CHAPTER 5 – QUANTUM INFORMATION
1. Quantum Information
The amplitude of a signal can represent information. As an example, consider a semiconductor
diode acting as a light detector. The higher the intensity of the incoming light, the higher the electric
current generated by the diode. If the intensity is not too high, the responsivity R(I) of the diode
depends linearly on the light intensity I. The electrical signal is analog in nature and can take any
value between zero and a maximum value.
Although this is intuitive, analog signals are difficult to store and process in practice because
infinitely many values are possible. Signal processing becomes much simpler when signals are
expressed in discrete steps. For this purpose we use bits. A bit can take only two values: 0 or 1.
Digital information therefore consists of long sequences of bits.
With 16 bits, for example, 2^16 = 65536 different intensity values can be represented. This principle
underlies modern electronics and data storage.
1.1 Bit versus Qubit
A classical bit is strictly binary: it is either 0 or 1. A quantum bit (qubit), however, can exist in a
superposition of states:
ψ = α|0■ + β|1■
Here |0■ and |1■ are orthogonal basis states, and α and β are complex amplitudes. The state must
be normalized:
|α|² + |β|² = 1
Before measurement, a qubit exists in both states simultaneously. Upon measurement, the
wavefunction collapses into either |0■ or |1■.
1.2 Implementation of a Qubit
A qubit requires a system with two orthogonal states. Examples include spin-up and spin-down
states of a fermion, or the polarization of a photon. In this course, photon polarization is used.
Horizontal polarization |H■ and vertical polarization |V■ form one basis. A general photon state can
be written as:
ψ = α|H■ + β|V■
Measurement forces the photon into one of these states with probabilities |α|² and |β|².
Other possible bases are diagonal/anti-diagonal polarization and left/right circular polarization.
1.3 Poincaré Sphere
The Poincaré sphere provides a graphical representation of all polarization states of a photon. Each
basis corresponds to an axis, and any qubit state lies on the surface of the sphere. This is
analogous to the Bloch sphere used for spin-½ particles.
1.4 Measurement in a Basis
Measurement collapses the wavefunction into one of the basis states. Repeating the same
measurement yields the same result with certainty. Measuring in a different basis introduces
randomness.
For example, a photon in a right-circular polarization state |R■ measured in the horizontal/vertical
basis yields |H■ or |V■ with equal probability.
1.5 Quantum Key Distribution (BB84 Protocol)
Quantum mechanics can be used to securely distribute encryption keys. The BB84 protocol was
proposed by Bennett and Brassard in 1984.
Alice wants to send Bob a secret key without eavesdropping.
Step 1: Alice and Bob choose two bases: - H/V basis: |H■ = 0, |V■ = 1 - D/A basis: |D■ = 0, |A■ =
1
Step 2: Alice prepares a random sequence of bits in randomly chosen bases and sends the
photons to Bob.
Step 3: Bob measures each photon in a randomly chosen basis.
If Bob uses the correct basis, he measures the correct bit with 100% certainty. If not, the result is
random.
Step 4: Bob publicly announces which basis he used for each measurement. Alice tells him which
were correct. Only those bits are kept as the key.
Step 5: Detecting Eavesdropping
If an eavesdropper (Eve) intercepts and measures photons, she must guess the basis. Wrong
guesses collapse the wavefunction incorrectly, introducing errors.
Alice and Bob publicly compare a subset of their key. If errors are detected, eavesdropping has
occurred.
The probability that Eve remains undetected after N comparisons is 0.75^N.
1.6 Quantum Entanglement
Entanglement occurs when two particles share a joint wavefunction. Measuring one immediately
determines the state of the other, regardless of distance.
Entanglement can be used for secure communication, though it cannot transmit information faster
than light. This phenomenon is related to the EPR paradox and Bell’s theorem.
1.7 Experimental Systems
Qubits can be implemented using atomic transitions, NV centers in diamond, photon polarization,
and entangled photon pairs generated via cascade emission.
1.8 Summary
Classical bits are binary, while qubits can exist in superpositions. Measurement collapses quantum
states and can alter them. Photon polarization provides a practical qubit implementation. Quantum
mechanics enables secure key distribution through protocols such as BB84.
1.9 Further Reading
Based on “Introduction to Quantum Information” by Stephen M. Barnett.