[Paper Review] Getting Information on Independently Prepared Quantum States -- When Are Individual Measurements as Powerful as Joint Measurements?

Chi‐Hang Fred Fung, H. F. Chau|ArXiv.org|Aug 20, 2008

Quantum Information and Cryptography3 citations

TL;DR

This paper proves that for quantum systems with independently prepared subsystems, individual measurements on each subsystem can extract at least as much Shannon mutual information as any joint measurement. The authors construct such individual measurements from a given joint POVM using trace operations and conditional probabilities, demonstrating that collective attacks in quantum key distribution are no more powerful than individual attacks for accessing the raw key.

ABSTRACT

Given a composite quantum system in which the states of the subsystems are independently (but not necessarily identically) prepared, we construct separate measurements on the subsystems from any given joint measurement such that the former always give at least as large information as the latter. This construction offers new insights into the understanding of measurements on this type of composite systems. Moreover, this construction essentially proves the intuition that separate measurements on the subsystems are sufficient to extract the maximal information about the separately prepared subsystems, thus making a joint measurement unnecessary. Furthermore, our result implies that individual attacks are as powerful as collective attacks in obtaining information on the raw key in quantum key distribution.

Motivation & Objective

• To determine whether individual measurements on independently prepared quantum subsystems can extract as much information as joint measurements.
• To construct explicit individual POVMs from a given joint POVM such that information gain is preserved or improved.
• To provide operational and theoretical justification for the intuition that separate measurements are sufficient for maximal information extraction.
• To clarify the implications for quantum key distribution, particularly regarding the equivalence of individual and collective attacks on the raw key.
• To offer a constructive alternative to existing proofs of accessible information additivity, yielding single measurements per subsystem rather than ensembles.

Proposed method

• For a two-subsystem system, derive the effective POVM on one subsystem by tracing out the other, conditioned on all possible states of the second subsystem.
• Use the joint POVM elements $M_b$ and the prior probabilities $p^{(k)}_{a_k}$ to compute the individual measurement operators via $M^{(k)}_{b_k} = \operatorname{Tr}_{\bar{k}}[(\phi^{(k)}_{a_k} \otimes \mathbb{I}) M_b] p^{(k)}_{a_k}$.
• Generalize the construction to $K > 2$ subsystems by iteratively applying the conditional measurement principle across all subsystems.
• Prove that the mutual information between input states and measurement outcomes is non-decreasing under this construction using the data processing inequality.
• Provide a second construction method based on projected measurements and conditional states, offering intuitive operational meaning.
• Use the Csiszár measure as a potential extension to other information measures, showing that similar constructions may be possible under convexity conditions.

Experimental results

Research questions

• RQ1 Can individual measurements extract at least as much information as joint measurements when subsystems are independently prepared?
• RQ2 Is there a constructive method to derive individual POVMs from a given joint POVM that preserve or improve information gain?
• RQ3 What is the operational meaning of the derived individual measurements in terms of conditional state projections?
• RQ4 How does this result affect the equivalence of individual and collective attacks in quantum key distribution?
• RQ5 Can this construction be generalized to other information measures beyond Shannon mutual information?

Key findings

• For independently prepared quantum subsystems, individual measurements can extract at least as much Shannon mutual information as any joint measurement.
• The constructed individual POVMs are explicitly derived from the joint POVM using partial traces and prior probabilities, ensuring no loss of information.
• The construction works for arbitrary Hilbert space dimensions, any number of subsystems, and non-identical input state distributions.
• The result implies that in quantum key distribution protocols like BB84, individual attacks are as powerful as collective attacks in accessing the raw key.
• The second construction method provides an intuitive explanation: knowing the state of other subsystems projects the measurement on a given subsystem into a conditional POVM.
• The method avoids ensemble-based measurements, yielding a single individual measurement per subsystem, unlike previous approaches based on Wootters’ proof.