Information-geometric reconstruction of quantum theory

In this paper, we show how the framework of information geometry, the natural geometry of discrete probability distributions, can be used to derive the quantum formalism. The derivation rests upon a few elementary features of quantum phenomena, such as complementarity and global gauge invariance. When appropriately formulated within an information-geometric framework, and combined with a novel information-theoretic principle, these features lead to the abstract quantum formalism for finite-dimensional quantum systems, and the result of Wigner’s theorem. By means of a correspondence principle, several correspondence rules of quantum theory, such as the canonical commutation relationships, are also systematically derived. The derivation suggests that information geometry is directly or indirectly responsible for many of the central structural features of the quantum formalism, such as the importance of square roots of probability and the occurrence of sinusoidal functions of phases in a pure quantum state. Global gauge invariance is shown to play a crucial role in the emergence of the formalism in its complex form.

Figure 1

An abstract experimental setup. In each run of the experiment, a physical system (such as a silver atom) is emitted from a source, passes a preparation step, undergoes an interaction, $\mathbf{I}$, and is then subject to a measurement. The preparation is implemented as a measurement, ${\mathbf{A}}^{\prime }$, which has $N_{A^{\prime }}$ possible outcomes, followed by the selection of those systems that yield some outcome $j$ $(j=1,2,\dots ,N_{A^{\prime }})$. The measurement $\mathbf{A}$ has $N_A$ possible outcomes.

Figure 2

Complementarity postulate. Probability tree showing the events that occur when measurement $\mathbf{A}$ is performed, and the corresponding outcomes (circled) that are observed. One of $2N$ possible events occurs, with probability $P_1,P_2,\dots ,P_{2N}$, respectively. The individual events are not resolved. Outcome $i$ is obtained whenever either event $2i-1$ or $2i$ is realized $(i=1,\dots ,N)$.

Figure 3

Simulation of measurement ${\mathbf{A}}^{\prime }$. A unitary transformation, $\mathsf{U}$, transforms the input state, $\mathsf{v}$, into $\mathsf{U}\mathsf{v}$. Measurement $\mathbf{A}$ is performed on this state, and yields an outcome and the outgoing state, $\tilde{\mathsf{v}}$, which is then transformed by the unitary transformation $\mathsf{V}$ into the output state $\mathsf{V}\tilde{\mathsf{v}}$.

Figure 4

An example of the application of the AVCP. (i) A classical experiment showing the measurements of $A$, $B$, and $C$ performed at times $t_A$, $t_B$, and $t_C$, respectively, with values denoted as $a$, $b$, and $c$, respectively. Here, $t_A=t_B=t_1$ and $t_C=t_2$. Suppose that one finds that the relation $c=f(a,b)$ holds for all initial states of the system. (ii) The corresponding quantum experiment. Three copies of the system are prepared in the same initial state, ${\mathsf{v}}_0$, at time $t_0$, and are placed in identical backgrounds. In this example, it is assumed that the operators $\mathsf{A}$ and $\mathsf{B}$ do not commute. Hence, by the AVCP, measurements $\mathbf{A}$ and $\mathbf{B}$ are performed on different copies of the system. Measurement $\mathbf{C}$ is performed on a separate copy of the system. In any given run of the experiment, the probabilities that measurements $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ yield values $a_i$, $b_j$, and $c_k$ $(i,j,k=1,\dots ,N)$ are $p_i$, $p_j^{\prime }$, and $p_k^{″}$, respectively. The AVCP then asserts that, provided the polynomial expansion of $f(a,b)$ contains no product terms involving $a$ and $b$, the relation $\overline{c}=\overline{f(a,b)}$ holds for all initial states, ${\mathsf{v}}_0$, of the system, where the average is taken over an infinite number of runs of the experiment; that is, ${\sum }_k{c}_k{p}_k^{″}={\sum }_{ij}f(a_i,b_j)p_i{p}_j^{\prime }$ for all ${\mathbf{v}}_0$.