Case Study 2: Quantum Entanglement as a Resource — From Teleportation to Cryptography

Introduction

For decades after the EPR paper, entanglement was viewed primarily as a philosophical curiosity — a puzzling feature of quantum mechanics that prompted debate but had no practical applications. This changed dramatically in the 1990s, when a series of breakthroughs revealed that entanglement is a resource — something that can be produced, distributed, quantified, and consumed to accomplish tasks that are impossible with classical physics alone.

This case study examines three landmark protocols — quantum teleportation, superdense coding, and quantum key distribution — all built on the Bell states introduced in Section 11.8.

Protocol 1: Quantum Teleportation (Bennett et al., 1993)

The Problem

Alice has a qubit $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ that she wants to transmit to Bob. She does not know $\alpha$ and $\beta$. She cannot simply measure the qubit and send the result, because measurement would destroy the state. Can she transmit the quantum state using only classical communication?

The Setup

Alice and Bob share a Bell state $|\Phi^+\rangle_{23} = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)_{23}$, prepared in advance. Alice holds qubits 1 (the unknown state) and 2 (her half of the Bell pair). Bob holds qubit 3.

The Protocol

Step 1: Write the total state.

$$|\Psi_{\text{total}}\rangle = |\psi\rangle_1 \otimes |\Phi^+\rangle_{23} = (\alpha|0\rangle_1 + \beta|1\rangle_1) \otimes \frac{1}{\sqrt{2}}(|00\rangle_{23} + |11\rangle_{23})$$

Expanding:

$$= \frac{1}{\sqrt{2}}(\alpha|000\rangle + \alpha|011\rangle + \beta|100\rangle + \beta|111\rangle)$$

Step 2: Rewrite qubits 1 and 2 in the Bell basis.

Using the inverse Bell basis relations:

$$|00\rangle = \frac{1}{\sqrt{2}}(|\Phi^+\rangle + |\Phi^-\rangle), \quad |11\rangle = \frac{1}{\sqrt{2}}(|\Phi^+\rangle - |\Phi^-\rangle)$$

$$|01\rangle = \frac{1}{\sqrt{2}}(|\Psi^+\rangle + |\Psi^-\rangle), \quad |10\rangle = \frac{1}{\sqrt{2}}(|\Psi^+\rangle - |\Psi^-\rangle)$$

After careful substitution and regrouping (we use subscripts 12 for the Bell basis of Alice's two qubits):

$$|\Psi_{\text{total}}\rangle = \frac{1}{2}\Big[|\Phi^+\rangle_{12}(\alpha|0\rangle_3 + \beta|1\rangle_3) + |\Phi^-\rangle_{12}(\alpha|0\rangle_3 - \beta|1\rangle_3)$$ $$+ |\Psi^+\rangle_{12}(\beta|0\rangle_3 + \alpha|1\rangle_3) + |\Psi^-\rangle_{12}(-\beta|0\rangle_3 + \alpha|1\rangle_3)\Big]$$

Step 3: Alice measures qubits 1 and 2 in the Bell basis.

Each outcome occurs with probability $\frac{1}{4}$:

Step 4: Classical communication.

Alice sends her measurement result (2 classical bits) to Bob. Bob applies the corresponding Pauli correction. He now has the original state $|\psi\rangle$.

Key Features

• The qubit is not cloned. Alice's original qubit is destroyed by her measurement. Teleportation respects the no-cloning theorem.
• No faster-than-light communication. Bob's qubit is in a random state until he receives Alice's classical message. The classical bits must travel at or below the speed of light.
• Entanglement is consumed. The shared Bell pair is destroyed in the process. Teleportation converts 1 ebit of entanglement + 2 classical bits into 1 qubit of quantum communication.
• The state can be unknown. Alice need not know $\alpha$ and $\beta$. The protocol works for any input state, including entangled qubits (teleporting half of an entangled pair).

Experimental Realizations

Quantum teleportation has been demonstrated with photons (Bouwmeester et al., 1997; Zeilinger group), trapped ions (Barrett et al., 2004), and over distances exceeding 1,400 km using the Micius satellite (Ren et al., 2017).

Protocol 2: Superdense Coding (Bennett & Wiesner, 1992)

The Problem

Alice wants to send Bob 2 classical bits of information. She can only transmit one qubit. Is this possible?

The Protocol

Alice and Bob share $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$. Alice encodes her 2-bit message by applying one of four operations to her qubit:

Alice sends her qubit to Bob. Bob now holds both qubits and performs a Bell measurement (projecting onto the Bell basis). Since the four Bell states are orthogonal, he can distinguish them perfectly and extract both classical bits.

The Resource Accounting

• Input: 1 ebit of pre-shared entanglement + 1 qubit of quantum communication
• Output: 2 classical bits transmitted
• Comparison: Without entanglement, 1 qubit can carry at most 1 classical bit (Holevo's bound)

Superdense coding is the "dual" of teleportation: teleportation converts entanglement + classical bits into quantum communication; superdense coding converts entanglement + quantum communication into extra classical communication capacity.

Protocol 3: Quantum Key Distribution (BB84 and E91)

BB84 (Bennett & Brassard, 1984)

The BB84 protocol does not directly use entanglement (Alice sends single qubits), but its security can be analyzed through an entanglement-based reformulation.

E91 (Ekert, 1991)

Ekert's protocol makes entanglement explicit:

Step 1: A source distributes Bell pairs $|\Psi^-\rangle$ to Alice and Bob.

Step 2: Each party independently and randomly chooses a measurement basis from a set of three angles.

Step 3: After measurements, they publicly announce their basis choices (but not results).

Step 4: When they chose the same basis, their results are perfectly anticorrelated — these become the secret key.

Step 5: When they chose different bases, they use the results to test the CHSH inequality. If Bell's inequality is violated to the quantum maximum, no eavesdropper has intercepted the particles.

Why Entanglement Guarantees Security

The key insight is monogamy of entanglement: if Alice and Bob's qubits are maximally entangled with each other, they cannot be entangled with anything else — including an eavesdropper Eve's system. Formally:

If $\hat{\rho}_{AB}$ has maximum entanglement, then for any extension to a system $ABE$, the state must be a product $\hat{\rho}_{ABE} = \hat{\rho}_{AB} \otimes \hat{\rho}_E$. Eve is completely uncorrelated with the key.

Any eavesdropping attempt introduces decoherence into the shared state, reducing the observed Bell inequality violation below $2\sqrt{2}$. Alice and Bob can detect this and abort the protocol.

Real-World Implementations

QKD systems are commercially available (ID Quantique, Toshiba) and have been deployed in metropolitan fiber networks (Geneva, Tokyo, Beijing). Satellite-based QKD using the Micius satellite has achieved intercontinental key distribution (Liao et al., 2018).

Entanglement as a Resource: The Bigger Picture

Quantifying Entanglement

The protocols above reveal that entanglement has quantitative value:

The "ebit" (entanglement bit) — one shared Bell pair — is the fundamental unit of entanglement. These resource conversions define the field of quantum Shannon theory.

Entanglement Distillation

Real-world entanglement is imperfect. Noise degrades Bell pairs into mixed states with less-than-maximal entanglement. Entanglement distillation protocols (Bennett et al., 1996) can convert many noisy pairs into fewer high-quality Bell pairs using only local operations and classical communication (LOCC). This is the quantum analogue of error correction.

The Entanglement Frontier

Current research pushes entanglement into new domains:

• Quantum computing: Multi-qubit entanglement is the engine of quantum speedups. Shor's algorithm and Grover's algorithm rely on generating and manipulating entangled states of many qubits.
• Quantum error correction: Logical qubits are encoded in entangled states of many physical qubits. The surface code, a leading error correction scheme, uses a lattice of entangled qubits.
• Quantum networks: A "quantum internet" would distribute entanglement between distant nodes, enabling teleportation, distributed quantum computing, and secure communication.
• Quantum sensing: Entangled states can achieve measurement precision beyond the standard quantum limit (the Heisenberg limit), with applications to gravitational wave detection (LIGO) and atomic clocks.

Discussion Questions

1. In quantum teleportation, the original qubit is destroyed. If Alice could teleport without destroying the original, what fundamental theorem of quantum mechanics would be violated?

2. Superdense coding sends 2 classical bits using 1 qubit + 1 ebit of entanglement. Is entanglement "free"? What resources are needed to create and distribute a Bell pair in the first place?

3. The E91 protocol detects eavesdropping through Bell inequality violations. Could an eavesdropper simply intercept both particles, prepare a new Bell pair, and send it on to Alice and Bob? Why or why not?

4. Entanglement is "monogamous" — a qubit maximally entangled with one partner cannot be entangled with another. Prove this for pure states using the Schmidt decomposition: if $|\Psi\rangle_{AB}$ has Schmidt rank $d_A$, show that no purification to $ABE$ can have system $A$ entangled with $E$.

5. The quantum internet would distribute entanglement over global distances. Given that photon loss in optical fibers limits direct transmission to about 100 km, how might "quantum repeaters" (which use entanglement swapping and distillation) extend this range? Sketch the basic idea using the teleportation protocol.

Connections to Chapter 11 Formalism

Every protocol in this case study is a direct application of the mathematics developed in Chapter 11. The tensor product is not just abstract formalism — it is the language in which quantum technology is designed and analyzed.