# Lower and upper bounds¶

Oriented Bell expressions are described by a Bell expression along with a direction <= or >=. Data about an oriented Bell expression is written using a BellExpression along with additional data in the special properties lower or upper, describing knowledge about the expr >= bound and expr <= bound directions respectively.

Each direction lower and upper can have the following properties:

bounds

Bounds corresponding to different sets of interest can be listed here through a mapping list. The mapping keys local, quantum and no-signaling are reserved for the usual accordingly named sets. The mapping values are given by a string expression made of:

• integers,
• infinity written inf,
• rational numbers written down using the format numerator/denominator,
• decimal numbers written using digits.digits,
• arithmetic operators +, -, *, /,
• parenthesis,
• intervals written down as [lb, ub], where lb, ub are expressions.

Here is an example of section specifying several bounds:

bounds:
local: 2
quantum: [-inf, 2.828427124746191]
nosignaling: 4


Here, the [-inf, 2.828427124746191] interval expression for the quantum bound interval signifies that we have obtained an approximation for the quantum bound through relaxations, so that the real upper bound is below 2.828427124746191. The -inf side of the interval could be replaced by finding an explicit state and measurements that achieve a quantum value.

keywords

Keywords can also be associated to an oriented Bell expression. For now, no keywords are reserved, nor is any keywords automatically computed for oriented expressions. Keywords here respect the same syntax as the ones for Bell expressions mentioned above.

Special keywords specify if a bound corresponds to a facet of some polytope. The keywords facet-polytope and not-facet-polytope can be used for that effect, where polytope corresponds to the name of a bound. The keywords (not-)facet-local and (not-)facet-no-signaling are reserved. If the facet-defining property is not known, the corresponding keyword is not present.

The following is a valid keywords section:

keywords: [facet-local]


Two orientations of a Bell expression can be specified in a single file by incorporating both a lower and an upper section in the file. This can provide a full decription of the bounds satisfied by a given Bell expression.