| Holevo quantity | |
|---|---|
| Name | Holevo quantity |
| Unit | bit (information) |
| Field | Quantum information theory |
| Introduced by | Alexander Holevo |
| Year | 1973 |
Holevo quantity
The Holevo quantity is an information-theoretic functional that upper-bounds the amount of classical information extractable from a quantum ensemble by any measurement. It plays a central role in quantum information theory and the study of quantum channel capacity, giving a bridge between quantum states and classical information limits. The quantity is important for protocols in quantum communication and for understanding limits imposed by quantum mechanics on classical information transmission.
Given an ensemble {p_i, ρ_i} of quantum states ρ_i with prior probabilities p_i on a finite-dimensional Hilbert space H, the Holevo quantity χ is defined as χ({p_i,ρ_i}) = S(ρ) − Σ_i p_i S(ρ_i), where ρ = Σ_i p_i ρ_i is the ensemble average (the average density operator) and S(·) denotes the von Neumann entropy S(σ) = −Tr(σ log σ). The definition parallels the classical mutual information but substitutes quantum entropy for Shannon entropy, linking to the notion of ensemble distinguishability. The first occurrence of this bound was given by Alexander Holevo in 1973 in his foundational paper on accessible information and measurement limits. The Holevo quantity is sometimes called the Holevo χ quantity or Holevo bound.
Operationally, χ upper-bounds the accessible information I_acc of the ensemble, defined as the maximum classical mutual information between the ensemble label and the outcome of any quantum measurement (a positive operator-valued measure, POVM). Formally, I_acc({p_i,ρ_i}) ≤ χ({p_i,ρ_i}). This inequality expresses that even optimal measurements performed by parties such as in quantum cryptography or classical communication over quantum channels cannot extract more classical bits than χ permits. The Holevo quantity thereby constrains tasks like state discrimination, classical encoding into quantum states (ensemble encoding), and information extraction in protocols studied at institutions such as IBM Research and Bell Labs.
Holevo's theorem states the upper bound I_acc ≤ χ. Important properties include: - Nonnegativity: χ ≥ 0 with equality when all ρ_i are identical. - Upper bound by log d: χ ≤ log d for states on a d-dimensional Hilbert space, reaching equality for ensembles of orthogonal pure states. - Subadditivity and concavity relations derived from properties of von Neumann entropy and strong subadditivity, the latter proven by Elliott Lieb and Mary Beth Ruskai and related to proofs by Lieb and Ruskai. - Additivity issues: χ is central in additivity conjectures for capacities of quantum channels such as the Holevo capacity; additivity questions were historically linked to counterexamples by Matthew Hastings. These mathematical constraints connect χ to inequalities like Araki–Lieb inequality and to channel capacities formalized by Claude Shannon-style coding theorems generalized to quantum settings by researchers including Bennett and Shor.
For simple ensembles, χ can be computed analytically: - Binary pure-state ensemble: For two pure states |ψ0⟩, |ψ1⟩ with prior probabilities p and 1−p, χ reduces to the difference between the entropy of the mixed state and zero (pure states have zero von Neumann entropy), yielding χ = S(p|ψ0⟩⟨ψ0| + (1−p)|ψ1⟩⟨ψ1|). - Orthogonal ensemble: For orthogonal pure states with uniform probabilities, χ = log n where n is the number of states. - Depolarized ensembles: For ensembles subject to a depolarizing channel, analytic expressions combine channel parameters and eigenvalues of the average state. Numerical computation typically requires diagonalization of ρ and evaluation of eigenvalue spectra; software libraries used in practice include QuTiP and numerical frameworks developed at Los Alamos National Laboratory and university groups. For large ensembles or high-dimensional systems, convex optimization and semidefinite programming methods are employed.
The Holevo quantity determines the one-shot upper bound for classical information transmitted by quantum states and appears in the expression for the classical capacity of a quantum channel: the Holevo capacity χ(Φ) = max_{ensembles} χ({p_i, Φ(ρ_i)}) for channel Φ. The regularized classical capacity involves limits over multiple channel uses and is connected to additivity questions resolved in part by the work of Hastings and further developed by Peter Shor, John Preskill, and Charles Bennett. In practical quantum communication systems—such as optical quantum key distribution experiments at institutions like NIST and implementations by companies pursuing quantum key distribution—χ guides coding strategies, modulation of quantum states, and trade-offs between encoding complexity and reliable transmission rates.
Related measures include accessible information I_acc, quantum mutual information I(A:B) = S(ρ_A) + S(ρ_B) − S(ρ_AB) for bipartite states, and coherent information used for quantum capacity. The Holevo quantity bounds classical information extractable from ensembles, while quantum mutual information quantifies total correlations between subsystems; both are expressible in terms of von Neumann entropy and appear in proofs of capacities and privacy bounds by Devetak and Winter. Other extensions include the entanglement-assisted classical capacity (the Bennett–Shor–Smolin–Thapliyal theorem), which involves quantum mutual information, and measures of accessible information under restricted measurements (local operations and classical communication, LOCC). Studies in resource theories and operational tasks such as state discrimination, hypothesis testing, and entanglement distillation often refer to χ as a central limiting quantity.
Category:Quantum information theory Category:Entropy