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Maximally non-projective measurements are not always symmetric informationally complete

Gabriele Cobucci, Raphael Brinster, Shishir Khandelwal, Hermann Kampermann, Dagmar Bruß, Nikolai Wyderka, Armin Tavakoli

Abstract

Standard quantum measurements are projective. However, the full scope of quantum measurements is represented by positive operator-valued measures (POVMs) and many of these break the limitations of projective measurements as resources in quantum information. It is therefore natural to consider how accurately an experimenter with access only to projective measurements and classical processing can simulate POVMs. The most well-known class of non-projective measurements is called symmetric informationally complete (SIC). Such measurements are both ubiquitous in the broader scope of quantum information theory and known to be the most strongly non-projective measurements in qubit systems. Here, we show that beyond qubit systems, the SIC property is in general not associated with the most non-projective measurement. For this, we put forward a semidefinite programming criterion for detecting genuinely non-projective measurements. This method allows us to determine quantitative simulability thresholds for generic POVMs and to put forward a conjecture on which qutrit and ququart measurements that are most strongly non-projective.

Maximally non-projective measurements are not always symmetric informationally complete

Abstract

Standard quantum measurements are projective. However, the full scope of quantum measurements is represented by positive operator-valued measures (POVMs) and many of these break the limitations of projective measurements as resources in quantum information. It is therefore natural to consider how accurately an experimenter with access only to projective measurements and classical processing can simulate POVMs. The most well-known class of non-projective measurements is called symmetric informationally complete (SIC). Such measurements are both ubiquitous in the broader scope of quantum information theory and known to be the most strongly non-projective measurements in qubit systems. Here, we show that beyond qubit systems, the SIC property is in general not associated with the most non-projective measurement. For this, we put forward a semidefinite programming criterion for detecting genuinely non-projective measurements. This method allows us to determine quantitative simulability thresholds for generic POVMs and to put forward a conjecture on which qutrit and ququart measurements that are most strongly non-projective.

Paper Structure

This paper contains 9 sections, 1 theorem, 58 equations, 2 figures, 2 tables.

Table of Contents

  1. Redundancy of post-processing in projective simulation
  2. Dual SDP
  3. Witness bounds
  4. Projective simulation of noisy Hesse SIC-POVM
  5. Simulable decompositions and generalisations of flagged SIC-POVMs
  6. Flagged SIC2
  7. Flagged Hesse SIC
  8. Worst case noise
  9. Result tables

Key Result

Lemma 1

Let $\mathbf{E} = \{E_a\}_{a=1}^n$ be a $d$-dimensional POVM and $\mathbf{E}_f = \{E^a_f\}_{a=1}^{n+1}$ the $(d+1)$-dimensional flagged POVM with effects where $0_d$ denotes the $d\times d$-dimensional zero matrix. Then $\tilde{v}^*(\mathbf{E}_f) \geq \tilde{v}^*(\mathbf{E})$.

Figures (2)

  • Figure 1: The projective simulability threshold visibility $v^*(\mathbf{E}_\theta)$ defined in Eq. \ref{['eq:vstar']}, for projective simulation of depolarised qutrit SIC-POVMs constructed from fiducial vector $|\varphi^{(\theta)}\rangle$ in Eq. \ref{['eq:sic3fiducial']}.
  • Figure 2: The threshold visibility $v^*$ from Eq. \ref{['eq:vstar']} obtained for various POVMS in $2\leq d\leq 6$. Black dots represent SIC-POVMs, while green ones fSIC-POVMs (the number in the flag corresponds to the dimension of the embedded SIC-POVM), for which the visibility thresholds are calculated in supplemental_material. The blue lines represent upper bounds on the visibility obtained for SIC-POVMs in $d=5,6$ using the witness method, while the red line denotes a lower bound in $d=5$ found by explicit construction.

Theorems & Definitions (2)

  • Lemma 1
  • proof