Entanglement

Entanglement#

Non-local correlations between quantum systems. Bell states, EPR paradox, Bell inequalities, and why Einstein called it “spooky action at a distance.”

Quantum entanglement is a correlation between quantum systems that has no classical analog. When two qubits are entangled, the state of the pair cannot be described by specifying the state of each qubit independently. Measuring one qubit instantaneously determines the outcome of measuring the other, regardless of the distance separating them.

Entanglement is not merely a theoretical curiosity. It is the resource that powers quantum teleportation, superdense coding, quantum key distribution, and the speedups in many quantum algorithms.

Definition#

A pure state \(|\psi_{AB}\rangle\) of a bipartite system is entangled if it cannot be written as a tensor product:

\[|\psi_{AB}\rangle \neq |\phi_A\rangle \otimes |\chi_B\rangle \quad \text{for any } |\phi_A\rangle, |\chi_B\rangle\]

States that can be written as tensor products are called separable or product states.

Detecting Entanglement#

\[ad \neq bc\]

This is the condition that the \(2 \times 2\) coefficient matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) has rank greater than 1.

The Bell States#

The four maximally entangled two-qubit states form the Bell basis:

\[|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) \qquad |\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)\]

\[|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle) \qquad |\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)\]

Properties of the Bell states:

They form a complete orthonormal basis for two qubits
Each qubit individually is in a maximally mixed state: \(\rho_A = \frac{I}{2}\)
Measuring both qubits in the same basis always produces perfectly correlated (\(\Phi^\pm\)) or anti-correlated (\(\Psi^\pm\)) outcomes
The sign (\(\pm\)) determines the relative phase, which affects correlations when measured in other bases

Monogamy of Entanglement

If two qubits are maximally entangled with each other, neither can be entangled with any third party. This monogamy property, quantified by the Coffman-Kundu-Wootters inequality, is the foundation of quantum key distribution: if Alice and Bob share a maximally entangled pair, no eavesdropper can share any of that entanglement.

The EPR Argument#

In 1935, Einstein, Podolsky, and Rosen (EPR) argued that quantum mechanics is incomplete. Their reasoning:

Alice and Bob share \(|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\)
If Alice measures her qubit and gets \(|0\rangle\), Bob’s qubit is instantly in state \(|0\rangle\)
Alice could instead measure in the \(X\)-basis; if she gets \(|+\rangle\), Bob’s qubit is instantly \(|+\rangle\)
Alice’s choice of measurement basis determines Bob’s state, but Alice and Bob can be arbitrarily far apart
EPR concluded that either (a) quantum mechanics is incomplete and there are “hidden variables” predetermining the outcomes, or (b) measurement involves “spooky action at a distance”

EPR favored option (a): the outcomes are predetermined by hidden variables that quantum mechanics fails to describe.

Bell’s Theorem (1964)#

John Bell proved that any local hidden variable theory makes predictions that satisfy a specific inequality, and quantum mechanics violates it. This means EPR’s option (a) is ruled out by experiment.

The CHSH Inequality#

The most commonly tested form is the Clauser-Horne-Shimony-Holt (CHSH) inequality. Alice chooses between measurements \(A_1\) or \(A_2\); Bob chooses between \(B_1\) or \(B_2\). Each measurement yields \(\pm 1\). Define:

\[S = \langle A_1 B_1 \rangle + \langle A_1 B_2 \rangle + \langle A_2 B_1 \rangle - \langle A_2 B_2 \rangle\]

Local hidden variable bound: \(|S| \leq 2\)

Quantum mechanics prediction: \(|S| \leq 2\sqrt{2} \approx 2.828\)

The maximum quantum value \(2\sqrt{2}\) is called the Tsirelson bound. It is achieved by measuring the Bell state \(|\Phi^+\rangle\) with optimal measurement angles.

Optimal Measurement Settings#

The quantum maximum is achieved when:

Alice measures at angles \(0\) and \(\pi/4\) from the \(Z\)-axis
Bob measures at angles \(\pi/8\) and \(-\pi/8\)

This gives:

\[S = 2\sqrt{2} > 2\]

The violation of the CHSH inequality has been confirmed in thousands of experiments, culminating in the loophole-free tests of 2015 (see Experimental Quantum Mechanics).

Entanglement as a Resource#

Entanglement enables several quantum information protocols:

Protocol	What it does	Entanglement consumed
Quantum teleportation	Transmit qubit using 1 ebit + 2 classical bits	1 Bell pair
Superdense coding	Transmit 2 classical bits using 1 qubit + 1 ebit	1 Bell pair
QKD (E91)	Generate shared secret key	1 Bell pair per key bit
MBQC	Measurement-based quantum computation	Cluster state

Entanglement Cannot Transmit Information

Despite the instantaneous correlations, entanglement alone cannot transmit information faster than light. Bob’s local measurement statistics are completely random (\(50/50\) for any measurement basis) regardless of what Alice does. Only when Alice sends her measurement results through a classical channel can Bob extract useful information. This is consistent with special relativity.

Quantifying Entanglement#

For a pure bipartite state \(|\psi_{AB}\rangle\), entanglement is quantified by the von Neumann entropy of the reduced density matrix:

\[E(|\psi_{AB}\rangle) = S(\rho_A) = -\text{Tr}(\rho_A \log_2 \rho_A)\]

where \(\rho_A = \text{Tr}_B(|\psi_{AB}\rangle\langle\psi_{AB}|)\).

For maximally entangled states: \(E = 1\) ebit (entropy bit). For product states: \(E = 0\).

Entanglement

Contents