Universal adjointation of isometry operations using conversion of quantum supermaps

Identification of possible transformations of quantum objects including quantum states and quantum operations is indispensable in developing quantum algorithms. Universal transformations, defined as input-independent transformations, appear in various quantum applications. Such is the case for universal transformations of unitary operations. However, extending these transformations to non-unitary operations is nontrivial and largely unresolved. Addressing this, we introduce <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext class="MJX-tex-mathit" mathvariant="italic">isometry adjointation</mml:mtext></mml:mrow></mml:math> protocols that transform an input isometry operation into its adjoint operation, which include both unitary operation and quantum state transformations. The paper details the construction of parallel and sequential isometry adjointation protocols, derived from unitary inversion protocols using quantum combs and the (dual) Clebsch-Gordan transforms, and achieving optimal approximation error. This error is shown to be independent of the output dimension of the isometry operation. In particular, we explicitly obtain an asymptotically optimal parallel protocol achieving an approximation error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03F5;</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="normal">&amp;#x0398;</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math> is the input dimension of the isometry operation and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> is the number of calls of the isometry operation. The research also extends to isometry inversion and universal error detection, employing semidefinite programming to assess optimal performances. The findings suggest that the optimal performance of general protocols in isometry adjointation and universal error detection is not dependent on the output dimension, and that indefinite causal order protocols offer advantages over sequential ones in isometry inversion and universal error detection.