[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