QCQI – Chapter 10 Detailed Summary: Entanglement, Entropy & Information Tasks

Course Notes (MSU)

Learning Objectives¶

• Use von Neumann entropy $S(\rho)$, mutual information $I(A{:}B)$, and conditional entropy $S(A|B)$.

• State and apply key inequalities: subadditivity, Araki–Lieb, and strong subadditivity (SSA).

• Characterize and manipulate entanglement under LOCC: Schmidt form, majorization, concentration/dilution, distillation.

• Compute mixed-state entanglement measures in small systems: concurrence, entanglement of formation, negativity.

• Implement and analyze canonical information tasks: teleportation, dense coding, and entanglement swapping.

Entropy and Information¶

von Neumann entropy $S(\rho)=-\operatorname{Tr}\,\rho\log_2\rho$; for pure bipartite $|\psi\rangle_{AB}$ with Schmidt coefficients $\{\lambda_i\}$, the entanglement entropy is $S(\rho_A)=S(\rho_B)=H(\{\lambda_i\})$.

Mutual information $I(A{:}B)=S(\rho_A)+S(\rho_B)-S(\rho_{AB})$ quantifies total correlations. Conditional entropy $S(A|B)=S(\rho_{AB})-S(\rho_B)$ may be negative in quantum theory (signature of entanglement).

Relative entropy $S(\rho\Vert\sigma)=\operatorname{Tr}\,\rho(\log\rho-\log\sigma)$ is nonnegative and monotone under CPTP maps (data-processing).

Eureka! Negative $S(A|B)$ (impossible classically) flags that $A$ can be predicted from $B$ better than classically—entanglement supplies the “missing bits.”

Key inequalities¶

SSA $\iff$ conditional mutual information $I(A{:}C|B)\ge 0$.

Eureka! SSA is the “triangle inequality” of quantum information: correlations cannot increase when discarding systems.

Schmidt Decomposition & Majorization¶

Any pure bipartite state has $|\psi\rangle_{AB}=\sum_i \sqrt{\lambda_i}|i\rangle_A|i\rangle_B$. Under LOCC, $|\psi\rangle\to|\phi\rangle$ is possible iff $\boldsymbol{\lambda}^\psi$ is majorized by $\boldsymbol{\lambda}^\phi$ (Nielsen’s theorem).

Eureka! LOCC cannot sharpen the Schmidt spectrum. Conversions are governed by a single preorder: majorization.

Entanglement Manipulation¶

Concentration (pure states)¶

From many copies of a partially entangled state, LOCC can produce nearly $n H(\{\lambda\})$ Bell pairs (typical subspace argument).

Dilution¶

Conversely, Bell pairs can be consumed to create copies of a target entangled state at rate $H(\{\lambda\})$.

Distillation (mixed states)¶

Protocols such as BBPSSW/DEJMPS use bilateral CNOTs and measurement to increase fidelity with a Bell state, keeping pairs on certain outcomes.

Eureka! Pure-state entanglement is fungible: rate equals the entropy of entanglement. Mixed-state entanglement is subtler: distinct notions (distillable entanglement, entanglement cost) need not coincide.

Separability Tests & Mixed-State Measures¶

PPT criterion: If $\rho^{T_B}\ngeq 0$ then $\rho$ is entangled (negativity $\mathcal{N}(\rho)=\frac{\|\rho^{T_B}\|_1-1}{2}$ quantifies this). For $2\times2$ and $2\times3$, PPT $\Leftrightarrow$ separable.

Two-qubit concurrence (Wootters): $C(\rho)=\max(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4)$, where $\{\lambda_i\}$ are the decreasing square roots of eigenvalues of $\rho\,\tilde\rho$ with $\tilde\rho=(\sigma_y\otimes\sigma_y)\rho^\ast(\sigma_y\otimes\sigma_y)$. Entanglement of formation $E_F(\rho)=h\!\left(\frac{1+\sqrt{1-C^2}}{2}\right)$ with $h$ binary entropy.

Eureka! For two qubits, “how entangled?” has a closed form: the concurrence.

Information Tasks¶

Teleportation¶

Consume one Bell pair and two classical bits to transfer an unknown qubit using a Bell measurement and Pauli corrections.

Dense Coding¶

Send two classical bits by transmitting one qubit given a shared Bell pair, via local Pauli encodings and a Bell measurement at the receiver.

Entanglement Swapping¶

Bell-measure two halves of independent Bell pairs to entangle the distant, previously uncorrelated ends.

Eureka! Entanglement is a resource convertible into communication tasks: qubits $\leftrightarrow$ cbits when EPR pairs are available.

Pseudo-code Aids¶

Pseudo-code: Wootters Concurrence (two qubits)

Input: density matrix $\rho$ $\tilde\rho \leftarrow (\sigma_y\otimes\sigma_y)\rho^\ast(\sigma_y\otimes\sigma_y)$ $\lambda \leftarrow \text{eigs of } \rho\tilde\rho$ (sorted decreasing as square roots) $C \leftarrow \max(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4)$

Pseudo-code: Negativity

Input: $\rho_{AB}$ $\rho^{T_B} \leftarrow$ partial transpose on $B$ $\mathcal{N} \leftarrow (\|\rho^{T_B}\|_1-1)/2$

Pseudo-code: BBPSSW-like One-Round Distillation (concept)

Given: two noisy pairs $\rho^{\otimes 2}$ Both sides apply CNOT(source$\to$target); measure target in $Z$; keep source iff outcomes agree Optionally twirl $\rho\to$ Bell-diagonal before repeating

Pseudo-code: Teleportation (Pauli frame)

Share $|\Phi^+\rangle_{AB}$; sender holds qubit $Q$ to send Bell-measure $Q$ and $A$ to get bits $b_1b_2$; send to $B$ Receiver applies $X^{b_2}Z^{b_1}$ on $B$ to recover state

Schematics¶

Teleportation (concept)¶

Dense Coding (concept)¶

Hands-on Notebook (Multi-Backend)¶

Run: L14_QCQI_Ch10_Entanglement_Entropy_and_Information_Tasks

In the first cell set backend to one of: cirq, pennylane, braket, pyquil, or qiskit.

Mini-Lab Ideas¶

1. Compute $S(\rho)$, $I(A{:}B)$, $S(A|B)$ for Bell, product, and Werner states; verify inequalities numerically.

2. Implement concurrence and negativity for random two-qubit states; compare $E_F$ vs. $\mathcal{N}$ ordering.

3. Simulate one round of BBPSSW on Werner states and plot fidelity improvement vs. parameter $p$.

4. Build teleportation and dense coding circuits in your favorite framework and verify success by state tomography / counts.