuring a system frequently enough can freeze its evolution, effectively preventing transitions between states.

Entanglement

Two or more particles are entangled when their joint quantum state cannot be written as a product of individual states. Measuring one particle instantly constrains what you'll find when measuring the other, regardless of separation distance.

• The EPR paradox highlights this non-locality: entangled particles seem to share information faster than light, though no usable signal is transmitted
• Entanglement is quantified by measures like entanglement entropy (von Neumann entropy of the reduced density matrix) and concurrence
• It enables protocols like quantum teleportation and superdense coding

For statistical mechanics, entanglement matters because subsystems of a larger quantum system in a pure state are generally entangled with each other, producing mixed reduced density matrices and non-zero entanglement entropy.

Density matrices

The density matrix is the central tool for quantum statistical mechanics. It handles both pure and mixed states in a single formalism and connects naturally to ensemble averages and thermodynamic quantities.

Definition and properties

The density matrix is defined as:

$\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|$

where $p_i$ is the probability of the system being in pure state $|\psi_i\rangle$, with $\sum_i p_i = 1$.

Four properties you should know:

1. Hermitian: $\rho^\dagger = \rho$
2. Unit trace: $\text{Tr}(\rho) = 1$ (probabilities sum to 1)
3. Positive semidefinite: all eigenvalues $\geq 0$ (no negative probabilities)
4. Purity test: $\text{Tr}(\rho^2) = 1$ for pure states, $\text{Tr}(\rho^2) < 1$ for mixed states

The quantity $\text{Tr}(\rho^2)$ is called the purity and gives you a quick diagnostic for how mixed a state is.

Reduced density matrix

When you have a composite system $AB$ and you only care about subsystem $A$, you obtain the reduced density matrix by tracing out the degrees of freedom of $B$:

$\rho_A = \text{Tr}_B(\rho_{AB})$

This operation discards information about $B$ while preserving all predictions for measurements on $A$ alone. Even if the full system $AB$ is in a pure state, the reduced density matrix $\rho_A$ can be mixed, which is exactly what happens when $A$ and $B$ are entangled.

Reduced density matrices are essential for analyzing open quantum systems, where a small system interacts with a large environment.

Von Neumann entropy

The von Neumann entropy is the quantum generalization of Shannon entropy:

$S(\rho) = -\text{Tr}(\rho \log \rho) = -\sum_i \lambda_i \log \lambda_i$

where $\lambda_i$ are the eigenvalues of $\rho$. (The logarithm is typically base $e$ in physics, giving entropy in nats, or base 2 for bits in information theory.)

• $S = 0$ for a pure state (complete information)
• $S = \log d$ for a maximally mixed state in a $d$-dimensional Hilbert space (maximum ignorance)
• For a bipartite pure state, $S(\rho_A) = S(\rho_B)$, and this quantity measures the entanglement between $A$ and $B$

In statistical mechanics, von Neumann entropy connects directly to thermodynamic entropy when you use the canonical density matrix.

Partial trace operation

The partial trace is the mathematical procedure that produces reduced density matrices. For a product of operators acting on subsystems $A$ and $B$:

$\text{Tr}_B(|a\rangle\langle b| \otimes |c\rangle\langle d|) = |a\rangle\langle b| \cdot \langle d|c\rangle$

The factor $\langle d|c\rangle = \text{Tr}(|c\rangle\langle d|)$ is just the inner product on subsystem $B$. For a general operator, you expand it in a product basis and apply this rule term by term.

The partial trace preserves both the trace and Hermiticity of the original operator. It's the unique operation that correctly reproduces all local measurement statistics on the retained subsystem.

Quantum ensembles

Quantum ensembles bridge microscopic quantum mechanics and macroscopic statistical mechanics. They provide the framework for studying collective behavior of quantum systems, especially in thermal equilibrium.

Statistical mixture of states

A statistical mixture represents a collection of quantum systems, each prepared in some pure state $|\psi_i\rangle$ with classical probability $p_i$. This is distinct from a superposition: in a mixture, the system is definitely in one of the pure states, but you don't know which one.

The density matrix formalism handles this naturally. The same density matrix $\rho$ can correspond to different ensembles of pure states (this non-uniqueness is a key feature, not a bug). What matters physically is $\rho$ itself, not the particular decomposition into pure states.

Statistical mixtures arise naturally when a quantum system is in contact with a thermal reservoir, where the environment introduces classical uncertainty about the system's state.

Ensemble average

The expectation value of an observable $A$ in a state described by $\rho$ is:

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

This formula generalizes the pure-state result $\langle \psi|A|\psi\rangle$ and provides the direct link between quantum mechanics and measurable macroscopic quantities. All thermodynamic properties in quantum statistical mechanics are computed as ensemble averages using this trace formula.

The linearity of the trace ensures that ensemble averages respect the linearity of quantum mechanics: $\langle A + B \rangle = \langle A \rangle + \langle B \rangle$ always holds.

Quantum vs classical ensembles

The off-diagonal elements of the density matrix in a given basis represent quantum coherences. These are what distinguish a quantum ensemble from a classical probability distribution over the same set of states.

Time evolution

Time evolution describes how quantum states and density matrices change, which is necessary for predicting the dynamics of quantum systems in statistical mechanics.

Schrödinger equation for states

The time evolution of a pure state is governed by the Schrödinger equation:

$i\hbar \frac{\partial}{\partial t}|\psi(t)\rangle = H|\psi(t)\rangle$

where $H$ is the Hamiltonian operator. For a time-independent Hamiltonian, the solution is:

$|\psi(t)\rangle = U(t)|\psi(0)\rangle, \quad U(t) = e^{-iHt/\hbar}$

This evolution is unitary, meaning it preserves normalization, inner products, and superposition structure. No information is lost during unitary evolution.

Von Neumann equation for matrices

The density matrix evolves according to the von Neumann equation:

$i\hbar \frac{\partial \rho}{\partial t} = [H, \rho]$

where $[H, \rho] = H\rho - \rho H$ is the commutator. This is equivalent to the Schrödinger equation for pure states but also handles mixed states. It preserves the trace, Hermiticity, and positivity of $\rho$.

Note the sign difference from the Heisenberg equation of motion for operators: $i\hbar \dot{A} = [A, H]$. The density matrix evolves in the Schrödinger picture, opposite to operators in the Heisenberg picture.

For open quantum systems interacting with an environment, the von Neumann equation is replaced by master equations (like the Lindblad equation) that include non-unitary dissipative terms.

Unitary transformations

A unitary transformation satisfies $U^\dagger U = UU^\dagger = I$. Key properties:

• Generated by Hermitian operators: $U = e^{-iHt/\hbar}$
• Preserve inner products: $\langle U\psi | U\phi \rangle = \langle \psi | \phi \rangle$
• Preserve eigenvalue spectra and probability distributions
• Compose as $U_2 U_1$ for sequential operations

Unitary evolution describes closed quantum systems. When a system interacts with an environment, the combined system still evolves unitarily, but the subsystem alone undergoes non-unitary evolution, leading to decoherence.

Quantum statistical mechanics

This is where quantum mechanics meets thermodynamics. The density matrix formalism makes this connection precise.

Canonical ensemble in quantum systems

A quantum system in thermal equilibrium with a heat bath at temperature $T$ is described by the canonical density matrix:

$\rho = \frac{1}{Z}e^{-\beta H}$

where $\beta = 1/(k_B T)$ and $Z$ is the partition function. This density matrix is diagonal in the energy eigenbasis, with each eigenstate $|n\rangle$ weighted by the Boltzmann factor $e^{-\beta E_n}$.

This form can be derived by maximizing the von Neumann entropy subject to a fixed average energy constraint, which is the quantum version of the classical maximum entropy principle.

In the high-temperature limit ($k_B T$ much larger than the energy level spacing), quantum effects become negligible and you recover the classical canonical ensemble.

Partition function for quantum states

The quantum partition function is:

$Z = \text{Tr}(e^{-\beta H}) = \sum_n e^{-\beta E_n}$

where the sum runs over all energy eigenstates (counting degeneracies). This single function generates all equilibrium thermodynamic quantities through derivatives.

The partition function encodes quantum effects like discrete energy levels and degeneracy. For example, a quantum harmonic oscillator with energies $E_n = \hbar\omega(n + 1/2)$ gives $Z = e^{-\beta\hbar\omega/2}/(1 - e^{-\beta\hbar\omega})$, which differs significantly from the classical result at low temperatures.

Quantum thermodynamic quantities

All standard thermodynamic quantities follow from the partition function:

• Internal energy: $U = -\frac{\partial}{\partial \beta} \ln Z$
• Free energy: $F = -k_B T \ln Z$
• Entropy: $S = k_B(\ln Z + \beta U)$, equivalently $S = -k_B \text{Tr}(\rho \ln \rho)$
• Heat capacity: $C = \frac{\partial U}{\partial T} = -k_B \beta^2 \frac{\partial U}{\partial \beta}$

Quantum effects produce deviations from classical predictions, especially at low temperatures. The most famous example is the vanishing of heat capacity as $T \to 0$ (consistent with the third law of thermodynamics), which classical statistical mechanics fails to predict.

Applications

Quantum states and density matrices are foundational not only for statistical mechanics but also for quantum information science and the study of open systems.

Quantum information theory

Quantum information theory uses quantum states as information carriers. A qubit is a two-level quantum system whose state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ can encode information in superposition, unlike a classical bit.

Entanglement serves as a resource: quantum teleportation transfers a quantum state using shared entanglement plus classical communication, while superdense coding sends two classical bits using one qubit and shared entanglement. Quantum error correction protects fragile quantum information from decoherence by encoding logical qubits into entangled states of multiple physical qubits.

Quantum computing basics

Quantum computing exploits superposition and entanglement to perform certain computations faster than any classical computer.

• Quantum gates are unitary operations on qubits (analogous to classical logic gates)
• Quantum circuits chain gates together to implement algorithms
• Shor's algorithm factors large integers exponentially faster than known classical methods
• Grover's algorithm searches unsorted databases with a quadratic speedup
• Measurement at the end of a computation collapses the quantum state to classical bits, so algorithms must be designed to make the correct answer likely

Decoherence and open systems

Decoherence is the loss of quantum coherence caused by interactions with the environment. It's the primary mechanism behind the quantum-to-classical transition: a system that starts in a pure superposition becomes effectively mixed as information leaks into the environment.

Decoherence is modeled using quantum master equations, the most common being the Lindblad equation, which adds non-unitary terms to the von Neumann equation. Strategies to combat decoherence include quantum error correction codes and decoherence-free subspaces (subspaces of the Hilbert space that are immune to certain types of environmental noise).

Measurement theory

Measurement theory formalizes how information is extracted from quantum systems. Different measurement frameworks offer different trade-offs between information gain and disturbance.

Projective measurements

A projective measurement is described by a complete set of orthogonal projection operators $\{P_i\}$ satisfying $\sum_i P_i = I$ and $P_i P_j = \delta_{ij} P_i$.

1. The probability of outcome $i$ is $p_i = \text{Tr}(P_i \rho)$
2. After obtaining outcome $i$, the state updates to $\rho' = \frac{P_i \rho P_i}{p_i}$
3. Repeating the same measurement immediately gives the same result (repeatability)

Projective measurements cause maximal disturbance to the state. They correspond to measuring a Hermitian observable, where the $P_i$ project onto its eigenspaces.

POVM measurements

A Positive Operator-Valued Measure (POVM) generalizes projective measurements. It's described by a set of positive operators $\{E_i\}$ satisfying $\sum_i E_i = I$, but the $E_i$ need not be orthogonal or idempotent.

• Probability of outcome $i$: $p_i = \text{Tr}(E_i \rho)$
• POVMs can have more outcomes than the dimension of the Hilbert space
• They're essential for optimal quantum state discrimination (distinguishing non-orthogonal states as well as possible)
• A POVM doesn't specify the post-measurement state; for that, you need the full measurement operators

Weak measurements

Weak measurements extract partial information while minimally disturbing the quantum state. The measurement operators are close to the identity, meaning the system is only slightly perturbed.

They allow observation of weak values, which can lie outside the eigenvalue spectrum of the observable and reveal subtle quantum effects. Weak measurements have been used to study quantum paradoxes (like Hardy's paradox) and enable novel quantum feedback and control protocols.

Quantum correlations

Quantum correlations go beyond what classical physics allows and are central to both foundational questions and practical quantum technologies.

Classical vs quantum correlations

A key subtlety: quantum correlations can exist even in separable (unentangled) mixed states. Entanglement is not the only form of non-classical correlation.

Quantum discord

Quantum discord captures quantum correlations that go beyond entanglement. It's defined as the difference between two classically equivalent expressions for mutual information that become inequivalent in quantum mechanics:

$D(A:B) = I(A:B) - J(A:B)$

where $I$ is the quantum mutual information and $J$ is the classical mutual information obtained after optimizing over measurements on one subsystem.

Discord is nonzero for most quantum states, including some separable states. It plays a role in the DQC1 model of quantum computation, which achieves speedup using highly mixed states with almost no entanglement but nonzero discord.

Bell inequalities

Bell inequalities set upper bounds on correlations achievable by any local hidden variable theory. The most commonly tested form is the CHSH inequality:

$|\langle AB\rangle + \langle AB'\rangle + \langle A'B\rangle - \langle A'B'\rangle| \leq 2$

Quantum mechanics predicts violations up to $2\sqrt{2} \approx 2.83$ (the Tsirelson bound). Experimental violations have been confirmed in increasingly loophole-free tests, ruling out local hidden variable explanations of quantum mechanics.

Bell inequality violations are not just of foundational interest. They're the basis for device-independent quantum cryptography and certified randomness generation, where security guarantees come from the observed violation itself rather than trust in the devices.