An operator-based bound on information and disturbance in quantum measurements

Quantum measurements can be described by operators that assign conditional probabilities to different outcomes while also describing unavoidable physical changes to the system. Here, we point out that operators describing information gain at minimal disturbance can be expanded into a set of unitary operators representing experimentally distinguishable patterns of disturbance. The observable statistics of disturbance defines a tight upper bound on the information gain of the measurement.

💡 Research Summary

The paper investigates the fundamental trade‑off between information gain and disturbance in quantum measurements by focusing on the operator description of the measurement process rather than on information‑theoretic quantities alone. Starting from the Kraus representation of a measurement, the authors restrict themselves to minimally disturbing measurements, which can be described solely by self‑adjoint (Hermitian) operators ˆMₘ that are diagonal in the eigenbasis {|a⟩} of the observable being measured. The diagonal elements of ˆMₘ are the square roots of the conditional probabilities p(m|a) that quantify how the measurement outcome m depends on the input eigenstate a.

The central technical step is to expand each ˆMₘ in a basis of orthogonal unitary operators drawn from the Heisenberg‑Weyl group. In a d‑dimensional Hilbert space the unitary set {ˆU(k)}ₖ₌₀^{d‑1}, defined by ˆU(k)=∑ₐ e^{i2πka/d}|a⟩⟨a|, forms a complete orthogonal basis under the Hilbert‑Schmidt inner product. Consequently, any Hermitian measurement operator can be written as ˆMₘ = (1/d)∑ₖ Cₖ ˆU(k), where the complex coefficients Cₖ are the discrete Fourier transform of the square roots of the conditional probabilities: Cₖ = (1/d)∑ₐ e^{-i2πka/d}√p(m|a). This representation casts the measurement as a coherent superposition of experimentally distinguishable unitary “disturbance patterns”.

To make the disturbance observable, the authors introduce a complementary basis {|b⟩} related to {|a⟩} by a discrete Fourier transform. In this basis each unitary ˆU(k) simply shifts the index: ˆU(k)|b⟩ = |b+k⟩ (mod d). Acting with ˆMₘ on an input state |b⟩ yields ˆMₘ|b⟩ = ∑ₖ Cₖ |b+k⟩, and the joint probability of obtaining outcome m together with the shifted basis state |b+k⟩ is p(m,b+k|b) = |Cₖ|². Thus the absolute values |Cₖ| are directly accessible experimentally by measuring the distribution of outcomes in the complementary basis; they encode the pattern of disturbance induced by the measurement.

The information side is treated via Bayesian updating. With a uniform prior over the eigenstates, the posterior probability p(a|m) = p(m|a)/∑_{a’}p(m|a’) quantifies how well the measurement outcome identifies the input a. The authors show that, given only the observable magnitudes |Cₖ|, the maximal possible value of any conditional probability p(m|a) is bounded by (1/d)∑ₖ|Cₖ|, which is achieved when all phases of the Cₖ align. Translating this bound to the posterior yields the central result:

p(a|m) ≤ (1/d) ∑ₖ