Sharma, Sazim and Pati (EPL 127, 50004 (2019)) proved that the information gained by a measuring apparatus about a pure system state is upper bounded by the system's initial coherence and satisfies a web of trade-off relations with the apparatus's final coherence, initial mixedness, apparatus-environment entanglement, entropy exchange, and disturbance. All of these derivations, however, assume that the evolved apparatus states correlated with distinct system outcomes are mutually orthogonal, i.e. that the measurement records are perfectly distinguishable. Two issues are explicitly left open: (i) which of the bounds and trade-offs survive, and in what corrected form, for general non-orthogonal (finitely distinguishable) apparatus records; and (ii) under what conditions the final apparatus coherence and the final apparatus-environment entanglement can be made simultaneously small, as maximal information gain requires. Both issues remain open in the later literature. The problem asks for the completion of this coherence-based resource accounting of information gain in quantum measurement.

Background

A quantum measurement can be modeled at the reversible level as a unitary interaction that correlates a measured system with an apparatus. In the model introduced by V. Vedral [Physical Review Letters 90, 050401 (2003)] and adopted by Sharma-Sazim-Pati, the system SS (MM-dimensional) starts in a pure state ∣ΨS⟩=∑isi∣si⟩|\Psi_S\rangle = \sum_i s_i |s_i\rangle written in the eigenbasis of the measured observable, and the apparatus AA (NN-dimensional) starts in a generally mixed state ρA=∑iai∣ai⟩⟨ai∣\rho_A = \sum_i a_i |a_i\rangle\langle a_i|, with aia_i the eigenvalues and {∣ai⟩}\{|a_i\rangle\} the corresponding eigenbasis of ρA\rho_A (the basis in which apparatus coherences are evaluated). A unitary UU acting on S⊗AS \otimes A conditions the apparatus evolution on the system eigenstates, ∣sj⟩⊗∣ai⟩↦∣sj⟩⊗∣a~ij⟩|s_j\rangle \otimes |a_i\rangle \mapsto |s_j\rangle \otimes |\tilde a_{ij}\rangle, where ∣a~ij⟩|\tilde a_{ij}\rangle is the evolved apparatus (pointer) state correlated with outcome jj; after the interaction the apparatus carries a record of the outcome. Vedral showed that the information extractable from the apparatus equals the classical correlations built between system and apparatus and is bounded by their entanglement.

Several information-theoretic quantities quantify this process. The information gain ImI_m is the Holevo quantity of the ensemble of post-interaction apparatus states: for an ensemble {pi,ρi}\{p_i, \rho_i\} the Holevo quantity is χ=S(∑ipiρi)−∑ipiS(ρi)\chi = S(\sum_i p_i \rho_i) - \sum_i p_i S(\rho_i), the standard upper bound on the classical information accessible about which member of the ensemble was prepared (here, which outcome occurred), with SS the von Neumann entropy. Quantum coherence - the resource embodied in superposition relative to a fixed basis - is quantified by the relative entropy of coherence CR(ρ)=S(ρD)−S(ρ)C_R(\rho) = S(\rho_D) - S(\rho), where ρD\rho_D is ρ\rho with all off-diagonal entries removed in the chosen basis. The coherent information Ic(S′⟩A′)=S(ρA′)−S(ρS′A′)I_c(S' \rangle A') = S(\rho_{A'}) - S(\rho_{S'A'}) measures the net distinct quantum information transmitted from system to apparatus. When the initially mixed apparatus is purified by an environment EE through ∣ΨAE⟩=∑iai ∣ai⟩∣ei⟩|\Psi_{AE}\rangle = \sum_i \sqrt{a_i}\, |a_i\rangle |e_i\rangle, the entropy exchange SeS_e is the entropy the interaction generates in the environment (Se=S(ρA′E)=S(ρS′)S_e = S(\rho_{A'E}) = S(\rho_{S'})), and the residual apparatus-environment quantum correlation can be quantified by the relative entropy of entanglement ER(ρ)=min⁡σ separableS(ρ∥σ)E_R(\rho) = \min_{\sigma \,\mathrm{separable}} S(\rho \| \sigma). The disturbance caused to the apparatus by the interaction is D(ρA,Φ)=S(ρA)−Ic(ρA,Φ)D(\rho_A, \Phi) = S(\rho_A) - I_c(\rho_A, \Phi), where Φ\Phi is the completely positive trace-preserving map that the interaction induces on the apparatus.

Within this model, Sharma-Sazim-Pati [EPL 127, 50004 (2019)] proved, under the assumption that the evolved apparatus states correlated with distinct outcomes are mutually orthogonal, ⟨a~ij∣a~ik⟩=δjk\langle \tilde a_{ij} | \tilde a_{ik}\rangle = \delta_{jk} (perfectly distinguishable records), that the information gain equals the coherent information, Im=Ic(S′⟩A′)I_m = I_c(S' \rangle A'), that the initial system coherence caps the gain, CR(∣ΨS⟩⟨ΨS∣)≥ImC_R(|\Psi_S\rangle\langle\Psi_S|) \ge I_m, and the trade-offs

Im+CR(ρA′)+S(ρA)≤log⁡N,Im≤Se,Im+CR(ρS′)≤log⁡M,I_m + C_R(\rho_{A'}) + S(\rho_A) \le \log N, \qquad I_m \le S_e, \qquad I_m + C_R(\rho_{S'}) \le \log M,

ER(ρA′E)+Im+CR(ρA′)≤log⁡N,D(ρA,Φ)+Im+CR(ρA′)≤2log⁡N,E_R(\rho_{A'E}) + I_m + C_R(\rho_{A'}) \le \log N, \qquad D(\rho_A, \Phi) + I_m + C_R(\rho_{A'}) \le 2\log N,

with MM and NN the system and apparatus dimensions, following the source's conventions (the source attributes the weaker bound Im+S(ρA)≤log⁡NI_m + S(\rho_A) \le \log N to Vedral's original analysis). The last two relations say that a large information gain forces the final apparatus coherence and the final apparatus-environment entanglement to be jointly small; the paper obtains only partial intuition for this from complementarity relations between coherence and entanglement, notably the Xi-Li-Fan relation CR(ρA)+E(ρAE)≤log⁡NC_R(\rho_A) + E(\rho_{AE}) \le \log N for pure bipartite states [Scientific Reports 5, 10922 (2015)] and earlier coherence-disturbance trade-offs [Sharma-Pati, Physical Review A 97, 062308 (2018)].

The orthogonality assumption is an idealization: real apparatuses leave only finitely distinguishable records. The authors state that the analysis "can be extended to more general measurement scenario in which the evolved apparatus states are not strictly orthogonal to each other", and that it is "prima facie" not clear "whether at the same time the final coherence of the apparatus and the final entanglement between apparatus and environment will be small". Both extensions are left open.

Problem Statement

Consider the measurement model of Sharma-Sazim-Pati: a pure MM-dimensional system state ∣ΨS⟩=∑isi∣si⟩|\Psi_S\rangle = \sum_i s_i |s_i\rangle; an initially mixed NN-dimensional apparatus ρA=∑iai∣ai⟩⟨ai∣\rho_A = \sum_i a_i |a_i\rangle\langle a_i| with {∣ai⟩}\{|a_i\rangle\} the eigenbasis of ρA\rho_A, optionally purified by an environment EE via ∣ΨAE⟩=∑iai ∣ai⟩∣ei⟩|\Psi_{AE}\rangle = \sum_i \sqrt{a_i}\, |a_i\rangle |e_i\rangle; and a unitary UU acting on S⊗AS \otimes A only, implementing ∣sj⟩⊗∣ai⟩↦∣sj⟩⊗∣a~ij⟩|s_j\rangle \otimes |a_i\rangle \mapsto |s_j\rangle \otimes |\tilde a_{ij}\rangle (primes on S′S' and A′A' below denote the states after the interaction). Let ImI_m denote the information gain (the Holevo quantity of the post-interaction apparatus ensemble), CRC_R the relative entropy of coherence - in the eigenbasis {∣si⟩}\{|s_i\rangle\} of the measured observable for system states and the eigenbasis {∣ai⟩}\{|a_i\rangle\} for apparatus states, following the source's convention of fixing a basis for the apparatus - Ic(S′⟩A′)=S(ρA′)−S(ρS′A′)I_c(S' \rangle A') = S(\rho_{A'}) - S(\rho_{S'A'}) the coherent information, SeS_e the entropy exchange, ERE_R the relative entropy of entanglement, and D(ρA,Φ)=S(ρA)−Ic(ρA,Φ)D(\rho_A, \Phi) = S(\rho_A) - I_c(\rho_A, \Phi) the apparatus disturbance. The published results - the identity Im=Ic(S′⟩A′)I_m = I_c(S' \rangle A') and the inequalities CR(∣ΨS⟩⟨ΨS∣)≥ImC_R(|\Psi_S\rangle\langle\Psi_S|) \ge I_m, Im+CR(ρA′)+S(ρA)≤log⁡NI_m + C_R(\rho_{A'}) + S(\rho_A) \le \log N, Im≤SeI_m \le S_e, Im+CR(ρS′)≤log⁡MI_m + C_R(\rho_{S'}) \le \log M, ER(ρA′E)+Im+CR(ρA′)≤log⁡NE_R(\rho_{A'E}) + I_m + C_R(\rho_{A'}) \le \log N, and D(ρA,Φ)+Im+CR(ρA′)≤2log⁡ND(\rho_A, \Phi) + I_m + C_R(\rho_{A'}) \le 2\log N - are all derived under the orthogonality assumption ⟨a~ij∣a~ik⟩=δjk\langle \tilde a_{ij} | \tilde a_{ik}\rangle = \delta_{jk}.

The problem - one objective, namely completing the coherence-based resource accounting of information gain in quantum measurement without the perfectly-distinguishable-record idealization - has two linked parts, both of which are required deliverables.

(1) Non-orthogonal records. Remove the orthogonality assumption: for arbitrary evolved apparatus states, determine which of the above relations survive and in what form, and establish the correct generalized bounds relating ImI_m, the initial and final apparatus coherence, the apparatus mixedness, the entropy exchange, the apparatus-environment entanglement, and the apparatus disturbance, including how the record overlaps ⟨a~ij∣a~ik⟩\langle \tilde a_{ij}|\tilde a_{ik}\rangle enter.

(2) Joint coherence-entanglement behavior. Characterize when the final apparatus coherence CR(ρA′)C_R(\rho_{A'}) and the final apparatus-environment entanglement ER(ρA′E)E_R(\rho_{A'E}) can both be made arbitrarily small (driven toward zero) - the regime required for maximal information gain.

References

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