[Paper Review] Error-disturbance relations for finite dimensional systems

Asger C. Ipsen|arXiv (Cornell University)|Nov 1, 2013

Quantum Mechanics and Applications16 references3 citations

TL;DR

This paper proposes operational error-disturbance relations for finite-dimensional quantum systems using calibration and variation error metrics to quantify the trade-off between measurement accuracy and disturbance. For qubits, it derives tight inequalities showing that joint measurement of non-commuting observables is fundamentally limited by quantum uncertainty, with explicit bounds depending on the overlap between eigenbases.

ABSTRACT

We propose an error-disturbance relation for general observables on finite dimensional Hilbert spaces based on operational notions of error and disturbance. For two-dimensional systems we derive tight inequalities expressing the trade-off between accuracy and disturbance.

Motivation & Objective

• To formulate error-disturbance relations for finite-dimensional quantum systems based on operational definitions of error and disturbance.
• To address the lack of consensus in quantifying disturbance when measuring non-commuting observables, particularly in the context of joint measurements.
• To derive tight, quantitative trade-off inequalities between measurement error and disturbance for two-dimensional systems (qubits).
• To compare two distinct error metrics—calibration error and variation error—on their implications for optimal joint measurement schemes.
• To establish bounds on the joint measurability of observables that depend on the overlap between their eigenbases.

Proposed method

• Defines error and disturbance using two metrics: calibration error $ d_c(M,P) = \sup_a (1 - \tr(M_a P_a)) $, and variation error $ d_v(M,P) = \sup_\rho \frac{1}{2} \sum_a |\tr(M_a\rho) - \tr(P_a\rho)| $.
• Models joint measurements via Positive Operator-Valued Measures (POVMs) $ F_{ab} $, with marginal POVMs $ M_a = \sum_b F_{ab} $ and $ \bar{M}_a = \sum_b F_{ba} $.
• Uses the operator norm equivalence $ d_v(M,P) = \sup_{X \in \mathcal{F}} \| M(X) - P(X) \| $ to express variation error in terms of operator norms.
• Derives bounds on matrix elements of $ \bar{M}_1 $ using Cauchy-Schwarz and positivity constraints on operators, leading to inequalities involving $ \epsilon = d_\beta(M,P) $ and $ \bar{\epsilon} = d_\beta(\bar{M},\bar{P}) $.
• Constructs a function $ I(\epsilon, \bar{\epsilon}; t, t') $ that must be non-negative for valid joint measurements, with $ t = |\langle \bar{1}|1\rangle| $, to derive constraints on $ \epsilon $ and $ \bar{\epsilon} $.
• Combines bounds across all basis vector pairs to define feasible regions $ H $ and $ H' $, leading to a lower bound on the sum $ \epsilon + \bar{\epsilon} $.

Experimental results

Research questions

• RQ1 What is the tightest possible trade-off between measurement error and disturbance for two non-commuting observables in a finite-dimensional Hilbert space?
• RQ2 How do different operational definitions of error—calibration vs. variation—lead to different optimal joint measurement strategies?
• RQ3 Can explicit bounds on joint measurability be derived that depend on the overlap between the eigenbases of the observables?
• RQ4 How does the presence of mutual unbiased bases affect the error-disturbance trade-off?
• RQ5 Is it possible to derive a general error-disturbance relation that remains valid even with post-measurement error correction?

Key findings

• For qubits, the paper derives tight error-disturbance inequalities that quantify the fundamental trade-off between accuracy and disturbance in joint measurements of non-commuting observables.
• The calibration error $ d_c $ and variation error $ d_v $ lead to inequivalent optimal joint measurement schemes, indicating that the choice of error metric affects the achievable trade-off.
• A general bound is derived in the form $ I(\epsilon, \bar{\epsilon}; t, t') \geq 0 $, where $ t = |\langle \bar{1}|1\rangle| $, which constrains the possible values of error and disturbance.
• The bound is non-trivial as long as the overlap $ |\langle \bar{1}|1\rangle||\langle \bar{1}|2\rangle| > 0 $, showing that disturbance cannot be eliminated when observables are not perfectly aligned.
• The sum $ d_c(M,P) + d_c(\bar{M},\bar{P}) $ is bounded from below by the infimum of $ \epsilon + \bar{\epsilon} $ over the intersection of feasible regions $ H \cap H' $, providing a global lower bound on total error.
• The derived relations remain valid even when error correction is applied after measurement, indicating robustness to post-processing.