Quantum Information

Table of Contents

• Quantum Interference
• Polarization and Wave Plates
• Dense Coding
• Quantum Teleportation
• Entanglement Swapping
• Quantum Key Distribution (QKD)

Quantum Interference

Beam Splitter and Superposition

A beam splitter (BS) creates a quantum superposition of paths rather than splitting a photon physically.

For a 50/50 beam splitter:

\[ | \text{in} \rangle \rightarrow \frac{1}{\sqrt{2}} | \text{transmitted} \rangle + \frac{i}{\sqrt{2}} | \text{reflected} \rangle \]

• The coefficients are probability amplitudes
• Probabilities are obtained by:

\[ P = |\text{amplitude}|^2 \]

Single Beam Splitter Experiment

A photon incident on a beam splitter is detected at one of two detectors:

\[ P(D_1) = P(D_2) = \frac{1}{2} \]

• The photon is never split between detectors
• This appears as classical randomness

Multiple Paths Without Interference

With multiple beam splitters and independent paths:

\[ P(D_1) = P(D_2) = P(D_3) = P(D_4) = \frac{1}{4} \]

• Probabilities distribute evenly
• No interference occurs when paths are independent

Mach–Zehnder Interferometer

A Mach–Zehnder interferometer consists of:

1. Beam splitter (creates superposition)
2. Mirrors (redirect paths)
3. Second beam splitter (recombines paths)

Interference of Amplitudes

The total amplitude at a detector is:

\[ A_{\text{total}} = A_1 + A_2 \]

The probability is:

\[ P = |A_1 + A_2|^2 \]

This differs from classical addition:

\[ P \neq |A_1|^2 + |A_2|^2 \]

Example: Perfect Interference

For one detector:

\[ A = \frac{i}{2} + \frac{i}{2} = i \]

\[ P = |i|^2 = 1 \]

For the other detector:

\[ A = \frac{i^2}{2} + \frac{1}{2} = 0 \]

\[ P = 0 \]

Result:

• All photons arrive at one detector
• No photons arrive at the other

Role of Phase

A phase shift modifies a path:

\[ |\psi\rangle \rightarrow e^{i\phi} |\psi\rangle \]

Total amplitude becomes:

\[ A = A_1 + e^{i\phi} A_2 \]

• Interference depends on relative phase
• Phase determines constructive or destructive interference

Constructive and Destructive Interference

Constructive interference:

\[ |A_1 + A_2|^2 \text{ is maximized} \]

Destructive interference:

\[ A_1 + A_2 = 0 \]

Conditions for Interference

Interference occurs when:

• Paths are indistinguishable
• Phases are coherent

Blocking a path removes interference because only one amplitude remains.

Conceptual Takeaways

• Quantum systems combine amplitudes, not probabilities
• Interference arises from complex phase relationships
• A photon behaves as a superposition of paths
• Measurement removes interference by destroying coherence

Polarization and Wave Plates

Polarization as a Qubit

Photon polarization is a two-level quantum system:

\[ |H\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad |V\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]

General state:

\[ |\psi\rangle = \alpha |H\rangle + \beta |V\rangle \]

with normalization:

\[ |\alpha|^2 + |\beta|^2 = 1 \]

Jones Vector Representation

Polarization is represented as:

\[ \begin{bmatrix} E_x \\ E_y \end{bmatrix} \]

This encodes:

• Amplitude
• Relative phase

Linear Polarization

At angle \(\theta\):

\[ |\theta\rangle = \begin{bmatrix} \cos\theta \\ \sin\theta \end{bmatrix} \]

Special Polarization States

Diagonal:

\[ |D\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 1 \end{bmatrix} \]

Anti-diagonal:

\[ |A\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ -1 \end{bmatrix} \]

Circular Polarization

Right circular:

\[ |R\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ i \end{bmatrix} \]

Left circular:

\[ |L\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ -i \end{bmatrix} \]

• Circular polarization arises from a phase difference of \(\pm \frac{\pi}{2}\)

Polarizing Beam Splitter (PBS)

An optical element that separates light into two orthogonal polarizations:

• Transmits \(|H\rangle\)
• Reflects \(|V\rangle\)

Measurement probabilities:

\[ P(H) = |\alpha|^2, \quad P(V) = |\beta|^2 \]

After measurement, the state collapses to one basis state.

Half-Wave Plate (HWP)

A half-wave plate introduces a phase shift between orthogonal components.

Matrix form:

\[ HWP(\theta) = \begin{bmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{bmatrix} \]

Important Cases

\(\theta = 0^\circ\)

\[ HWP = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \]

• Leaves \(|H\rangle\) unchanged
• Adds phase \(-1\) to \(|V\rangle\)

\(\theta = 45^\circ\)

\[ HWP = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

• Swaps \(|H\rangle \leftrightarrow |V\rangle\)

\(\theta = 22.5^\circ\)

\[ HWP = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \]

• Equivalent to the Hadamard transformation

\[ |H\rangle \rightarrow |D\rangle, \quad |V\rangle \rightarrow |A\rangle \]

Physical Interpretation of HWP

A half-wave plate:

1. Splits the polarization into two orthogonal components
2. Introduces a phase difference of \(\pi\)
3. Recombines the components

This results in a rotation of polarization.

Key Observations

• The transformation depends on \(2\theta\), not \(\theta\)
• Wave plates perform unitary operations
• They act as quantum gates on polarization states

Conceptual Takeaways

• Polarization is a quantum two-level system
• Phase determines the difference between linear and circular states
• Measurement projects onto a basis and destroys superposition
• Wave plates implement controlled transformations of quantum states

Dense Coding

Overview

Dense coding is a quantum communication protocol that allows sending two classical bits by transmitting only one qubit, using a shared entangled pair.

Key idea:

• Classical limit: 1 qubit → 1 bit
• With entanglement: 1 qubit → 2 bits

Initial Setup

Alice and Bob share an entangled Bell state:

\[ |\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) \]

Ownership:

• Alice holds qubit 1
• Bob holds qubit 2

This shared entanglement is the key resource.

Bell States

The four maximally entangled Bell states:

\[ |\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) \]

\[ |\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle) \]

\[ |\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle) \]

\[ |\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle) \]

Interpretation:

• \( \Phi \): correlated bits (same values)
• \( \Psi \): anti-correlated bits (opposite values)
• \( + / - \): relative phase difference

Encoding by Alice

Alice encodes 2 classical bits by applying one of four operations to her qubit:

\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

\[ X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

\[ Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \]

\[ XZ = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]

Mapping:

Key idea:

• Alice does not send two bits directly
• She transforms the shared entangled state

Transmission

After encoding:

• Alice sends her qubit to Bob
• Bob now has both qubits

Only one qubit is physically transmitted

Decoding by Bob

Bob performs a Bell-state measurement.

• Each Bell state corresponds to a unique 2-bit message
• If all four states are distinguishable:

\[ I = \log_2(4) = 2 \text{ bits} \]

This exceeds the classical limit.

Information Capacity

Ideal case:

\[ 2 \text{ classical bits per qubit} \]

Classical limit:

\[ 1 \text{ bit per qubit} \]

Entanglement doubles the capacity.

Physical Implementation (Linear Optics)

In practice, photons are used with polarization:

• \( |H\rangle \) = horizontal
• \( |V\rangle \) = vertical

Bell states become:

\[ |\Phi^+\rangle = \frac{1}{\sqrt{2}}(|HH\rangle + |VV\rangle) \]

\[ |\Phi^-\rangle = \frac{1}{\sqrt{2}}(|HH\rangle - |VV\rangle) \]

\[ |\Psi^+\rangle = \frac{1}{\sqrt{2}}(|HV\rangle + |VH\rangle) \]

\[ |\Psi^-\rangle = \frac{1}{\sqrt{2}}(|HV\rangle - |VH\rangle) \]

Alice’s Encoding with Wave Plates

Alice uses half-wave plates (HWPs):

• No HWP → \( |\Psi^+\rangle \)
• HWP(0°) → phase flip
• HWP(45°) → bit flip
• Both → combined transformation

These implement the operators \( I, X, Z, XZ \)

Bell-State Measurement (BSA)

Bob uses:

• Beam splitter (BS)
• Polarizing beam splitters (PBS)
• Detectors

The measurement outcome depends on photon behavior:

• Interference at BS
• Polarization splitting at PBS

Key Physical Effects

Bosonic Symmetry

Photons are identical particles:

\[ |H_1 V_2\rangle = |V_2 H_1\rangle \]

This leads to interference effects.

Hong–Ou–Mandel Interference

At a beam splitter:

• Some states → photons bunch (same output)
• Some states → photons separate (different outputs)

This determines which Bell states can be distinguished.

Detection Patterns

From the Bell-state analyzer:

• \( |\Psi^-\rangle \): photons exit different ports
• \( |\Psi^+\rangle \): same port, different detectors
• \( |\Phi^\pm\rangle \): same detector pattern

Key limitation:

• Cannot distinguish \( |\Phi^+\rangle \) vs \( |\Phi^-\rangle \)

Practical Limitation

Linear optics cannot fully distinguish all Bell states.

Result:

• Only 3 distinguishable outcomes

\[ I_{\text{max}} = \log_2(3) \approx 1.58 \text{ bits} \]

Improved Bell-State Analyzer

Using auxiliary entangled photons:

• Near-deterministic discrimination
• \( \sim 75% \) success rate

Still limited by linear optics constraints.

Conceptual Takeaways

• Entanglement enables higher communication capacity
• Information is encoded in global quantum states, not local bits
• Alice modifies a shared state, not her qubit alone
• Measurement requires joint operations on both qubits
• Physical implementations impose real limitations

Common Misconceptions

• Alice does not “store two bits in one qubit”
• Bob cannot decode without receiving Alice’s qubit
• Entanglement alone does not transmit information
• Without entanglement → dense coding is impossible

Quantum Teleportation

Overview

Quantum teleportation is a protocol that transfers an unknown quantum state from one location to another using:

• Entanglement
• Classical communication

Key idea:

• The quantum state is not copied
• The original is destroyed
• The state is reconstructed at the destination

The Problem

Given an unknown qubit:

\[ |\psi\rangle = \alpha|0\rangle + \beta|1\rangle \]

Goal:

\[ |\psi\rangle_A \rightarrow |\psi\rangle_B \]

Naively, one might try:

\[ |\psi\rangle |0\rangle \rightarrow |\psi\rangle |\psi\rangle \]

This is impossible.

No-Cloning Theorem

Unknown quantum states cannot be copied.

Reason:

Linearity implies:

\[ C(\alpha|0\rangle + \beta|1\rangle) = \alpha C|0\rangle + \beta C|1\rangle \]

But cloning would require:

\[ (\alpha|0\rangle + \beta|1\rangle)(\alpha|0\rangle + \beta|1\rangle) \]

These expressions are not equal for general \( \alpha, \beta \).

Conclusion:

• Cloning is impossible
• State must be transferred, not duplicated

Initial Setup

Three qubits:

• Qubit 1: unknown state \( |\psi\rangle \) (Alice)
• Qubit 2: entangled (Alice)
• Qubit 3: entangled (Bob)

Shared Bell state:

\[ |\Phi^+\rangle_{23} = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) \]

Ownership:

• Alice → qubits 1 and 2
• Bob → qubit 3

Total Initial State

\[ |\Psi\rangle = |\psi\rangle_1 \otimes |\Phi^+\rangle_{23} \]

\[ = (\alpha|0\rangle + \beta|1\rangle) \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) \]

This is a three-qubit system.

Bell States

\[ |\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle) \]

\[ |\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle) \]

These form a complete basis for two qubits.

Bell Basis Expansion

The total state can be rewritten as:

\[ |\Psi\rangle = \frac{1}{2} \Big(|\Phi^+\rangle_{12} (\alpha|0\rangle + \beta|1\rangle)_3 - |\Phi^-\rangle_{12} (\alpha|0\rangle - \beta|1\rangle)_3 - |\Psi^+\rangle_{12} (\alpha|1\rangle + \beta|0\rangle)_3 - |\Psi^-\rangle_{12} (\alpha|1\rangle - \beta|0\rangle)_3\Big) \]

Key insight:

• Alice’s measurement outcome determines Bob’s state

Measurement by Alice

Alice performs a Bell-state measurement on qubits 1 and 2.

Result:

• One of four Bell states
• Produces 2 classical bits

After measurement:

• Original state is destroyed
• Bob’s qubit collapses into a related state

Conditional States at Bob

Bob has a modified version of \( |\psi\rangle \).

Classical Communication

Alice sends 2 classical bits to Bob.

Important:

• Without this, Bob cannot recover the state
• No faster-than-light communication

Correction by Bob

Bob applies an operation depending on Alice’s result:

After correction:

\[ |\psi\rangle_B = \alpha|0\rangle + \beta|1\rangle \]

Final Result

• Bob obtains the original quantum state
• Alice’s state is destroyed

\[ |\psi\rangle_A \rightarrow |\psi\rangle_B \]

No duplication occurs.

Physical Implementation (Optical Systems)

Qubits represented by polarization:

• \( |H\rangle \) = horizontal
• \( |V\rangle \) = vertical

Components:

• Beam splitter (BS)
• Polarizing beam splitters (PBS)
• Detectors

Bell-state analyzer:

• Can distinguish only some Bell states

Practical Limitation

Linear optics can distinguish:

\[ |\Psi^+\rangle, \quad |\Psi^-\rangle \]

But not:

\[ |\Phi^+\rangle, \quad |\Phi^-\rangle \]

Result:

\[ P_{\text{success}} = \frac{1}{2} \]

Teleportation is probabilistic in practice.

Conceptual Flow

1. Entanglement shared
2. Alice entangles unknown with Bell pair
3. Alice measures (destroys original state)
4. Classical bits sent to Bob
5. Bob applies correction
6. State reconstructed

Conceptual Takeaways

• Quantum information cannot be copied
• Entanglement enables state transfer
• Classical communication is required
• Information is transferred, not particles
• Measurement redistributes quantum information

Common Misconceptions

• Teleportation does not move matter
• Entanglement alone does not transmit information
• The state does not exist in two places
• Classical communication is essential

Entanglement Swapping

Overview

Entanglement swapping is a protocol that creates entanglement between two particles that have never interacted.

Key idea:

• Two independent entangled pairs are prepared
• A joint measurement is performed on one particle from each pair
• This creates entanglement between the remaining two particles

Initial Setup

Two Bell pairs:

\[ |\Phi^+\rangle_{12} = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) \]

\[ |\Phi^+\rangle_{34} = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) \]

Ownership:

• Alice → qubit 1
• Bob → qubit 4
• Middle station → qubits 2 and 3

Initially:

• (1,2) are entangled
• (3,4) are entangled
• (1,4) are not entangled

Total Initial State

\[ |\Psi\rangle = |\Phi^+\rangle_{12} \otimes |\Phi^+\rangle_{34} \]

\[ = \frac{1}{2} (|0000\rangle + |0011\rangle + |1100\rangle + |1111\rangle) \]

This is a four-qubit system.

Bell States

\[ |\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle) \]

\[ |\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle) \]

These form a complete basis for two qubits.

Bell Basis Expansion (Key Step)

Rewrite the total state in terms of Bell states of qubits (2,3):

\[ |\Psi\rangle = \frac{1}{2} \Big( |\Phi^+\rangle_{23} |\Phi^+\rangle_{14} + |\Phi^-\rangle_{23} |\Phi^-\rangle_{14} + |\Psi^+\rangle_{23} |\Psi^+\rangle_{14} + |\Psi^-\rangle_{23} |\Psi^-\rangle_{14} \Big) \]

Key insight:

• The system is a superposition of correlated Bell states
• Each Bell state of (2,3) is paired with the same Bell state of (1,4)

Measurement at the Middle Station

A Bell-state measurement is performed on qubits 2 and 3.

Result:

• One of four Bell states is obtained
• The system collapses to the corresponding term

After measurement:

• Original entanglement (1–2 and 3–4) is destroyed

Resulting State of (1,4)

Depending on the measurement outcome:

Result:

• Qubits (1,4) become entangled
• Type of entanglement depends on measurement outcome

Classical Communication (Optional Correction)

If a specific Bell state is required:

• Measurement result must be sent to Alice or Bob
• A correction operation can be applied

This is similar to teleportation:

• Operations: \( I, X, Z, XZ \)

Final Result

\[ (1,2), (3,4) ;\rightarrow; (1,4) \]

• Entanglement is transferred
• Qubits 1 and 4 are now entangled
• They never interacted directly

Physical Interpretation

• Entanglement is not a local property
• It is a property of the global quantum state

Measurement:

• Does not create entanglement
• It selects one correlation pattern already present

Physical Implementation (Optical Systems)

Qubits represented by polarization:

• \( |H\rangle \), \( |V\rangle \)

Procedure:

• Two entangled photon pairs are generated
• Photons 2 and 3 interfere at a beam splitter
• A Bell-state measurement is performed

Practical Limitation

Linear optics:

• Cannot distinguish all Bell states

Typically distinguishable:

\[ |\Psi^+\rangle, \quad |\Psi^-\rangle \]

Result:

\[ P_{\text{success}} = \frac{1}{2} \]

Entanglement swapping is probabilistic in practice.

Conceptual Flow

1. Prepare two entangled pairs
2. Bring qubits (2,3) together
3. Rewrite system in Bell basis
4. Perform Bell-state measurement on (2,3)
5. Collapse system to one term
6. Qubits (1,4) become entangled

Conceptual Takeaways

• Entanglement can be created between non-interacting particles
• Measurement redistributes quantum correlations
• Entanglement is a global property
• Basis choice determines how correlations are revealed
• Essential for quantum communication networks

Common Misconceptions

• Measurement creates entanglement (it does not)
• Particles must interact to become entangled
• Entanglement is stored locally in particles
• The process transmits information instantly

Quantum Key Distribution

Overview

Quantum Key Distribution (QKD) is a method for securely sharing a secret key between two parties using quantum mechanics.

Key idea:

• Security is based on physical laws, not computational hardness
• Measurement of quantum states disturbs them
• Eavesdropping can be detected

Classical Motivation

The one-time pad provides perfect secrecy:

\[ C = M \oplus K \]

• \( M \): message
• \( K \): secret key
• \( C \): ciphertext

Problem:

• Alice and Bob must already share a secure key
• Key distribution is difficult

QKD addresses this problem by enabling Alice and Bob to establish a shared secret key whose security is guaranteed by quantum mechanics, provided they already share an authenticated public classical channel; in other words, QKD reduces the key-distribution problem to authentication rather than eliminating all trust assumptions.

Quantum Advantage

Two fundamental principles:

No-Cloning Theorem

Unknown quantum states cannot be copied.

Measurement Disturbance

Measuring a quantum state generally changes it.

Conclusion:

• An eavesdropper cannot observe communication without being detected

BB84 Protocol

Alice uses two orthogonal bases:

Rectilinear Basis (+)

\[ |0\rangle = |H\rangle, \quad |1\rangle = |V\rangle \]

Diagonal Basis (×)

\[ |0\rangle = |D\rangle, \quad |1\rangle = |A\rangle \]

Basis Relationship

The bases are related by superposition:

\[ |D\rangle = \frac{1}{\sqrt{2}}(|H\rangle + |V\rangle) \]

\[ |A\rangle = \frac{1}{\sqrt{2}}(|H\rangle - |V\rangle) \]

Key consequence:

• Measuring in the wrong basis gives random results

Protocol Steps

1. Alice generates random bits
2. Alice randomly chooses a basis (+ or ×)
3. Alice sends encoded qubits
4. Bob randomly chooses measurement bases
5. Bob measures each qubit

Measurement Outcomes

• Same basis → deterministic result
• Different basis → random result (50/50)

Sifting Process

After transmission:

• Alice and Bob publicly announce bases only
• They discard all mismatched cases

Result:

• Remaining bits form the sifted key

Key Efficiency

Approximately:

\[ 50% \text{ of bits are discarded} \]

Eavesdropping (Intercept-Resend Attack)

Eve:

1. Intercepts qubit
2. Measures in random basis
3. Resends qubit

Effect of Eavesdropping

If Eve uses the wrong basis:

• She disturbs the state
• Bob may receive incorrect value

Even when:

• Alice and Bob use the same basis

This introduces errors.

Error Detection

Alice and Bob:

1. Reveal a subset of bits
2. Compare results

If error rate is high:

• Eavesdropping is detected
• Key is discarded

Key Insight

Security arises because:

• Information gain ⇒ disturbance
• Disturbance ⇒ detectable errors

B92 Protocol

B92 is a simplified QKD protocol using:

• Only two non-orthogonal states

Encoding

Alice sends:

• Bit 0 → \( |H\rangle \)
• Bit 1 → \( |D\rangle \)

These states are not orthogonal:

\[ \langle H | D \rangle \neq 0 \]

Measurement Bases

Bob randomly measures in:

• H/V basis
• D/A basis

Measurement Behavior

If Alice sends \( |H\rangle \)

• H/V → always H
• D/A → random

If Alice sends \( |D\rangle \)

• D/A → always D
• H/V → random

Conclusive vs Inconclusive Results

A result is conclusive if it rules out one possibility.

Conclusive Results

• Detect V → must be \( |D\rangle \) → bit = 1
• Detect A → must be \( |H\rangle \) → bit = 0

Inconclusive Results

• Detect H or D → cannot determine bit → discard

Sifting Process

Bob:

• Announces positions of conclusive results

Alice:

• Keeps corresponding bits

Result:

• Shared secret key

Efficiency

• Many measurements are discarded
• Lower efficiency than BB84

Key Insight

Security arises from:

• Inability to perfectly distinguish non-orthogonal states

Conceptual Comparison

Conceptual Takeaways

• Quantum mechanics enables secure key distribution
• Measurement disturbs quantum states
• Eavesdropping introduces detectable errors
• BB84 uses basis incompatibility
• B92 uses non-orthogonal states
• Security is physical, not computational

Common Misconceptions

• Randomness alone provides security
• QKD transmits the key directly (it generates it)
• Eavesdropping can be hidden without errors
• Non-orthogonal states can be perfectly distinguished