Operational foundations for complementarity and uncertainty relations

The so-called preparation uncertainty that occurs in the quantum world can be understood well in purely operational terms, and its existence in any given theory, perhaps differently than in quantum mechanics, can be verified by examining only measurement statistics. Namely, one says that uncertainty occurs in some theory when for some pair of observables, there is no preparation that would exhibit deterministic statistics for both of them. However, the right-hand side of the uncertainty relation is not operational anymore if we do not insist that it is just the minimum of the left-hand side for a given theory. For example, in quantum mechanics, it is some function of two observables that must be computed within the quantum formalism. Also, while joint nonmeasurability of observables is an operational notion, the complementarity in Bohr's sense (i.e., in terms of information needed to describe the system) has not yet been expressed in purely operational terms. Here we propose a general operational framework that provides answers to the above issues. We introduce an operational definition of complementarity and further postulate that complementary observables have to exhibit uncertainty, which means that we propose to put the (operational) complementarity as the right-hand side of the uncertainty relation. In particular, we identify two different notions of uncertainty and complementarity for which the above principle holds in the quantum-mechanical realm. We also introduce postulates for the general measures of uncertainty and complementarity. In order to define quantifiers of complementarity we first turn to the simpler notion of independence that is defined solely in terms of the statistics of two observables. Importantly, for clean and extremal observables—i.e., ones that cannot be simulated irreducibly by other observables—any measure of independence reduces to the proper complementary measure. Finally, as an application of our general framework we define a number of complementarity indicators and show that they can be used to state uncertainty relations. One of them, assuming some natural symmetries, leads to the Tsirelson bound for the Clauser-Horne-Shimony-Holt (CHSH) inequality. Lastly, we show that for a single system a variant of information causality called the information content principle, under the above symmetries, can be interpreted as an uncertainty relation in the above sense.

Figure 1

The statistics set is embedded into the Cartesian product of two simplices $S={\mathrm{\Delta }}_n\times {\mathrm{\Delta }}_n$. (a) Dichotomic observables ($n=2$): Simplices are one-dimensional and the axes represent probabilities of a single outcome for each observable. Their Cartesian product is a square, and $S_{X,Y}$ is its convex subset. (b) Observables with three outputs ($n=3)$ give rise to two-dimensional simplices. Their Cartesian product and the set $S_{X,Y}$ cannot be visualized.

Figure 2

The red implications hold by definition and, thus, hold in any theory. The blue ones hold for sharp projective observables in quantum theory. We further postulate the blue implications for clean, sharp, and extremal observables in physical theories.

Figure 3

The statistics set for (a) most independent observables, (b) intermediate case, and (c) the same observables.

Figure 4

The statistics sets for quantum binary observables. Circle is for most complementary (e.g., ${\sigma }_x$ and ${\sigma }_z$, diagonal for both being ${\sigma }_z$).

Figure 5

Various sets $S_{X,Y}$. (a) $S_{X,Y}$ is equal to full square—the so-called “square bit.” (b) One observable is completely noisy—reports no information. (c) Both observables are not sharp; i.e., there is no state that would give deterministic outcome for any of them. (d), (e) Both observables are sharp. (f) Quantum-mechanical observables.

Figure 6

Uncertainty for two-outcome observables. (a) No corner included, hence we have uncertainty for any state. (b) One corner included—represents preparation that has no uncertainty for both observables; exclusion still holds. (c) Two corners included, so for two preparations no uncertainty; still exclusion holds. (d) No uncertainty and no exclusion, since opposite corners are included. (e) Classical case (the same observables)—no uncertainty. (f) Generic quantum observables: both uncertainty and exclusion.

Figure 7

Uncertainty principle in quantum case.

Figure 8

The rescaling measure of independence (${\mathrm{Ind}}_r$) of the statistics set for binary-outcome observables $X,Y$. The statistics set is presented in gray.

Figure 9

The statistics set of two quantum observables ${\sigma }_z$ and $n_x{\sigma }_x+n_z{\sigma }_z$ is presented in gray. The semimajor and semiminor axes are denoted by a and b, respectively.

Figure 10

Examples of a classical bit and a “diamond.” (a) Independence of two identical observables is zero, since it is obtained as an image of perfectly correlated probability distributions. (b) Independence of observables for which the statistics constitutes a diamond is equal to 1. Preimage is a square that contains the center of the product of the simplices, which has independence 1.

Figure 11

The statistics set $S_{X,Y}$ possesses the symmetry under the reflection of the diagonal of the square. State of the system $P_{a_1{a}_2}$ is described by the pair of probabilities $(q_X\left(2\right|P),q_Y\left(2\right|P))$.

Figure 12

An arbitrary statistics set $S_{X,Y}$ for two observables $B_{1,2}$ of Bob's system. The four different preparations $P_{A_i\pm }$ depending on Alice's measurement choice and outcome are presented by their coordinates. The four preparations should satisfy the no-signaling conditions (65).

Figure 13

We assume that the minimum distance between the corners and the points on $S_{X,Y}$, which lie on the boundary of $S$, is $t$. This, together with the fact that $S_{X,Y}$ touches all four edges of the square, imposes that the boundaries of $S_{X,Y}$ cannot be closer to the center than the dotted lines presented here. This implies that a square of length $t$ will fit inside $S_{X,Y}$.