Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank

• Main Authors: Metod Saniga, Henri de Boutray, Frédéric Holweck, Alain Giorgetti
• Format: Article
• Language: English
• Published: MDPI AG 2021-09-01
• Series: Mathematics
• Subjects: N-qubit observables; binary symplectic polar spaces; distinguished sets of doilies; geometric hyperplanes; Veldkamp lines
• Online Access: https://www.mdpi.com/2227-7390/9/18/2272

We study certain physically-relevant subgeometries of binary symplectic polar spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>2</mn><mi>N</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> of small rank <i>N</i>, when the points of these spaces canonically encode <i>N</i>-qubit observables. Key characteristics of a subspace of such a space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>2</mn><mi>N</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> are: the number of its negative lines, the distribution of types of observables, the character of the geometric hyperplane the subspace shares with the distinguished (non-singular) quadric of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>2</mn><mi>N</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> and the structure of its Veldkamp space. In particular, we classify and count polar subspaces of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>2</mn><mi>N</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> whose rank is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> features three negative lines of the same type and its <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>’s are of five different types. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> is endowed with 90 negative lines of two types and its <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>’s split into 13 types. A total of 279 out of 480 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>’s with three negative lines are composite, i.e., they all originate from the two-qubit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. Given a three-qubit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> and any of its geometric hyperplanes, there are three other <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>’s possessing the same hyperplane. The same holds if a geometric hyperplane is replaced by a ‘planar’ tricentric triad. A hyperbolic quadric of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> is found to host particular sets of seven <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>’s, each of them being uniquely tied to a Conwell heptad with respect to the quadric. There is also a particular type of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>’s, a representative of which features a point each line through which is negative. Finally, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>7</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> is found to possess 1908 negative lines of five types and its <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>’s fall into as many as 29 types. A total of 1524 out of 1560 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>’s with 90 negative lines originate from the three-qubit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. Remarkably, the difference in the number of negative lines for any two distinct types of four-qubit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>’s is a multiple of four.