bibkey: zhouwuchau2016errordisturbance
authors: S.S. Zhou; Shengjun Wu; H.F. Chau
year: 2016
title: "State-independent error-disturbance trade-off for measurement operators"
doi: 10.1016/j.physleta.2016.03.046
url: https://arxiv.org/abs/1509.03812v1
claim: "For rank-one projective measurements A, A' and B on C^d, the state-independent error of replacing A by A', the disturbance that A' causes to B, and their overall error Delta(A,A',B) are maxima over states of infinity-norm distances between outcome distributions. The paper proves trade-offs for qubits and for mutually unbiased A and B, and conjectures, with the error between A and B relaxed to ignore outcome labels, that eps(A,A')+eta(A',B) and Delta(A,A',B) are both at least min{eps(A,B), eta(A,B)} in every dimension."
strata_touched:
  - D5/S3/Quantum/Measurement/ErrorDisturbanceEndpointRefutation
license: citation-only
triage: anchor

Zhou, Wu and Chau, State-independent error-disturbance trade-off for measurement operators

S. S. Zhou, S. Wu and H. F. Chau, arXiv:1509.03812v1 [quant-ph]; Physics Letters A 380(22–23), 1918–1924 (2016). Quotations are from the v1 TeX source, with the source’s macros \ket, \bra and \braket kept as written.

Section II measures the deviation between outcome distributions by $D(x,y)={\max }_i|x_i-y_i|$ and defines, for a state $\rho$ and the state ${\rho }^{\prime }={\sum }_i⟨a_i^{\prime }|\rho |a_i^{\prime }⟩|a_i^{\prime }⟩⟨a_i^{\prime }|$ after the measurement ${\mathcal{A}}^{\prime }$, $${\epsilon }_{\rho }(\mathcal{A},{\mathcal{A}}^{\prime })=D(P_{\mathcal{A}}(\rho ),P_{{\mathcal{A}}^{\prime }}(\rho )),{\eta }_{\rho }({\mathcal{A}}^{\prime },\mathcal{B})=D(P_{\mathcal{B}}(\rho ),P_{\mathcal{B}}({\rho }^{\prime })),$$ the state-independent error and disturbance $\epsilon (\mathcal{A},{\mathcal{A}}^{\prime })={\max }_{\rho }{\epsilon }_{\rho }$ and $\eta ({\mathcal{A}}^{\prime },\mathcal{B})={\max }_{\rho }{\eta }_{\rho }$, and

the \textbf{state-independent overall error} $\Delta (\mathcal{A},{\mathcal{A}}^{\prime },\mathcal{B})={\max }_{\rho }\left({\epsilon }_{\rho }(\mathcal{A},{\mathcal{A}}^{\prime })+{\eta }_{\rho }({\mathcal{A}}^{\prime },\mathcal{B})\right).$

Before the conjecture the error between $\mathcal{A}$ and $\mathcal{B}$ is relaxed:

ε(A,B)=minB​ε(A,B), where the minimum is over all possible projective measurements with rank one projectors B~ corresponding to the measurement operation B.

Conjecture 3 (label thm:conj):

For all $d\ge 2$, the following trade-off between error and disturbance holds: $\epsilon (\mathcal{A},{\mathcal{A}}^{\prime })+\eta ({\mathcal{A}}^{\prime },\mathcal{B})\ge f(\mathcal{A},\mathcal{B})$ and $\Delta (\mathcal{A},{\mathcal{A}}^{\prime },\mathcal{B})\ge f(\mathcal{A},\mathcal{B})$, where $f(\mathcal{A},\mathcal{B})=\min \{\epsilon (\mathcal{A},\mathcal{B}),\eta (\mathcal{A},\mathcal{B})\}$. Moreover, the equalities are both attained either when for all $i$, $|a_i^{\prime }⟩=|a_i⟩$, or when for all $i$, $|a_i^{\prime }⟩=|b_i⟩$.

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Source: https://arxiv.org/abs/1509.03812v1

• DOI: https://doi.org/10.1016/j.physleta.2016.03.046 (Physics Letters A 380(22–23), 1918–1924, May 2016; title, authors, journal, volume, issue and pages from Crossref).
• URL: https://arxiv.org/abs/1509.03812v1. The e-print is a gzipped tar archive; its member new_912arxiv.tex has 971 lines and md5 224f302edfcd0f7e887b2c9695c86591. Locations:
  • the post-measurement state, l. 175–176;
  • the state-dependent error and disturbance, l. 185 and l. 190; the infinity-norm distance, l. 196;
  • the state-independent error, disturbance and overall error, l. 203, l. 208 and l. 220;
  • Theorem 1 (qubits), l. 457; Theorem 2 (mutually unbiased bases), l. 552;
  • the relaxed error, l. 614–620; Conjecture 3, l. 623–639.
• The journal text was not read; that it states the same conjecture is not verified here.