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Communication channels

According to the quantum theory, quantum objects manifest themselves via their influence on classical objects (more exactly, on classically described degrees of freedom). Every object admits a quantum description, but some objects may be described classically for all practical purposes, since their thermal fluctuations hide their quantal properties. These are called classical objects. Macroscopic bodies (more exactly, their coordinates) under usual conditions are classical. Digital information in computers is also classical.

A communication channel may be thought of as a chain of physical objects and physical interactions between adjacent objects. If all objects in the chain are quantal, the channel is called quantal. If at least one object in the chain is classical, the channel is called classical.

For example, newspapers, television, mobile phones and the Internet implement only classical channels. Quantum channels are usually implemented by sending a particle (photon, electron) or another microscopic object (ion) from a nonclassical source to a nonclassical detector through a low-noise medium.

Classical communication (that is, communication through a classical channel) can create shared randomness, but cannot create entanglement. Moreover, entanglement creation is impossible when Alice's apparatus A is connected to a source S by a quantum channel but Bob's apparatus B is connected to S by a classical channel. Here is an explanation.

The classical channel S-B is a chain containing a classical object C. By assumption, no chain of interactions connects A and B (via S, or otherwise) bypassing C. Therefore A and B are conditionally independent given a possible state c of C. The response y_A of A to x_A given c need not be a function g_A(c,x_A) of c and x_A (uniqueness is not guaranteed), but still, we may choose one of possible responses y_A and let g_A(c,x_A) = y_A (so-called uniformization). Similarly, g_B(c,x_B) = y_B. Now, given c, the two one-time functions f_A(x_A) = g_A(c,x_A) and f_B(x_B) = g_B(c,x_B) lead to a possible disagreement of Alice and Bob (on the intersection of the row and the column) by the argument used before (in the section "Example"). A more thorough analysis shows that the classical bound on the winning probability, deduced before from the counterfactual definiteness, holds also in the case treated here.

S B p

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Schröder–Bernstein (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) property is a collective term for mathematical properties structurally similar to the Schröder-Bernstein theorem of set theory. It is not a well-defined mathematical notion but the following pattern noted in several well-defined notions:

If X is similar to a part of Y and also Y is similar to a part of X then X and Y are similar (to each other).

The general pattern

A mathematical property is said to be a Schröder–Bernstein property if it is formulated in the following form.

If X is similar to a part of Y and also Y is similar to a part of X then X and Y are similar (to each other).

In order to be specific one should decide

• what kind of mathematical objects are X and Y,
• what is meant by "a part",
• what is meant by "similar".

A Schröder–Bernstein property is a joint property of

• a class of objects,
• a binary relation "be a part of",
• a binary relation "be similar".

Instead of the relation "be a part of" one may use a binary relation "be embeddable into" interpreted as "be similar to some part of". Then a Schröder–Bernstein property takes the following form.

If X is embeddable into Y and Y is embeddable into X then X and Y are similar.

The same in the language of category theory:

If objects X, Y are such that X injects into Y (more formally, there exists a monomorphism from X to Y) and also Y injects into X then X and Y are isomorphic (more formally, there exists an isomorphism from X to Y).

Not all statements of this form are true (see "Examples" below). A problem of deciding, whether a Schröder–Bernstein property (for a given class and two relations) holds or not, is called a Schröder–Bernstein problem. A theorem that states a Schröder–Bernstein property (for a given class and two relations), thus solving the Schröder–Bernstein problem in the affirmative, is called a Schröder–Bernstein theorem (for the given class and two relations), not to be confused with the classical (Cantor–)Schröder–Bernstein theorem.

Examples

In the classical (Cantor-)Schröder–Bernstein theorem,

• objects are sets (maybe infinite),
• "a part" is interpreted as a subset,
• "similar" is interpreted as equinumerous.

A Schröder–Bernstein property can fail. For example, assume that

• objects are triangles,
• "a part" means a triangle inside the given triangle,
• "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles").

Then the statement fails badly: every triangle X evidently is similar to some triangle inside Y, and the other way round; however, X and Y need no be similar.

The Schröder–Bernstein theorem for measurable spaces states the Schröder–Bernstein property for the following case:

• objects are measurable spaces,
• "a part" is interpreted as a measurable subset treated as a measurable space,
• "similar" is interpreted as isomorphic.

It has a noncommutative counterpart, the Schröder–Bernstein theorem for operator algebras.

Banach spaces violate the Schröder–Bernstein property; here

• objects are Banach spaces,
• "a part" is interpreted as a subspace or a complemented subspace,
• "similar" is interpreted as linearly homeomorphic.

Many other Schröder–Bernstein problems related to various spaces and algebraic structures (groups, rings, fields etc) are discussed by informal groups of mathematicians (see the external links page).

References

Srivastava, S.M. (1998), A Course on Borel Sets, Springer, ISBN 0387984127.

Gowers, W.T. (1996), "A solution to the Schroeder-Bernstein problem for Banach spaces", Bull. London Math. Soc. 28: 297–304.

Casazza, P.G. (1989), "The Schroeder-Bernstein property for Banach spaces", Contemp. Math. 85: 61–78. Bold text