Exchange-symmetrized qudit Bell bases and Bell-state distinguishability

Entanglement of qudit pairs, with single-particle Hilbert space dimension <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>d</a:mi></a:math>, has important potential for quantum information processing, with applications in cryptography, algorithms, and error correction. For a pair of qudits of arbitrary even dimension <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>d</b:mi></b:math>, we introduce a generalized Bell basis with definite symmetry under exchange of internal states between the two particles. We show that no complete exchange-symmetrized basis can exist for odd <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>d</c:mi></c:math>. This framework extends prior work on exchange-symmetrized hyperentangled qubit bases, where <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>d</d:mi></d:math> is a power of 2. For our exchange-symmetrized basis, we show that measurement devices restricted to linear evolution and local measurement (LELM) can unambiguously distinguish <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mn>2</e:mn><e:mi>d</e:mi><e:mo>−</e:mo><e:mn>1</e:mn></e:mrow></e:math> qudit Bell states for any even <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>d</f:mi></f:math>. This achieves the upper bound in general for reliable Bell-state distinguishability via LELM and augments previously known limits for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>d</g:mi><g:mo>=</g:mo><g:msup><g:mn>2</g:mn><g:mi>n</g:mi></g:msup></g:mrow></g:math> and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>d</h:mi><h:mo>=</h:mo><h:mn>3</h:mn></h:mrow></h:math>. This result is relevant to near-term realizations of quantum communication protocols.