Experimental qudit state estimation via point tomography

Point tomography is an approach to the problem of state estimation, which is arguably the most efficient and simple method for modern high-precision quantum information experiments. In this scenario, the experimenter knows the target state that their device should prepare, except that intrinsic systematic errors will create small discrepancies in the state actually produced. By introducing a kind of informationally complete measurement, dubbed Fisher-symmetric measurements, point tomography determines deviations from the expected state with optimal efficiency. Moreover, the number of outcomes of a measurement saturating the Gill-Massar limit for reconstructing <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>d</a:mi></a:math>-dimensional quantum states can be greatly reduced from <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mo>∼</b:mo><b:mn>4</b:mn><b:mi>d</b:mi><b:mo>−</b:mo><b:mn>3</b:mn></b:mrow></b:math> to only <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mn>2</c:mn><c:mi>d</c:mi><c:mo>−</c:mo><c:mn>1</c:mn></c:mrow></c:math> outcomes. Thus, improving the measurement overhead as the dimension increases. Here we demonstrate the experimental viability of point tomography. Using a modern photonic platform constructed with state-of-the-art multicore optical fiber technology, we generate four-dimensional quantum states and implement seven-outcome Fisher-symmetric measurements. Our experimental results exhibit the main feature of point tomography, namely, a precision close to the Gill-Massar limit with a single few-outcome measurement. Specifically, we achieved a precision of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mo>(</d:mo><d:mn>3.81</d:mn><d:mo>±</d:mo><d:mn>0.05</d:mn><d:mo>)</d:mo><d:mo>/</d:mo><d:mi>N</d:mi></d:mrow></d:math>, while the Gill-Massar limit for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>d</e:mi><e:mo>=</e:mo><e:mn>4</e:mn></e:mrow></e:math> is <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mn>3</f:mn><f:mo>/</f:mo><f:mi>N</f:mi></f:mrow></f:math> (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>N</g:mi></g:math> being the ensemble size).