3. Quantum Entanglement

A pure state $|\psi\rangle_{AB}$ of a composite system AB is separable if:

Otherwise, it is entangled.

For mixed states, $\hat{\rho}_{AB}$ is separable if:

A convex combination of product states

Key insight: Entangled states exhibit correlations that cannot be explained by any local hidden variable theory

The four maximally entangled states of two qubits:

Properties:

• Form an orthonormal basis (Bell basis)
• Maximally entangled: reduced density matrices are maximally mixed
• Cannot be written as product states
• Measurement of one qubit instantly determines the other's state

Einstein, Podolsky, and Rosen (1935) argued that quantum mechanics is incomplete:

Consider two particles in state $|\Psi^-\rangle$ (singlet state):

EPR argument:

1. Measuring spin of particle A along any axis gives random result
2. This measurement instantly determines spin of B along same axis
3. Since particles are separated, measurement on A cannot affect B (locality)
4. Therefore, B must have had definite spin all along (reality)
5. But quantum mechanics doesn't describe these "elements of reality" → incomplete!

Resolution: Bell's theorem shows locality + realism incompatible with quantum predictions. Experiments confirm quantum mechanics; nature is non-local!

Any pure state $|\psi\rangle_{AB}$ can be written as:

Where $\{|i\rangle_A\}$ and $\{|i\rangle_B\}$ are orthonormal bases, and $\lambda_i \geq 0$, $\sum_i \lambda_i = 1$

Properties:

• Schmidt rank: Number of nonzero $\lambda_i$
• Separable iff Schmidt rank = 1
• Maximally entangled iff all $\lambda_i$ equal

The $\lambda_i$ are eigenvalues of both $\hat{\rho}_A$ and $\hat{\rho}_B$:

The von Neumann entropy of the reduced density matrix quantifies entanglement:

Properties for pure bipartite state $|\psi\rangle_{AB}$:

• $S_A = S_B$ (symmetry)
• $S_A = 0$ iff state is separable (no entanglement)
• $S_A = \ln d$ for maximally entangled state in $d \times d$ system

Example: Bell state $|\Phi^+\rangle$

Maximal entanglement for two qubits

Several measures quantify entanglement beyond entropy:

1. Concurrence (for two qubits):

Where $\lambda_i$ are eigenvalues of $\hat{\rho}(\sigma_y \otimes \sigma_y)\hat{\rho}^*(\sigma_y \otimes \sigma_y)$

Range: $C \in [0, 1]$, with 0 = separable, 1 = maximally entangled

2. Entanglement of Formation:

Minimum average entanglement needed to prepare $\hat{\rho}$

3. Negativity:

Based on partial transpose; easy to compute but not always monotone

Unlike classical correlations, entanglement cannot be freely shared:

If A is maximally entangled with B, it cannot be entangled with C

Example: Three qubits cannot all be pairwise maximally entangled

Implications:

• Fundamental constraint on quantum networks
• Security of quantum key distribution
• Structure of quantum error-correcting codes

Three-qubit entangled states come in inequivalent classes:

1. GHZ state:

Maximum violation of local realism; but tracing out one qubit gives separable state

2. W state:

Robust: any two qubits remain entangled after tracing out third

These states are inequivalent under local operations and classical communication (LOCC)

LOCC (Local Operations and Classical Communication):

Operations that cannot increase entanglement:

• Local unitary operations
• Local measurements
• Classical communication of measurement results

Entanglement distillation:

Convert many copies of weakly entangled pairs into fewer maximally entangled pairs using LOCC

Entanglement dilution:

Reverse process: create many weakly entangled pairs from few maximally entangled pairs

Transfer an unknown quantum state using entanglement and classical communication:

Protocol:

1. Alice and Bob share Bell pair $|\Phi^+\rangle_{AB}$
2. Alice has unknown state $|\psi\rangle_C = \alpha|0\rangle + \beta|1\rangle$ to send to Bob
3. Alice performs Bell measurement on C and her half of pair, gets 2 classical bits
4. Alice sends 2 bits to Bob classically
5. Bob applies appropriate unitary based on bits received
6. Bob's qubit is now in state $|\psi\rangle$

Key features:

• Original state $|\psi\rangle_C$ is destroyed (no cloning theorem)
• No faster-than-light communication (classical channel required)
• Consumes one Bell pair (entanglement as resource)

• Quantum cryptography: BB84 protocol, quantum key distribution
• Quantum computing: Entanglement enables quantum algorithms (Shor, Grover)
• Quantum metrology: Entangled states beat classical precision limits
• Quantum networks: Quantum internet based on distributed entanglement
• Fundamental tests: Bell inequality violations, loophole-free tests
• Condensed matter: Entanglement in ground states, topological order
• Black hole physics: ER=EPR conjecture linking entanglement to geometry