[Quantum Information] Overview of Quantum Theory

   Overview of Quantum Theory

Notes from RWTH Aachen University course 
“Quantum Information” Summer semester 2020
professor: Wegewijs, Maarten

• 🧐 Isn’t information independent of its physical realization?

  • ❌ We cannot think about information processing independent of its physical realization: underlying statistical model

• 🧐 Can we store infinitely more information already in a single quantum bit?

  • ⁉️ Quantum states are not trivially accessible.

• 🧐 Can we copy information?

  • ❌ by linearity of quantum physics.

  No-cloning theorem

• 🧐 Can we communicate quantum information without copying?

  • ⭕️ same by linearity of quantum physics

  Quantum teleportation

• 🧐 What is non-classical about quantum states / information?

• ❎ measured outcome in \(Z\)-basis of \(1/\sqrt{2}(|01\rangle+|10\rangle)\) is fully (anti-)correlated \(\Rightarrow\) not the reason, because classical correlated state also happens
  • \(|1_2\rangle\) happends when \(|0_1\rangle\) is measured
• ✅ \(A\) and \(B\) measure in arbitrary basis, outcome is still fully (anti-)correlated. \(\Rightarrow\) Not possible classically

Preparation

• vector \(|\psi\rangle\)
  • probabilistic: Stochastic state = Probability vector
  • open: Mixed states = Density operato
  • complex Hilbert space
  • \(\langle\psi|\psi\rangle=1\)

Evolution

• Unitary \(U\)
  • probabilistic: Stochastic channel = Transition probability matrix
  • open: Quantum channel
  • \(U^\dagger U=UU^\dagger=\mathbb{I}\) \(\Rightarrow\) probability conservation

Conditioned Evolution

• open: Quantum instrument

Ovservation

• covector \(E_m\)
  • probabilistic: Stochastic measurement = Conditional probability vector
  • open: Mixing of measurements = Effect operator
  • orthogonal projection on Hilbert subspace \(P_m=|\psi\rangle\langle\psi|=P_m^2\)
    \[p_m=|\langle m|U|\psi\rangle|^2=\|P_mU|\psi\rangle\|^2\]

對 \(m\) 的measurement (測量 \(m\) 出現的機率)
\(\langle m | = P_m = |m\rangle\langle m|=P_m^2\)

• \[ |\psi\rangle = e^{i\varphi}|\psi\rangle\ \ \ \forall \varphi \in \mathbb{R} \]

相同 state 可以用不同 vector 表示
\(|\psi\rangle = i|\psi\rangle = -|\psi\rangle\)

• Linear:

  • \(\langle v|\)\(\alpha_1\)\(w_1+\)\(\alpha_2\)\(w_2\rangle=\)\(\alpha_1\)\(\langle v|w_1\rangle\)+\(\alpha_2\)\(\langle v|w_2\rangle\)

• Anti-linear:

  • \(\langle v|w\rangle^*=\langle w|v\rangle\)
  • \(\langle\alpha_1\)\(w_1+\)\(\alpha_2\)\(w_2|v\rangle=\)\(\alpha_1^*\)\(\langle w_1|v\rangle\)+\(\alpha_2^*\)\(\langle w_2|v\rangle\)

• Projection operator
  \[\sum_i|i\rangle\langle i|=\mathbb{I}\]

• measurements

  • Observable = Hermitian operator \(H=H^\dagger\)
  • \(\Rightarrow\) has uniquely spectral decomposition
  • \[M=\sum_i m_i |i\rangle\langle i|=\sum_i m_i P_i = \sum_m m P_m\]
  • probability for outcome \(m\)
    \[p_m=\langle\psi|P_m|\psi\rangle=\|P_m|\psi\rangle\|^2\]
  • post-measurement state for outcome \(m\)
    \[|\psi_m\rangle=\frac{1}{\sqrt{p_m}}P_m|\psi\rangle\]
  • non-degenerate spectrum: measure specific \(m\)
    degenerate spectrum: get expectation of all outcomes
  • example:
    • Observable
      \[Z=|0\rangle\langle 0|-|1\rangle\langle 1|=P_0-P_1\]
      • \(+1\) with eigenvector \(|0\rangle\)
        \(-1\) with eigenvector \(|1\rangle\)
    • measurement on \(|\psi\rangle=\alpha|0\rangle+\beta|1\rangle\)
      \[\frac{P_0|\psi\rangle}{\|P_0|\psi\rangle\|}=\frac{\alpha}{|\alpha|}|0\rangle\sim|0\rangle\text{ with probability }\|P_0|\psi\rangle\|^2=|\alpha|^2 \\ \frac{P_1|\psi\rangle}{\|P_1|\psi\rangle\|}=\frac{\beta}{|\beta|}|1\rangle\sim|1\rangle\text{ with probability }\|P_1|\psi\rangle\|^2=|\beta|^2\]

• non-commuting measurements-quantum uncertainty

• \(Z\) and \(X\) cannot be measured simultaneously becasue they Fail to commute.
  \[[Z,X]=2iY\]

Readings

Nielsen and Chuang

 2.1-2.2.5 Introduction to quantum mechanics

 1.2 Qubits

 1.3 Quantum computation

Preskill

 1 Introduction

Wilde (advanced)

 3 The Noiseless Quantum Theory