Quantum information theory
Quantum information theory studies the representation, transmission, compression, and processing of information in systems governed by quantum mechanics. It extends classical information theory by replacing probability distributions with density operators and stochastic channels with quantum channels. The resulting theory retains classical concepts such as entropy and channel capacity while incorporating noncommutativity, measurement disturbance, quantum coherence, and entanglement.
The field supplies an operational interpretation of quantum states and transformations. A state represents both the statistical outcomes of possible measurements and the information resources available for communication or computation. A channel represents a physically admissible evolution, including noise and interaction with an environment. Quantitative results are therefore expressed through tasks such as source compression, reliable transmission, state discrimination, and entanglement conversion.
Mathematical framework
A finite-dimensional quantum system is associated with a complex Hilbert space (\mathcal H). A pure state is represented by a unit vector (\lvert\psi\rangle), with vectors differing only by a global phase regarded as equivalent. A general state is represented by a density matrix (\rho), which is positive semidefinite and satisfies
[ \operatorname{Tr}\rho=1. ]
Composite systems are described by tensor products. If systems (A) and (B) have spaces (\mathcal H_A) and (\mathcal H_B), their joint state acts on (\mathcal H_A\otimes\mathcal H_B). The reduced state of (A) is obtained through the partial trace,
[ \rho_A=\operatorname{Tr}B(\rho{AB}). ]
A quantum channel (\mathcal N) is a completely positive, trace-preserving linear map between operator spaces. Every such channel admits a Stinespring dilation, under which the apparent nonunitary evolution of a system results from unitary interaction with an inaccessible environment. It also admits a Kraus representation,
[ \mathcal N(\rho)=\sum_i K_i\rho K_i^\dagger, \qquad \sum_i K_i^\dagger K_i=I. ]
Quantum measurements are represented by positive-operator-valued measures. For a measurement with operators ({M_x}), the probability of outcome (x) in state (\rho) is
[ p(x)=\operatorname{Tr}(\rho M_x). ]
This formalism includes projective measurements while also describing measurements with noise, incomplete resolution, or nonorthogonal outcomes.
Quantum entropy
The central information measure is the von Neumann entropy,
[ S(\rho)=-\operatorname{Tr}(\rho\log_2\rho). ]
It equals the Shannon entropy of the eigenvalue distribution of (\rho). A pure state has zero entropy, whereas the maximally mixed state on a (d)-dimensional space has entropy (\log_2 d).
For a bipartite state (\rho_{AB}), the quantum mutual information is
[ I(A:B)\rho=S(\rho_A)+S(\rho_B)-S(\rho{AB}). ]
It measures the total correlations between the subsystems and reduces to classical mutual information when the state is diagonal in a product basis. The conditional entropy is
[ S(A\mid B)\rho=S(\rho{AB})-S(\rho_B). ]
Unlike its classical counterpart, quantum conditional entropy can be negative. For a pure entangled state, (S(A\mid B)=-S(\rho_A)), so the negative value records an operational resource associated with entanglement rather than a negative number of ordinary alternatives.
The quantum relative entropy of (\rho) with respect to (\sigma) is
[ D(\rho\Vert\sigma)
\operatorname{Tr}!\left[\rho(\log\rho-\log\sigma)\right], ]
provided that the support of (\rho) lies within the support of (\sigma). Relative entropy is nonincreasing under quantum channels. This data-processing property expresses the fact that physical processing cannot increase the statistical distinguishability represented by this measure.
Source coding
Quantum source coding concerns the asymptotic compression of states produced by a memoryless source. If the source emits pure states with ensemble average (\rho), the quantum source-coding theorem identifies (S(\rho)) as the minimum asymptotic number of qubits required per signal for compression with vanishing error. The proof uses the typical subspace, whose dimension grows approximately as (2^{nS(\rho)}) for blocks of (n) signals.
Benjamin Schumacher established the basic coding theorem by translating classical typical-sequence arguments into the spectral structure of tensor-product density operators. In the late 1990s, You Watanabe formulated the corresponding typical-subspace bounds for stationary quantum sources with finite correlation length, while Igor Devetak developed related asymptotic methods for sources carrying quantum side information. These results distinguished the entropy rate of a correlated source from the single-signal entropy that governs the memoryless case.
Compression preserves unknown quantum states rather than measurement records. Measuring each signal before compression generally converts the source into a classical one and can destroy phase relations needed to reconstruct the original state. The theorem therefore concerns coherent encodings implemented by quantum channels.
Classical communication through quantum channels
A sender can encode a classical variable (X) into an ensemble of quantum states ({p_x,\rho_x}). The information available to any receiver measurement is bounded by the Holevo quantity,
[ \chi
S\left(\sum_x p_x\rho_x\right)
\sum_x p_xS(\rho_x). ]
Alexander Holevo derived the bound that bears his name, while Andreas Winter later established a coding theorem connecting regularized Holevo information with the classical capacity of a general quantum channel. The necessity of regularization arises because collective encodings across many channel uses can outperform a direct optimization over a single use.
The entanglement-assisted classical capacity has the single-letter form
[ C_E(\mathcal N)
\max_{\rho_A} I(A':B)_{\sigma}, ]
where (\lvert\phi\rangle_{A'A}) purifies (\rho_A) and
[ \sigma_{A'B}
(\operatorname{id}_{A'}\otimes\mathcal N) \bigl(\lvert\phi\rangle\langle\phi\rvert\bigr). ]
Shared entanglement does not itself transmit a message, but it changes the achievable rate when combined with a quantum channel. Superdense coding provides the elementary noiseless example of this relation.
Quantum communication
The quantum capacity (Q(\mathcal N)) is the highest asymptotic rate at which a channel can transmit unknown quantum states with vanishing error. Its principal information quantity is the coherent information,
[ I_c(\rho,\mathcal N)
S\bigl(\mathcal N(\rho)\bigr)
S\bigl((\operatorname{id}\otimes\mathcal N) (\lvert\psi\rangle\langle\psi\rvert)\bigr), ]
where (\lvert\psi\rangle) purifies (\rho). For a general channel, the capacity is given by a regularized optimization,
[ Q(\mathcal N)
\lim_{n\to\infty} \frac{1}{n} \max_{\rho^{(n)}} I_c!\left(\rho^{(n)},\mathcal N^{\otimes n}\right). ]
The expression lacks a general single-use reduction because coherent information is not additive for all channels. Certain channel classes have simpler formulas. For a degradable quantum channel, the environment’s output can be simulated from the receiver’s output, and the coherent-information optimization becomes single-letter.
The no-cloning theorem constrains quantum communication by forbidding the perfect copying of arbitrary unknown states. The theorem follows from linearity: a transformation that correctly copies two distinct nonorthogonal states cannot preserve their inner product. The related no-broadcasting theorem extends this limitation to mixed states and identifies commutativity as the condition under which a family of states can be broadcast.
Entanglement as information
Entanglement is treated as a resource because spatially separated parties cannot create it using only local operations and classical communication. For a bipartite pure state, the entropy of either reduced state gives the asymptotic rate of reversible conversion between copies of that state and maximally entangled pairs. Mixed states require distinct measures because their preparation cost and distillable yield need not coincide.
Quantum teleportation converts one shared entangled pair and two transmitted classical bits into the transfer of an unknown qubit state. Charles Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William Wootters derived the protocol as an application of entanglement combined with classical communication. The transmitted classical data reveal no complete description of the input, and the protocol does not produce an additional copy at the sender.
Entropic inequalities determine many restrictions on multipartite correlations. Strong subadditivity states that
[ S(AB)+S(BC)\geq S(B)+S(ABC). ]
The inequality implies the monotonicity of quantum relative entropy and supports the modern interpretation of quantum Markov structure. Equality characterizes states in which the correlations between (A) and (C) are mediated through a recoverable decomposition of (B).
Quantum error correction
Quantum error correction protects encoded information by distributing it across a larger Hilbert space. A code with projector (P) corrects an error set ({E_a}) precisely when the Knill–Laflamme conditions hold:
[ PE_a^\dagger E_bP=c_{ab}P. ]
The condition means that the environment acquires no information capable of distinguishing logical code states. Recovery therefore does not identify the encoded state; it identifies an error syndrome whose statistics are independent of the logical information.
Peter Shor constructed the first quantum code correcting an arbitrary single-qubit error, and Andrew Steane introduced a related code based on classical linear coding. Emanuel Knill and Raymond Laflamme subsequently gave the general algebraic criterion for exact correction. The stabilizer formalism, developed by Daniel Gottesman, organized broad families of codes through commuting subgroups of the Pauli group.
Quantum error correction connects operational communication rates with the structure of a noisy channel. Successful transmission is equivalent to approximately decoupling the encoded reference system from the channel environment. This decoupling viewpoint also links channel coding with privacy amplification and entanglement distillation.
Cryptographic applications
Quantum cryptography uses restrictions on state discrimination and copying to establish information-theoretic security. In quantum key distribution, measurement statistics are used to estimate the correlations between legitimate participants and any external system. A sufficiently low estimated disturbance permits classical error correction followed by privacy amplification, producing a shared key that is nearly independent of an adversary’s quantum state.
The BB84 protocol, introduced by Charles Bennett and Gilles Brassard, uses two incompatible bases for encoding classical bits. Artur Ekert later related key distribution to entanglement and the violation of a Bell inequality. Security proofs express secrecy through trace distance or conditional entropy rather than through the computational difficulty of reversing a classical function.
Relation to computation
Quantum computation can be treated as controlled information processing by quantum channels. The circuit model represents an algorithm as a sequence of local unitary gates followed by measurement. David Deutsch gave an early universal formulation, and Peter Shor demonstrated that quantum circuits can factor integers in polynomial time using the quantum Fourier transform.
Information-theoretic analysis separates computational speed from communication capacity. A quantum computer does not evade the Holevo bound by storing exponentially many amplitudes, because those amplitudes are not independently accessible through measurement. Its computational behavior instead depends on coherent transformations that cause amplitudes associated with different computational paths to interfere before the final measurement.