Macroscopic Superposition States in Isolated Quantum Systems

Abstract

For any choice of initial state and weak assumptions about the Hamiltonian, large isolated quantum systems undergoing Schrödinger evolution spend most of their time in macroscopic superposition states. The result follows from von Neumann’s 1929 Quantum Ergodic Theorem. As a specific example, we consider a box containing a solid ball and some gas molecules. Regardless of the initial state, the system will evolve into a quantum superposition of states with the ball in macroscopically different positions. Thus, despite their seeming fragility, macroscopic superposition states are ubiquitous consequences of quantum evolution. We discuss the connection to many worlds quantum mechanics.

Appendix A

Writing $c_{j_1,j_2}=c'_{j_1,j_2}+ic''_{j_1,j_2}$, where $c'_{j_1,k_1}, c''_{j_1,k_1}\in {\mathbb {R}}$,

$$\begin{aligned} \sum _{j_1=1}^{n_1} \sum _{j_2=1}^{n_2} (c^{\prime 2}_{1,j_1,k_1} +c^{\prime \prime 2}_{j_1,k_1}) =1, \end{aligned}$$

(16)

and using

$$\begin{aligned}&{{\,\mathrm{Re}\,}}{(\rho _1)_{j_1,k_1}} =\sum _{j_2=1}^{n_2} (c'_{j_1,j_2}c'_{k_1,j_2} +c''_{j_1,j_2}c''_{k_1,j_2}), \end{aligned}$$

(17)

$$\begin{aligned}&{{\,\mathrm{Im}\,}}{(\rho _1)_{j_1,k_1}} =\sum _{j_2=1}^{n_2} (c''_{j_1,j_2}c'_{k_1,j_2} -c'_{j_1,j_2}c''_{k_1,j_2}), \end{aligned}$$

(18)

we find

$$\begin{aligned} \Vert \nabla _c {{\,\mathrm{Re}\,}}{(\rho _1)_{j_1,k_1}}\Vert ^2&=\Vert \left( \delta _{l_1,j_1} c'_{k_1,l_2} +c'_{j_1,l_2}\delta _{l_1,k_1}, \delta _{l_1,j_1} c''_{k_1,l_2} +c''_{j_1,l_2}\delta _{l_1,k_1} \right) _{1\le l_1\le n_1,1\le l_2\le n_2}\Vert ^2 \nonumber \\&=\sum _{l_2=1}^{n_2} \left( c^{\prime 2}_{j_1,l_2} +c^{\prime \prime 2}_{j_1,l_2} +c^{\prime 2}_{k_1,l_2} +c^{\prime \prime 2}_{k_1,l_2} \right) +2\delta _{j_1,k_1}\sum _{l_2=1}^{n_2} \left( c^{\prime 2}_{j_1,l_2} +c^{\prime \prime 2}_{j_1,l_2} \right) \nonumber \\&=\sum _{l_2=1}^{n_2} \left( |c_{j_1,l_2}|^2 +|c_{k_1,l_2}|^2 \right) +2\delta _{j_1,k_1}\sum _{l_2=1}^{n_2} |c_{j_1,l_2}|^2 \end{aligned}$$

(19)

$$\begin{aligned} \Vert \nabla _c {{\,\mathrm{Im}\,}}{(\rho _1)_{j_1,k_1}}\Vert ^2&=\Vert \left( c''_{j_1,l_2}\delta _{l_1,k_1} -\delta _{l_1,j_1}c''_{k_1,l_2}, \delta _{l_1,j_1}c'_{k_1,l_2} -c'_{j_1,l_2}\delta _{l_1,k_1} \right) _{1\le l_1\le n_1,1\le l_2\le n_2}\Vert ^2 \nonumber \\&=\sum _{l_2=1}^{n_2} \left( c^{\prime 2}_{j_1,l_2} +c^{\prime \prime 2}_{j_1,l_2} +c^{\prime 2}_{k_1,l_2} +c^{\prime \prime 2}_{k_1,l_2} \right) -2\delta _{j_1,k_1}\sum _{l_2=1}^{n_2} \left( c^{\prime 2}_{j_1,l_2} +c^{\prime \prime 2}_{j_1,l_2} \right) \nonumber \\&=\sum _{l_2=1}^{n_2} \left( |c_{j_1,l_2}|^2 +|c_{k_1,l_2}|^2 \right) -2\delta _{j_1,k_1}\sum _{l_2=1}^{n_2} |c_{j_1,l_2}|^2. \end{aligned}$$

(20)

As a result,

$$\begin{aligned}&\Vert \nabla _c {{\,\mathrm{Re}\,}}{(\rho _1)_{j_1,j_1}}\Vert ^2 \le 4, \end{aligned}$$

(21)

$$\begin{aligned}&\Vert \nabla _c {{\,\mathrm{Re}\,}}{(\rho _1)_{j_1,k_1}}\Vert ^2 \le 1, \quad j_1\not =k_1, \end{aligned}$$

(22)

$$\begin{aligned}&\Vert \nabla _c {{\,\mathrm{Im}\,}}{(\rho _1)_{j_1,k_1}}\Vert ^2 \le 1, \quad j_1\not =k_1. \end{aligned}$$

(23)

For bounds (8) and (9) on the off-diagonal terms in $\rho _1$ we use a somewhat stronger inequality as the median ${\mathsf {M}}{f}$ and expectation ${\mathsf {E}}{f}$ coincide: ${\mathsf {P}}{(|f- {\mathsf {M}}{f}|\ge \epsilon )} \le \exp ( - n_1 n_2 \epsilon ^2 \Vert f\Vert ^{-2}_{\text {L}} )$, Proposition A.0.5 in [9]. For probability of deviation from expectation we use Proposition 1.8 in [7] and the median result to obtain

$$\begin{aligned} {\mathsf {P}}{(|f-{\mathsf {E}}{f}|\ge \epsilon )} \le \exp {\left( -\frac{n_1 n_2 (\epsilon -\delta )^2}{\Vert f\Vert ^2_{\text {L}}}\right) } \end{aligned}$$

(24)

with $\delta = (\frac{\pi }{4 n_1 n_2})^{1/2} \Vert f\Vert _{\text {L}}$ and $\epsilon \ge \delta$. The specific bounds we obtain from concentration of measure are stronger than those previously given in the literature as expressions of Levy’s Lemma. They can be applied, for example, to relatively small quantum systems in the laboratory.