# Quantum Information

## Introduction

### Quantum Cryptography

Classical: use binary bits.

Quantum: Encode information through superpositions, entaglement, and wavefunction collapse (measurement).

#### No-cloning Theorem

No guarantee to make an exact replica of an arbitrary quantum state.

• Example - Two Level System

Why do we stop at a two level system for Quantum Computing. $$\{\Psi_a,\Psi_b\}$$.

Given an initial two particle state: $$|\Psi_a\rangle_1|\Psi_s\rangle_2$$ Let $$|\Psi_s\rangle$$ denote the starting state.

Let say we have $$\mathcal{U}|\Psi_a\rangle_1|\Psi_s\rangle_2=\Psi_a\rangle_1|\Psi_a\rangle_2$$.

Then, lets say that also does $$\mathcal{U}|\Psi_b\rangle_1|\Psi_s\rangle_2=\Psi_b\rangle_1|\Psi_b\rangle_2$$.

So, let $$|\Psi\rangle_1=\frac{1}{\sqrt{2}}(|\Psi_a\rangle_1+|\Psi_b\rangle_1)$$

Then, $$U\frac{1}{\sqrt{2}}(|\Psi_a\rangle_1+|\Psi_b\rangle_1)|\Psi_s\rangle_2=\frac{1}{\sqrt{2}}\left[|\Psi_a\rangle_1|\Psi_a\rangle_2+|\Psi_b\rangle_1|\Psi_b\rangle_2\right]$$ But we wanted, $$\frac{1}{\sqrt{2}}(|\Psi_a\rangle_1+|\Psi_b\rangle_1)\frac{1}{\sqrt{2}}(|\Psi_a\rangle_2+|\Psi_b\rangle_2)$$.

#### Example Encryption Scheme (Bennett and Brassard in 1984)

Using single photons, over 48km of Optical Fiber and 1.6km of free space (shown in 2000).

Alice: $$|V\rangle=1$$ and $$|H\rangle = 0$$.

Bob: $$|V\rangle=1$$ and $$|H\rangle = 0$$.

Assume Eve is inserted in the middle with another polarizer.

If Alice and Bob rotates their polarizers by 45 degrees, $$\frac{1}{\sqrt{2}}(|H\rangle\pm|V\rangle)$$, then Eve’s polarizer would make the transmittance be reduced.

Simple scheme would be that Alice and Bob exchange ahead of time which bits should be recorded and which are junk.

• Are there Security Risks associated with different Implementations

Different implementations to change states:

• Electrons use magnetic field
• Photons use optical components
• Atoms use E+M waves

#### Aside

• Shor’s Algorithm

#### Post Quantum Cryptography

• Lattice, CRYSTALS-KYBER, CRYSTALS-DILITHIUM

### Quantum Computing

Classical: input bits $$\Rightarrow$$ |Black box| $$\Rightarrow$$ output bits

QM: input qubits $$\Rightarrow$$ |Oracle (With an additional clock input for trigger operations)| $$\Rightarrow$$ output bits (measurements) (Note the Blackbox is called an oracle, and the input qubits are any QM two-level system)

Question: How is this clock any different from the clock in a classical black box.

$$|\Psi\rangle = \frac{\alpha}{\sqrt{\alpha^2+\beta^2}} |0\rangle + \frac{\beta}{\sqrt{\alpha^2+\beta^2}}|1\rangle$$

Hadamard gate: $$U_H=\frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\\1 & -1\end{pmatrix}, U_z=\begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}, U_{NOT X}=\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}$$.

Bloch Sphere Represenatation. $$|0\rangle=|+\rangle$$ is along $$+z$$ and $$|1\rangle=|-\rangle$$ is along $$-z$$. So, $$|s\rangle = \cos\theta/2|+\rangle + e^{i\varphi}\sin\theta/2|-\rangle$$.

2-qubit states. $$|\psi=C_{0,0}|0\rangle_{control}|0\rangle_{target}+C_{0,1}|0\rangle_{control}|1\rangle_{target}+C_{1,0}|1\rangle_{control}|0\rangle_{target}+C_{1,1}|1\rangle_{control}|1\rangle_{target}=\begin{pmatrix}C_{0,0}\\C_{0,1}\\C_{1,0}\\C_{1,1}\end{pmatrix}$$ $$U_{CNOT}=\begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{pmatrix}$$.

### Quantum Teleportation

Transfer of a quantum state.

### Bell states

Also called entangled states or EPR pairs.

$$|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|H\rangle_1|H\rangle_2\pm|V\rangle_1|V\rangle_2)$$

$$|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|H\rangle_1|V\rangle_2\pm|V\rangle_1|H\rangle_2)$$

Using a nonlinear crystal crystal so that: $$\omega=\omega_1+\omega_2$$ and \$\vec{k}=\vec{k}1+\vec{k}2, the energy and momentum are conserved.

Suppose we have Alice and Bob. Alice recieves $$|\Psi\rangle_1=C_H|H\rangle_1+C_V|V\rangle_1$$. Goal: Bob needs to output same photon.

Exchange EPR pair, $$|\Psi^-\rangle_{23}=\frac{1}{\sqrt{2}}(|H\rangle_2|V\rangle_3 - |V\rangle_2|H\rangle_3)$$. Photon 2 goes to Alice and Photon 3 goes to Bob. Alice sees $$|\Psi\rangle_{123}=|\Psi\rangle|\Psi^-\rangle$$. We can show: $$|\Psi\rangle_{123} = \frac{1}{2}[|\Phi^+\rangle_{12}(C_H|V\rangle_3-C_H|H\rangle_3)+|\Phi^-\rangle_{12}(C_H|V_3\rangle+C_V|H_3\rangle) + |\Psi^+\rangle_{12}(C_V|V\rangle_3-C_H|H\rangle_3)-|\Psi^-_{12}(C_H|H_3\rangle+C_V|V\rangle_3)]$$ Alice then measures Bell state. Say she gets $$|\varphi^-\rangle_{12}$$ so she gets $$|\Phi^-\rangle_{12}(C_H|V\rangle_3+C_V|H\rangle_3)$$. She then communicates over classical phone and says that she got the result she got. So Bob knows the state of his third state is similar to Alice’s original state $$|\Psi\rangle_1$$. Then Bob applies the unitary transformation that swaps the vertical and horizontal polarizations to recover the original $$|\Psi\rangle_1$$ in $$|\Psi\rangle_3$$.

## EPR

The product of EPR gave rose to multiple interpretations: Copenhagen (Spearheaded by Bohr), Hidden Variables (John Bell 1964, testable inequality, no hidden variables), Subjective Theories.

### Subjective Theories

• E.P. Wigner: Wavefunction collapse happens at human brain
• Everett: Multiverse - Each measurement creates universes of all possible universes, but we only exist in one of them (Where does the energy required come from, is this equivalent to hidden variables)

Created: 2024-05-30 Thu 21:16

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