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Quantum Information Theory is an extension of the ideas of quantum mechanics into information processing and transfer. It uses the qubit as the basic unit, which is able to be in superpositions of states. Differently from the classical bit, a qubit allows for interference and probabilities defined by quantum amplitudes. The interactions of more than one qubit lead to entanglement between qubits, with quantum correlation among states. These states are transformed with the help of quantum operations that include unitary transforms and gates. This set of elements constitutes the basis for quantum computing and quantum communication methods. The entropy and other information theoretical quantities are redefined using the von Neumann approach. The Holevo Bound sets the limits on extracting information in classical sense. Another unique aspect of quantum information is related to the no-cloning principle.

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.^[1]

A general qubit state can be written as

$| ψ ⟩ = α | 0 ⟩ + β | 1 ⟩,$

where $α, β \in ℂ$ and satisfy the normalization condition

$| α |^{2} + | β |^{2} = 1 .$

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 $| α |^{2}$
$| 1 ⟩$ with probability $| β |^{2}$

This probabilistic interpretation follows from the Born rule.

Bloch sphere

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

$| ψ ⟩ = \cos \frac{θ}{2} | 0 ⟩ + e^{i ϕ} \sin \frac{θ}{2} | 1 ⟩ .$

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:

$| ψ ⟩ \in ℋ_{2} \otimes ℋ_{2} .$

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.

Quantum operations

Quantum states evolve according to unitary transformations. A general evolution is given by

$| ψ ⟩ \to U | ψ ⟩,$

where $U$ is a unitary operator.

In quantum computing, these operations are implemented as quantum gates, which act on one or more qubits and form the building blocks of quantum circuits.

Examples include:

Pauli gates ( $X, Y, Z$ )
Hadamard gate
Controlled-NOT (CNOT) gate

These operations enable interference, entanglement, and algorithmic processing of quantum information.

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.^[2]

Definition

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

$| ψ ⟩ = | ψ_{A} ⟩ \otimes | ψ_{B} ⟩ .$

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:

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

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

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

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

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.^[3]

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:

$ρ_{A} = {T r}_{B} (ρ_{A B}) .$

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.^[4]

Definition

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

$S (ρ) = - T r (ρ \log ρ) .$

If $ρ$ has eigenvalues $λ_{i}$ , then

$S (ρ) = - \sum_{i} λ_{i} \log λ_{i} .$

Pure and mixed states

The entropy distinguishes between pure and mixed states:

For a pure state: $S (ρ) = 0$
For a mixed state: $S (ρ) > 0$

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 $ρ_{A B}$ , the entropy of a subsystem

$S (ρ_{A}) = S (ρ_{B})$

is a measure of entanglement.

Properties

The von Neumann entropy has several important properties:

Non-negativity: $S (ρ) \geq 0$
Unitary invariance: entropy is unchanged under unitary transformations
Subadditivity:

  $S (ρ_{A B}) \leq S (ρ_{A}) + S (ρ_{B})$

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.^[5]

Statement

Suppose a sender prepares quantum states $ρ_{i}$ with probabilities $p_{i}$ . The total state is

$ρ = \sum_{i} p_{i} ρ_{i} .$

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

$χ = S (ρ) - \sum_{i} p_{i} S (ρ_{i}),$

where $S (ρ)$ is the von Neumann entropy.

This quantity $χ$ is called the Holevo quantity.

Interpretation

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

$I \leq χ .$

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 $ρ_{i}$ are pure states, then $S (ρ_{i}) = 0$ , and the bound simplifies to

$χ = S (ρ) .$

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.^[6]

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.^[7]

Statement

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

$| ψ ⟩ \to | ψ ⟩ \otimes | ψ ⟩ .$

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

$| ψ ⟩ | 0 ⟩ \to | ψ ⟩ | ψ ⟩,$

and similarly for another state $| ϕ ⟩$ :

$| ϕ ⟩ | 0 ⟩ \to | ϕ ⟩ | ϕ ⟩ .$

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.^[8]

Table of contents (138 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

Full contents

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1. Foundations (7)

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2. Conceptual and interpretations (10)

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3. Mathematical structure and systems (11)

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4. Atomic and spectroscopy (11)

Physics:Quantum Atomic structure and spectroscopy
Physics:Quantum Hydrogen atom
Physics:Quantum Multi-electron atoms
Physics:Quantum Fine structure
Physics:Quantum Hyperfine structure
Physics:Quantum Isotopic shift
Physics:Quantum Zeeman effect
Physics:Quantum Stark effect
Physics:Quantum Spectral lines and series
Physics:Quantum Selection rules
Physics:Quantum Fermi's golden rule

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5. Wavefunctions and modes (8)

Physics:Quantum Wavefunction
Physics:Quantum Superposition principle
Physics:Quantum Eigenstates and eigenvalues
Physics:Quantum Boundary conditions and quantization
Physics:Quantum Standing waves and modes
Physics:Quantum Normal modes and field quantization
Physics:Number of independent spatial modes in a spherical volume
Physics:Quantum Density of states

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6. Quantum dynamics and evolution (9)

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7. Measurement and information (6)

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8. Quantum information and computing (6)

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9. Quantum optics and experiments (4)

Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
Physics:Quantum optics beam splitter experiments
Physics:Quantum Ultra fast lasers
Physics:Quantum Experimental quantum physics

Template:Quantum optics operators

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10. Open quantum systems (7)

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11. Quantum field theory (18)

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12. Statistical mechanics and kinetic theory (10)

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13. Condensed matter and solid-state physics (7)

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14. Plasma and fusion physics (9)

Physics:Quantum Plasma (fusion context)
Physics:Quantum Fusion reactions and Lawson criterion
Physics:Quantum Magnetic confinement fusion
Physics:Quantum Inertial confinement fusion
Physics:Quantum Plasma instabilities and turbulence
Physics:Quantum Tokamak
Physics:Quantum Tokamak core plasma
Physics:Quantum Tokamak edge physics and recycling asymmetries
Physics:Quantum Stellarator

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15. Timeline (8)

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16. Advanced and frontier topics (6)

References

↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.
↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.
↑ Bell, John S. (1987). Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press.
↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.
↑ Holevo, A. S. (1973). "Bounds for the quantity of information transmitted by a quantum communication channel". Problemy Peredachi Informatsii 9.
↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.
↑ Wootters, W. K.; Zurek, W. H. (1982). "A single quantum cannot be cloned". Nature 299.
↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.

Author: Harold Foppele

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[1] Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.

[2] Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.

[3] Bell, John S. (1987). Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press.

[4] Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.

[5] Holevo, A. S. (1973). "Bounds for the quantity of information transmitted by a quantum communication channel". Problemy Peredachi Informatsii 9.

[6] Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.

[7] Wootters, W. K.; Zurek, W. H. (1982). "A single quantum cannot be cloned". Nature 299.

[8] Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.

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Physics:Quantum information theory

Qubits and quantum states

Superposition

Bloch sphere

Multi-qubit systems

Physical significance

Quantum operations

Entanglement

Definition

Bell states

Measurement correlations

Reduced states

Physical significance

Quantum entropy

Definition

Pure and mixed states

Entropy and entanglement

Properties

Physical significance

Holevo bound

Statement

Interpretation

Special case

Physical significance

No-cloning theorem

Statement

Proof idea

Consequences

Applications

Physical significance

See also

Table of contents (138 articles)

Index

Full contents

References

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