Quantum Information Applications
Teleportation, no-cloning, superdense coding, and entanglement measures
Quantum Teleportation
Quantum teleportation, proposed by Bennett et al. (1993), allows the transfer of an unknown quantum state from Alice to Bob using shared entanglement and classical communication — without physically transmitting the quantum particle.
The Protocol (Step by Step)
Setup: Alice has an unknown qubit $|\psi\rangle_C = \alpha|0\rangle + \beta|1\rangle$that she wishes to send to Bob. They share a Bell pair $|\Phi^+\rangle_{AB} = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$.
The total three-qubit state is:
Step 1: Rewrite in the Bell basis for qubits C and A:
Step 2: Alice performs a Bell measurement on her two qubits (C and A), obtaining one of four results with equal probability $1/4$. She sends her 2-bit result to Bob classically.
Step 3: Bob applies the appropriate correction:
| Alice's Result | Bob's State | Bob's Correction |
|---|---|---|
| $|\Phi^+\rangle$ | $\alpha|0\rangle + \beta|1\rangle$ | $\mathbb{I}$ (nothing) |
| $|\Phi^-\rangle$ | $\alpha|0\rangle - \beta|1\rangle$ | $\hat{\sigma}_z$ |
| $|\Psi^+\rangle$ | $\alpha|1\rangle + \beta|0\rangle$ | $\hat{\sigma}_x$ |
| $|\Psi^-\rangle$ | $\alpha|1\rangle - \beta|0\rangle$ | $i\hat{\sigma}_y$ |
After Bob's correction, his qubit is in the state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$. Teleportation is complete.
No-Cloning Theorem
Teleportation destroys the original state, which is not a bug but a fundamental feature. The no-cloning theorem (Wootters, Zurek, and Dieks, 1982) states that it is impossible to create an exact copy of an arbitrary unknown quantum state.
Proof
Suppose a unitary cloning operation $\hat{U}$ exists such that for all states$|\psi\rangle$:
Apply this to two different states $|\phi\rangle$ and $|\psi\rangle$:
Taking the inner product of both sides (using unitarity: $\langle\phi|\langle 0|\hat{U}^\dagger\hat{U}|\psi\rangle|0\rangle = \langle\phi|\psi\rangle$):
This equation has only two solutions: $\langle\phi|\psi\rangle = 0$ or $\langle\phi|\psi\rangle = 1$. Therefore cloning works only for orthogonal or identical states — not for arbitrary states. $\square$
Superdense Coding
Superdense coding is the "dual" of teleportation: instead of using entanglement plus classical bits to transmit a quantum state, we use entanglement plus one qubit to transmit two classical bits.
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 Pauli gates to her qubit:
Alice sends her qubit to Bob, who performs a Bell measurement on both qubits. Since the four Bell states are orthogonal, he can perfectly distinguish all four messages, extracting 2 classical bits from 1 transmitted qubit.
Entanglement Swapping
Entanglement swapping creates entanglement between particles that have never interacted. Consider two pairs: Alice-Bob share $|\Phi^+\rangle_{12}$ and Charlie-Diana share $|\Phi^+\rangle_{34}$.
where the sum runs over the four Bell states. If Bob and Charlie perform a joint Bell measurement on particles 2 and 3, particles 1 and 4 (held by Alice and Diana, who may be far apart) are projected into an entangled Bell state — despite never having interacted!
Entanglement swapping is essential for quantum repeaters and long-distance quantum communication.
Entanglement Distillation and Purification
In practice, shared entangled pairs are degraded by noise. Entanglement distillation converts many copies of noisy (weakly entangled) pairs into fewer copies of nearly pure (maximally entangled) pairs using only local operations and classical communication (LOCC):
The optimal asymptotic rate is $m/n \to E_D(\hat{\rho})$, the distillable entanglement. For pure states,$E_D = S(\hat{\rho}_A)$, the entanglement entropy.
The reverse process, entanglement dilution, creates many copies of a target state from Bell pairs: $m$ Bell pairs $\to$ $n$ copies of $|\psi\rangle$, with rate $m/n \to E_F(|\psi\rangle) = S(\hat{\rho}_A)$.
Quantifying Entanglement: Key Measures
Concurrence
For two-qubit states, the concurrence provides a computable entanglement measure:
where $\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4$ are eigenvalues of $\hat{\rho}(\hat{\sigma}_y\otimes\hat{\sigma}_y)\hat{\rho}^*(\hat{\sigma}_y\otimes\hat{\sigma}_y)$. Range: $C \in [0, 1]$.
Entanglement of Formation
Minimum average entanglement over all pure-state decompositions of $\hat{\rho} = \sum_i p_i|\psi_i\rangle\langle\psi_i|$. For two qubits, $E_F$ is a monotonic function of the concurrence.
Negativity
where $\hat{\rho}^{T_A}$ is the partial transpose and $\mu_i$ are its eigenvalues. If $\mathcal{N} > 0$, the state is entangled (Peres criterion). Easy to compute for any dimension, though it can miss some entangled states (bound entangled states).
Multipartite Entanglement
Entanglement among three or more parties is far richer than the bipartite case. For three qubits, there are two fundamentally inequivalent classes of genuine tripartite entanglement:
GHZ States
Maximally entangled among all three qubits. Tracing out any one qubit produces a separable mixed state$\hat{\rho}_{AB} = \frac{1}{2}(|00\rangle\langle 00| + |11\rangle\langle 11|)$. The entanglement is "all or nothing" — fragile under particle loss.
W States
Robust under particle loss: tracing out any one qubit still leaves the other two entangled, with$\hat{\rho}_{AB} = \frac{1}{3}|00\rangle\langle 00| + \frac{2}{3}|\psi^+\rangle\langle\psi^+|$where $|\psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$.
GHZ and W states cannot be converted into each other by LOCC, even probabilistically. They represent distinct entanglement classes under stochastic LOCC (SLOCC).
Monogamy of Entanglement
A fundamental constraint on the distribution of entanglement: the Coffman-Kundu-Wootters (CKW) inequality:
where $C_{AB}$ is the concurrence between A and B, and $C_{A|BC}$ is the concurrence of A with the joint system BC. The residual $\tau_{ABC} = C_{A|BC}^2 - C_{AB}^2 - C_{AC}^2 \geq 0$ quantifies genuine tripartite entanglement (the 3-tangle).
Physical consequences of monogamy:
- If A is maximally entangled with B, it has zero entanglement with C or any other system
- Security of QKD: An eavesdropper cannot be entangled with the key without disturbing it
- Quantum error correction: Encodes information in entanglement patterns that errors cannot fully corrupt
- Black hole physics: Monogamy constrains the entanglement structure of Hawking radiation (firewall paradox)
Key Concepts Summary
- Entangled state: Cannot be written as $|\phi\rangle_A\otimes|\chi\rangle_B$; subsystems lack individual pure states
- Bell states: Four maximally entangled two-qubit states forming an orthonormal basis
- Schmidt decomposition: $|\psi\rangle_{AB} = \sum_i\sqrt{\lambda_i}|i\rangle_A|i\rangle_B$; Schmidt rank determines entanglement
- Entanglement entropy: $S = -\sum_i\lambda_i\ln\lambda_i$; zero for product states, $\ln d$ for maximally entangled
- Bell's theorem: No local hidden variable theory can reproduce quantum correlations; $|S_{\text{CHSH}}| \leq 2$ vs. $2\sqrt{2}$
- Teleportation: Transfer unknown state using 1 Bell pair + 2 classical bits; original destroyed
- No-cloning: Impossible to copy an arbitrary unknown quantum state
- Superdense coding: Transmit 2 classical bits using 1 qubit + shared entanglement
- Concurrence: Computable entanglement measure for two qubits, $C \in [0,1]$
- GHZ vs. W: Inequivalent classes of tripartite entanglement; fragile vs. robust
- Monogamy: $C_{A|BC}^2 \geq C_{AB}^2 + C_{AC}^2$; entanglement cannot be freely shared
Looking Ahead: Entanglement is not just a theoretical curiosity — it is the essential resource powering quantum technologies. Quantum computers exploit entanglement to achieve exponential speedups. Quantum communication networks use it for unconditionally secure cryptography. Quantum sensors harness it to surpass classical precision limits. Understanding and controlling entanglement is the central challenge of the quantum information revolution.