Quantum State Measurement of Quantum Entanglement: Measurement of Entangled States and Quantum State Evolution

Quantum entanglement remains one of the most profound phenomena in modern physics, describing a non-classical correlation between two or more particles regardless of the spatial distance separating them. When a measurement is performed on one particle of an entangled pair, the global state of the system is instantly altered. This non-locality—experimentally validated through Bell inequality tests—serves as the foundational pillar for quantum information science. Understanding the precise mechanics of how entangled states are measured and how quantum systems evolve post-measurement is critical for the advancement of quantum communication networks and computational architectures.
Consider a system prepared in a maximally entangled Bell state, such as the Einstein-Podolsky-Rosen (EPR) state $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|0\rangle_A|0\rangle_B + |1\rangle_A|1\rangle_B)$, where particles A and B are spatially separated. Measuring particle A represents a quintessential quantum operation. According to the Copenhagen interpretation, the system exists in a coherent superposition prior to measurement.

Once an observer chooses the computational basis ${|0\rangle, |1\rangle}$ to perform a projective measurement on particle A, the wave function undergoes instantaneous collapse. If the measurement yields the outcome $|0\rangle$ for particle A, the global state of the entire system immediately reduces to $|0\rangle_A|0\rangle_B$. Consequently, the state of particle B is definitively fixed to $|0\rangle$, even though no direct physical interaction or observation was applied to it.

Joint Probability Distributions and Correlation Analysis

To quantitatively describe the measurement characteristics of entangled states, we utilize the concepts of the reduced density matrix and joint probability distributions. Suppose we perform local measurements on both entangled particles, A and B, using the Pauli-Z operators $\sigma_z^A$ and $\sigma_z^B$, which correspond to the computational basis states $|0\rangle$ and $|1\rangle$ (yielding eigenvalues +1 and -1, respectively).

For the aforementioned EPR state, the joint measurement outcomes $(m_A, m_B)$ exhibit strictly correlated probabilities. Specifically:

• The probability of measuring (+1, +1) [A=0, B=0] is 50%;
• The probability of measuring (+1, -1) [A=0, B=1] is 0%;
• The probability of measuring (-1, +1) [A=1, B=0] is 0%;
• The probability of measuring (-1, -1) [A=1, B=1] is 50%.

This perfect correlation cannot be explained by any classical local hidden-variable theory, highlighting the fundamental distinction between quantum entanglement and classical statistical dependence. In experimental settings, researchers repeatedly measure these joint probability distributions to calculate the CHSH (Clauser-Horne-Shimony-Holt) inequality parameter. If the observed correlation strength exceeds the classical upper bound, it conclusively proves that the system exists in a genuine entangled state rather than a separable one.

Mathematical Formulation of Projective Measurements and Post-Selection

From the perspective of linear algebra, measuring a composite quantum system is mathematically equivalent to performing a projection operation within the system's Hilbert space. Consider a generic entangled state of a two-level system: $|\psi\rangle = \alpha|00\rangle + \beta|11\rangle$. If we wish to measure the probability amplitude of particle A being in state $|0\rangle_A$, we effectively map the system onto the subspace ${|0\rangle_A} \otimes \mathcal{H}_B$.

This process is described by the projection operator $P = |0\rangle_A\langle 0|_A \otimes I_B$, where $I_B$ is the identity operator on the Hilbert space of particle B. When a measurement is made and a specific outcome is recorded—for instance, post-selecting the scenario where particle A is found in $|0\rangle$—the state evolution strictly follows the normalization rule:

$$ |\psi'\rangle = \frac{P|\psi\rangle}{\sqrt{\langle\psi|P^\dagger P|\psi\rangle}} $$

Applying this to our example, if particle A is measured as $|0\rangle$, the unnormalized state becomes $\alpha|00\rangle$. Dividing by the square root of the norm (where the probability amplitude is $\alpha$) yields the final normalized state $|00\rangle$. This encapsulates the core mechanism of "measurement as evolution": once information is extracted and a specific subspace is confirmed, the coherence of the superposition collapses, and the quantum state of the remaining particle is definitively determined.

Partial Trace and Local Observables

In practical applications—such as remote quantum communication scenarios—we often cannot simultaneously access both entangled particles. Sometimes, an observer can only interact with a single subsystem. In such cases, we must compute the reduced density matrix of the individual particle, $\rho_A = \text{Tr}B(\rho{AB})$, to analyze its local properties.

For the pure entangled state $|\psi\rangle = |\Phi^+\rangle$, the global density matrix is $\rho_{AB} = |\psi\rangle\langle\psi|$. By performing a partial trace over the degrees of freedom of particle B, we obtain:

$$ \rho_A = \text{Tr}_B\left(\frac{|00\rangle\langle 00| + |11\rangle\langle 11|}{2}\right) = \begin{pmatrix} 0.5 & 0 \ 0 & 0.5 \end{pmatrix} $$

This result reveals a critical characteristic of entangled states: even though the global composite system is in a pure, perfectly coherent superposition, the local description of particle A is a maximally mixed state. This means that no single local measurement on particle A can reveal whether it is entangled with a distant partner. Only by statistically comparing its behavior with the remote measurement outcomes of particle B can the underlying entanglement be uncovered.

This phenomenon profoundly illustrates the quantum mechanical principle that the whole is greater than the sum of its parts. It also identifies a primary challenge in protocols like quantum teleportation: how to effectively manipulate and utilize global entanglement resources when restricted to local operations and classical communication.