Physics:Quantum information theory

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Quantum information theory: qubits, superposition, entanglement, quantum gates, and applications in computing and cryptography.

Qubits and quantum states

A qubit is the fundamental unit of quantum information, analogous to the classical bit but with fundamentally different properties. A qubit is described by a state vector in a two-dimensional complex Hilbert space.

A general qubit state can be written as

$|\psi ⟩=\alpha |0⟩+\beta |1⟩,$

where $\alpha ,\beta \in \mathbb{ℂ}$ and satisfy the normalization condition

$|\alpha {|}^2+|\beta {|}^2=1.$

Superposition

Unlike classical bits, which are either 0 or 1, a qubit can exist in a superposition of both states. Measurement yields:

• $|0⟩$ with probability $|\alpha {|}^2$
• $|1⟩$ with probability $|\beta {|}^2$

This probabilistic interpretation follows from the Born rule.

Bloch sphere

A qubit state can be represented geometrically on the Bloch sphere:

$|\psi ⟩=\cos \frac{\theta }{2}|0⟩+e^{i\varphi }\sin \frac{\theta }{2}|1⟩.$

This representation maps quantum states to points on the surface of a unit sphere.

Multi-qubit systems

Systems of multiple qubits are described by tensor products of Hilbert spaces. For two qubits:

$|\psi ⟩\in {{\mathscr{H}}}_2\otimes {{\mathscr{H}}}_2.$

This leads to richer structures, including entanglement.

Physical significance

Qubits:

• form the basis of quantum computation and communication,
• enable superposition and interference effects,
• provide the simplest example of quantum states in Hilbert space.

Entanglement

Entanglement is a uniquely quantum phenomenon in which the state of a composite system cannot be described as a product of states of its individual subsystems. It is one of the central features distinguishing quantum from classical information theory.

Definition

A two-qubit state is entangled if it cannot be written in the form

$|\psi ⟩=|{\psi }_A⟩\otimes |{\psi }_B⟩.$

Instead, entangled states involve correlations between subsystems that cannot be reduced to independent descriptions.

Bell states

The simplest examples of entangled states are the Bell states:

$|{\mathrm{\Phi }}^{+}⟩=\frac{1}{\sqrt{2}}(|00⟩+|11⟩),$

$|{\mathrm{\Phi }}^{-}⟩=\frac{1}{\sqrt{2}}(|00⟩-|11⟩),$

$|{\mathrm{\Psi }}^{+}⟩=\frac{1}{\sqrt{2}}(|01⟩+|10⟩),$

$|{\mathrm{\Psi }}^{-}⟩=\frac{1}{\sqrt{2}}(|01⟩-|10⟩).$

These states exhibit perfect correlations between measurement outcomes.

Measurement correlations

If two particles are entangled, a measurement on one immediately determines the outcome probabilities of measurements on the other, regardless of the spatial separation.

This behavior is consistent with quantum mechanics but cannot be explained by classical local hidden-variable theories.

Reduced states

Even when the total system is in a pure state, each subsystem may be described by a mixed state. This is obtained using the reduced density matrix:

${\rho }_A={\mathrm{T}\mathrm{r}}_B({\rho }_{AB}).$

This reflects the fact that subsystems of entangled systems do not have independent pure states.

Physical significance

Entanglement:

• is a key resource in quantum information processing,
• underlies quantum teleportation and superdense coding,
• enables violations of Bell inequalities,
• plays a central role in quantum cryptography and computing.

It is one of the most distinctive and powerful features of quantum mechanics.

Quantum entropy

Quantum entropy quantifies the amount of uncertainty or information contained in a quantum state. The central concept is the von Neumann entropy, which generalizes classical Shannon entropy to quantum systems.

Definition

For a quantum system described by a density operator $\rho$, the von Neumann entropy is defined as

$S(\rho )=-\mathrm{T}\mathrm{r}(\rho \log \rho ).$

If $\rho$ has eigenvalues ${\lambda }_i$, then

$S(\rho )=-\sum _i{\lambda }_i\log {\lambda }_i.$

Pure and mixed states

The entropy distinguishes between pure and mixed states:

• For a pure state: $S(\rho )=0$
• For a mixed state: $S(\rho )>0$

Thus, entropy measures the degree of statistical uncertainty in the system.

Entropy and entanglement

For a bipartite system, quantum entropy can be used to quantify entanglement. If a system is in a pure state ${\rho }_{AB}$, the entropy of a subsystem

$S({\rho }_A)=S({\rho }_B)$

is a measure of entanglement.

Properties

The von Neumann entropy has several important properties:

• Non-negativity: $S(\rho )\ge 0$
• Unitary invariance: entropy is unchanged under unitary transformations
• Subadditivity:

 $S({\rho }_{AB})\le S({\rho }_A)+S({\rho }_B)$  

These properties parallel those of classical information theory.

Physical significance

Quantum entropy:

• measures information content and uncertainty in quantum states,
• plays a central role in quantum thermodynamics,
• quantifies entanglement and correlations,
• is fundamental in quantum communication theory.

Holevo bound

The Holevo bound is a fundamental result in quantum information theory that limits the amount of classical information that can be extracted from a quantum system. It shows that even though quantum states can exist in superpositions, the accessible classical information is restricted.

Statement

Suppose a sender prepares quantum states ${\rho }_i$ with probabilities $p_i$. The total state is

$\rho =\sum _i{p}_i{\rho }_i.$

The amount of classical information that can be obtained about the index $i$ is bounded by

$\chi =S(\rho )-\sum _i{p}_{i}S({\rho }_i),$

where $S(\rho )$ is the von Neumann entropy.

This quantity $\chi$ is called the Holevo quantity.

Interpretation

The Holevo bound states that the mutual information between sender and receiver satisfies

$I\le \chi .$

Even if quantum states encode large amounts of information, measurement limits the amount of classical information that can be extracted.

Special case

If the states ${\rho }_i$ are pure states, then $S({\rho }_i)=0$, and the bound simplifies to

$\chi =S(\rho ).$

Thus, the accessible information is limited by the entropy of the ensemble.

Physical significance

The Holevo bound:

• limits the capacity of quantum communication channels,
• explains why qubits do not allow unlimited classical information storage,
• plays a central role in quantum communication and coding theory.

It is one of the key results connecting quantum mechanics with information theory.

No-cloning theorem

The no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state. This is a fundamental result of quantum mechanics with important consequences for quantum information theory.

Statement

There is no physical operation that can take an arbitrary quantum state $|\psi ⟩$ and produce two identical copies:

$|\psi ⟩\to |\psi ⟩\otimes |\psi ⟩.$

This holds for all possible quantum states.

Proof idea

The theorem follows from the linearity of quantum mechanics. Suppose a cloning operation exists such that

$|\psi ⟩|0⟩\to |\psi ⟩|\psi ⟩,$

and similarly for another state $|\varphi ⟩$:

$|\varphi ⟩|0⟩\to |\varphi ⟩|\varphi ⟩.$

Applying the same operation to a superposition leads to a contradiction, since linear evolution would give a different result than cloning each component separately.

Consequences

The no-cloning theorem implies:

• unknown quantum states cannot be copied perfectly,
• quantum information cannot be duplicated like classical information,
• measurement inevitably disturbs the system.

Applications

The theorem plays a crucial role in:

• quantum cryptography — ensuring secure communication (e.g. quantum key distribution),
• quantum communication — limiting information transfer strategies,
• quantum computing — constraining how information is processed and stored.

Physical significance

The no-cloning theorem highlights a fundamental difference between classical and quantum information. It ensures the security of quantum protocols and reflects the linear and unitary structure of quantum mechanics.

See also

Foundations

• Physics:Quantum basics
• Physics:Quantum mechanics
• Physics:Quantum mechanics measurements
• Physics:Quantum Mathematical Foundations of Quantum_Theory

Conceptual and interpretations

• Physics:Quantum Interpretations of quantum mechanics
• Physics:Quantum A Spooky Action at a Distance
• Physics:Quantum A Walk Through the Universe
• Physics:Quantum: The Secret of Cohesion: How Waves Hold Matter Together

Mathematical and solvable systems

• Physics:Quantum Exactly solvable quantum systems
• Physics:Quantum Formulas Collection
• Physics:Quantum A Matter Of Size

Symmetry and structure

• Physics:Quantum Symmetry in quantum mechanics
• Physics:Quantum Matter Elements and Particles

Atomic and spectroscopy

• Physics:Quantum Atomic structure and spectroscopy

Quantum wavefunctions and modes

• Physics:Number of independent spatial modes in a spherical volume

Quantum information and computing

• Physics:Quantum information theory
• Physics:Quantum Computing Algorithms in the NISQ Era
• Physics:Quantum_Noisy_Qubits

Quantum optics and experiments

• Physics:Quantum optics beam splitter experiments
• Physics:Quantum Ultra fast lasers
• Physics:Quantum Experimental quantum physics
• Template Quantum optics operators

Open quantum systems

• Physics:Quantum Open quantum systems

Quantum field theory

• Physics:Quantum field theory (QFT) basics

Timeline

• Physics:Quantum mechanics/Timeline
• Physics:Quantum_mechanics/Timeline/Quiz/

Advanced and frontier topics

• Physics:Quantum Supersymmetry
• Physics:Quantum Black hole thermodynamics
• Physics:Quantum Holographic principle
• Physics:Quantum gravity
• Physics:Quantum De Sitter invariant special relativity
• Physics:Quantum Doubly special relativity

References

Author: Harold Foppele