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{{Quantum book backlink|Quantum information and computing}}
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.
[[File:Quantum_information_theory_yellow.jpg|thumb|400px|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.<ref>{{cite book |last=Nielsen |first=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010}}</ref>

A general qubit state can be written as

<math>
|\psi\rangle = \alpha |0\rangle + \beta |1\rangle,
</math>

where <math>\alpha, \beta \in \mathbb{C}</math> and satisfy the normalization condition

<math>
|\alpha|^2 + |\beta|^2 = 1.
</math>

=== Superposition ===

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

* <math>|0\rangle</math> with probability <math>|\alpha|^2</math>
* <math>|1\rangle</math> with probability <math>|\beta|^2</math>

This probabilistic interpretation follows from the [[Wikipedia:Born rule|Born rule]].

=== Bloch sphere ===

A qubit state can be represented geometrically on the [[Wikipedia:Bloch sphere|Bloch sphere]]:

<math>
|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\frac{\theta}{2}|1\rangle.
</math>

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:

<math>
|\psi\rangle \in \mathcal{H}_2 \otimes \mathcal{H}_2.
</math>

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

<math>
|\psi\rangle \rightarrow U |\psi\rangle,
</math>

where <math>U</math> 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 (<math>X, Y, Z</math>)
* 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.<ref>{{cite book |last=Nielsen |first=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010}}</ref>

=== Definition ===

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

<math>
|\psi\rangle = |\psi_A\rangle \otimes |\psi_B\rangle.
</math>

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''':

<math>
|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle),
</math>

<math>
|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle),
</math>

<math>
|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle),
</math>

<math>
|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle).
</math>

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.<ref>{{cite book |last=Bell |first=John S. |title=Speakable and Unspeakable in Quantum Mechanics |publisher=Cambridge University Press |year=1987}}</ref>

=== 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:

<math>
\rho_A = \mathrm{Tr}_B(\rho_{AB}).
</math>

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.<ref>{{cite book |last=Nielsen |first=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010}}</ref>

=== Definition ===

For a quantum system described by a density operator <math>\rho</math>, the von Neumann entropy is defined as

<math>
S(\rho) = -\mathrm{Tr}(\rho \log \rho).
</math>

If <math>\rho</math> has eigenvalues <math>\lambda_i</math>, then

<math>
S(\rho) = -\sum_i \lambda_i \log \lambda_i.
</math>

=== Pure and mixed states ===

The entropy distinguishes between pure and mixed states:

* For a pure state: <math>S(\rho) = 0</math>
* For a mixed state: <math>S(\rho) > 0</math>

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 <math>\rho_{AB}</math>, the entropy of a subsystem

<math>
S(\rho_A) = S(\rho_B)
</math>

is a measure of entanglement.

=== Properties ===

The von Neumann entropy has several important properties:

* '''Non-negativity''': <math>S(\rho) \ge 0</math>
* '''Unitary invariance''': entropy is unchanged under unitary transformations
* '''Subadditivity''':
<math>
S(\rho_{AB}) \le S(\rho_A) + S(\rho_B)
</math>

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.<ref>{{cite journal |last=Holevo |first=A. S. |title=Bounds for the quantity of information transmitted by a quantum communication channel |journal=Problemy Peredachi Informatsii |volume=9 |year=1973}}</ref>

=== Statement ===

Suppose a sender prepares quantum states <math>\rho_i</math> with probabilities <math>p_i</math>. The total state is

<math>
\rho = \sum_i p_i \rho_i.
</math>

The amount of classical information that can be obtained about the index <math>i</math> is bounded by

<math>
\chi = S(\rho) - \sum_i p_i S(\rho_i),
</math>

where <math>S(\rho)</math> is the von Neumann entropy.

This quantity <math>\chi</math> is called the '''Holevo quantity'''.

=== Interpretation ===

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

<math>
I \le \chi.
</math>

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 <math>\rho_i</math> are pure states, then <math>S(\rho_i)=0</math>, and the bound simplifies to

<math>
\chi = S(\rho).
</math>

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.<ref>{{cite book |last=Nielsen |first=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010}}</ref>

=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.<ref>{{cite journal |last=Wootters |first=W. K. |last2=Zurek |first2=W. H. |title=A single quantum cannot be cloned |journal=Nature |volume=299 |year=1982}}</ref>

=== Statement ===

There is no physical operation that can take an arbitrary quantum state <math>|\psi\rangle</math> and produce two identical copies:

<math>
|\psi\rangle \rightarrow |\psi\rangle \otimes |\psi\rangle.
</math>

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

<math>
|\psi\rangle |0\rangle \rightarrow |\psi\rangle |\psi\rangle,
</math>

and similarly for another state <math>|\phi\rangle</math>:

<math>
|\phi\rangle |0\rangle \rightarrow |\phi\rangle |\phi\rangle.
</math>

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.<ref>{{cite book |last=Nielsen |first=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010}}</ref>

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum information theory|1}}

Physics:Quantum information theory - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

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