Visualization of superposition of macroscopically distinct states

We propose a method of visualizing superpositions of macroscopically distinct states in many-body pure states. We introduce a visualization function, which is a coarse-grained quasi joint probability density for two or more Hermitian additive operators. If a state contains superpositions of macroscopically distinct states, one can visualize them by plotting the visualization function for appropriately taken operators. We also explain how to efficiently find appropriate operators for a given state. As examples, we visualize four states containing superpositions of macroscopically distinct states: the ground state of the $XY$ model, that of the Heisenberg antiferromagnet, a state in Shor’s factoring algorithm, and a state in Grover’s quantum search algorithm. Although the visualization function can take negative values, it becomes nonnegative (hence, becomes a coarse-grained joint probability density) if the characteristic width of the coarse-graining function used in the visualization function is sufficiently large.

Figure 1

(Color online) $\Xi (M_x,M_y)$ with $W\to 0$ for the exact ground state of the $XY$ model on a two-dimensional square lattice with $N=14$. $c\delta \left(0\right)$ $[c∊\mathbb{R}]$ is represented by a vertical line with height $c$. Positive values are represented by (red) vertical lines, whereas negative values are represented by (blue) vertical lines.

Figure 2

The integral $I_{\widehat{M_x},{\widehat{M}}_y}$ of negative values of $\Xi (M_x,M_y)$ versus $W$ for the exact ground state of the $XY$ model on a two-dimensional square lattice with $N=14$.

Figure 3

(Color online) $\Xi (M_x,M_y)$ with $W=3.2$ for the exact ground state of the $XY$ model on a two-dimensional square lattice with $N=14$.

Figure 4

(Color online) $\Xi (M_x,M_y)$ with $W=2$ for the exact ground state of the $XY$ model on a two-dimensional square lattice with $N=14$. Positive-valued regions are colored red, whereas negative-valued regions are colored blue.

Figure 5

(Color online) $\Xi (M_x^{st},M_y^{st})$ with $W\to 0$ for the exact ground state of the Heisenberg antiferromagnet on a two-dimensional square lattice with $N=14$.

Figure 6

(Color online) $\Xi (M_x^{st},M_y^{st})$ with $W=2$ for the exact ground state of the Heisenberg antiferromagnet on a two-dimensional square lattice with $N=14$.

Figure 7

(Color online) $\Xi (M_x,M_y^{st})$ with $W\to 0$ for the state just after the modular exponentiation with $(I,\mathsf{x})=(21,5)$. Negative values are multiplied by 10, in order to make them more visible.

Figure 8

The integral $I_{{\widehat{M}}_x,{\widehat{M}}_y^{st}}$ of negative values of $\Xi (M_x,M_y^{st})$ versus $W$ for the state just after the modular exponentiation with $(I,\mathsf{x})=(21,5)$.

Figure 9

(Color online) $\Xi (M_x,M_y^{st})$ with $W=1.4$ for the state just after the modular exponentiation with $(I,\mathsf{x})=(21,5)$.

Figure 10

(Color online) $\Xi (M_x,M_y^{st})$ with $W=1$ for the state just after the modular exponentiation with $(I,\mathsf{x})=(21,5)$. In order to make negative regions more visible, negative values are multiplied by 10.

Figure 11

(Color online) $\Xi (M_{x-z},M_y)$ with $W\to 0$ for a state in Grover’s quantum search algorithm with $N=12$.

Figure 12

The integral $I_{{\widehat{M}}_{x-z},{\widehat{M}}_y}$ of negative values of $\Xi (M_{x-z},M_y)$ versus $W$ for a state in Grover’s quantum search algorithm with $N=12$.

Figure 13

(Color online) $\Xi (M_{x-z},M_y)$ with $W=2$ for a state in Grover’s quantum search algorithm with $N=12$.

Figure 14

The integral $I_{{\widehat{M}}_{x-z},{\widehat{M}}_y}$ of negative values of $\Xi (M_{x-z},M_y)$ versus $N$ with $W=N∕6$ and $W=N∕8$ for states in Grover’s quantum search algorithm. Lines are guides to the eye.

Figure 15

The integral $I_{{\widehat{M}}_{x-z},{\widehat{M}}_y}$ of negative values of $\Xi (M_{x-z},M_y)$ versus $N$ with $W=\sqrt{N}$ and $W=\sqrt{0.5N}$ for states in Grover’s quantum search algorithm.

Figure 16

(Color online) $\Xi (M_x,M_y)$ with $W=1.5$ for the separable state $\mid 0^{\otimes N}⟩$ with $N=14$.

Figure 17

The integral $I_{{\widehat{M}}_x,{\widehat{M}}_y}$ of negative values of $\Xi (M_x,M_y)$ versus $N$ with $W=N∕6$, $W=N∕8$, and $W=N∕10$ for separable states $\mid 0^{\otimes N}⟩$.

Figure 18

The integral $I_{{\widehat{M}}_x,{\widehat{M}}_y}$ of negative values of $\Xi (M_x,M_y)$ versus $N$ with $W=\sqrt{N}$ and $W=\sqrt{0.5N}$ for separable states $\mid 0^{\otimes N}⟩$.

Figure 19

(Color online) $\Xi (M_x,M_y)$ with $W=1.5$ for the cat state $\frac{1}{\sqrt{2}}\mid 0^{\otimes N}⟩+\frac{1}{\sqrt{2}}\mid 1^{\otimes N}⟩$ with $N=14$.

Figure 20

The integral $I_{{\widehat{M}}_x,{\widehat{M}}_y}$ of negative values of $\Xi (M_x,M_y)$ versus $N$ with $W=N∕6$, $W=N∕8$, and $W=N∕10$ for cat states $\frac{1}{\sqrt{2}}\mid 0^{\otimes N}⟩+\frac{1}{\sqrt{2}}\mid 1^{\otimes N}⟩$.

Figure 21

The integral $I_{{\widehat{M}}_x,{\widehat{M}}_y}$ of negative values of $\Xi (M_x,M_y)$ versus $N$ with $W=\sqrt{N}$ and $W=\sqrt{0.5N}$ for cat states $\frac{1}{\sqrt{2}}\mid 0^{\otimes N}⟩+\frac{1}{\sqrt{2}}\mid 1^{\otimes N}⟩$.