Bipartite quantum measurements with optimal single-sided distinguishability
Jakub Czartowski, Karol Życzkowski
Abstract
We analyse orthogonal bases in a composite <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>×</mml:mo><mml:mi>N</mml:mi></mml:math> Hilbert space describing a bipartite quantum system and look for a basis with optimal single-sided mutual state distinguishability. This condition implies that in each subsystem the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>N</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> reduced states form a regular simplex of a maximal edge length, defined with respect to the trace distance. In the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> of a two-qubit system our solution coincides with the elegant joint measurement introduced by Gisin. We derive explicit expressions of an analogous constellation for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math> and provide a general construction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>N</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> states forming such an optimal basis in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msub><mml:mo>⊗</mml:mo><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msub></mml:math>. Our construction is valid for all dimensions for which a symmetric informationally complete (SIC) generalized measurement is known. Furthermore, we show that the one-party measurement that distinguishes the states of an optimal basis of the composite system leads to a local quantum state tomography with a linear reconstruction formula. Finally, we test the introduced tomographical scheme on a complete set of three mutually unbiased bases for a single qubit using two different IBM machines.