Measurements and Mixed States

Measurements and Mixed States#

Generalized measurements (POVMs), quantum channels, Kraus operators, and their role in quantum ML circuits.

Projective measurements in the computational basis are the simplest way to extract classical information from a quantum state. Real quantum systems require a richer framework: generalized measurements (POVMs), density matrices for mixed states, and quantum channels for describing noise and open-system dynamics. These tools are essential for understanding how quantum ML circuits behave on real hardware.

Projective Measurements#

A projective measurement is defined by a set of orthogonal projectors \(\{P_m\}\) satisfying \(\sum_m P_m = I\) and \(P_m P_{m'} = \delta_{mm'} P_m\). When measuring a state \(|\psi\rangle\):

Outcome \(m\) occurs with probability \(p(m) = \langle\psi|P_m|\psi\rangle\)
The post-measurement state is \(\frac{P_m|\psi\rangle}{\sqrt{p(m)}}\)

For a single qubit measured in the computational basis, \(P_0 = |0\rangle\langle 0|\) and \(P_1 = |1\rangle\langle 1|\). This recovers Born’s rule from the previous chapter.

Measurement in Other Bases#

We are not limited to the computational basis. The Hadamard basis \(\{|+\rangle, |-\rangle\}\) where \(|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\) and \(|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)\) defines a different projective measurement. Measuring the state \(|0\rangle\) in this basis gives \(|+\rangle\) and \(|-\rangle\) each with probability \(1/2\).

The choice of measurement basis determines what information is extracted. In QML, measurement basis optimization is a key design variable.

Density Matrices#

Pure states \(|\psi\rangle\) describe quantum systems with complete knowledge. When a system is in state \(|\psi_i\rangle\) with classical probability \(p_i\), we need the density matrix:

\[\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|\]

This is a mixed state. Key properties of density matrices:

\(\rho\) is Hermitian: \(\rho = \rho^\dagger\)
\(\rho\) is positive semidefinite: \(\langle\phi|\rho|\phi\rangle \geq 0\) for all \(|\phi\rangle\)
Unit trace: \(\text{Tr}(\rho) = 1\)
Pure states satisfy \(\text{Tr}(\rho^2) = 1\); mixed states have \(\text{Tr}(\rho^2) < 1\)

Pure vs. mixed states on the Bloch sphere

Pure states live on the surface of the Bloch sphere. Mixed states live in the interior. The maximally mixed state \(\rho = I/2\) sits at the center, representing complete ignorance about the qubit.

The Partial Trace#

Given a composite system \(\rho_{AB}\) on \(\mathcal{H}_A \otimes \mathcal{H}_B\), the reduced density matrix on subsystem \(A\) is obtained by tracing out \(B\):

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

Even if \(\rho_{AB}\) is a pure entangled state, \(\rho_A\) is generally mixed. For example, tracing out one qubit of the Bell state \(|\phi^+\rangle\) yields \(\rho_A = I/2\), the maximally mixed state. This is why entangled subsystems appear random when examined locally.

Generalized Measurements (POVMs)#

A Positive Operator-Valued Measure (POVM) generalizes projective measurements. It is defined by a set of positive semidefinite operators \(\{E_m\}\) satisfying:

\[\sum_m E_m = I, \quad E_m \geq 0\]

The probability of outcome \(m\) is \(p(m) = \text{Tr}(E_m \rho)\). Unlike projective measurements, POVM elements need not be orthogonal, and there can be more outcomes than the dimension of the Hilbert space.

Important

POVMs are physically realizable: any POVM on a system can be implemented as a projective measurement on a larger system (Naimark’s theorem). In practice, this means adding ancilla qubits and performing a unitary followed by a projective measurement.

POVMs in QML#

POVMs are relevant to QML in several ways:

Optimal state discrimination: distinguishing between non-orthogonal quantum states (e.g., classifying quantum data) often requires POVMs rather than projective measurements.
Informationally complete measurements: POVMs with \(d^2\) elements (for \(d\)-dimensional systems) can reconstruct the full density matrix, useful for quantum state tomography.
Soft measurements: POVMs can extract partial information while minimally disturbing the state.

Quantum Channels#

The most general evolution of a quantum state (including noise, decoherence, and measurement) is described by a quantum channel: a completely positive, trace-preserving (CPTP) linear map \(\mathcal{E}: \rho \mapsto \mathcal{E}(\rho)\).

Kraus Representation#

Every quantum channel can be written in Kraus form:

\[\mathcal{E}(\rho) = \sum_k K_k \rho K_k^\dagger\]

where the Kraus operators \(\{K_k\}\) satisfy \(\sum_k K_k^\dagger K_k = I\). This representation is not unique, but all representations of the same channel are related by unitary transformations.

Common Noise Channels#

Channel	Kraus Operators	Physical Meaning
Bit flip	\(K_0 = \sqrt{1-p}\,I\), \(K_1 = \sqrt{p}\,X\)	Random NOT with probability \(p\)
Phase flip	\(K_0 = \sqrt{1-p}\,I\), \(K_1 = \sqrt{p}\,Z\)	Random phase with probability \(p\)
Depolarizing	\(K_0 = \sqrt{1-3p/4}\,I\), \(K_{1,2,3} = \sqrt{p/4}\,\sigma_{x,y,z}\)	Replaces state with \(I/2\) with probability \(p\)
Amplitude damping	\(K_0 = \begin{bmatrix}1 & 0 \\ 0 & \sqrt{1-\gamma}\end{bmatrix}\), \(K_1 = \begin{bmatrix}0 & \sqrt{\gamma} \\ 0 & 0\end{bmatrix}\)	Energy relaxation (\(T_1\) decay)

Impact on QML#

Noise channels are critical for understanding QML on real hardware:

Depolarizing noise washes out quantum correlations, reducing the effective expressibility of variational circuits.
Amplitude damping biases measurement outcomes toward \(|0\rangle\), introducing systematic errors in expectation value estimation.
Error mitigation techniques (zero-noise extrapolation, probabilistic error cancellation) attempt to undo the effects of noise channels without full error correction.

Warning

On current NISQ devices, noise limits circuit depth to roughly 10-100 two-qubit gates before the output is indistinguishable from random. This constraint shapes all practical QML algorithm design.

Von Neumann Entropy#

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

\[S(\rho) = -\text{Tr}(\rho \log_2 \rho) = -\sum_i \lambda_i \log_2 \lambda_i\]

where \(\lambda_i\) are the eigenvalues of \(\rho\). Properties:

\(S(\rho) = 0\) if and only if \(\rho\) is a pure state.
\(S(\rho) \leq \log_2 d\) with equality for the maximally mixed state.
For a bipartite pure state, \(S(\rho_A) = S(\rho_B)\) measures entanglement.