Decoherence as a High-Dimensional Geometrical Phenomenon

Abstract

We develop a mathematical formalism that allows to study decoherence with a great level generality, so as to make it appear as a geometrical phenomenon between reservoirs of dimensions. It enables us to give quantitative estimates of the level of decoherence induced by a purely random environment on a system according to their respectives sizes, and to exhibit some links with entanglement entropy.

Annex: Decoherence Estimated by the Entanglement Entropy with the Environment

We establish here the formula (3): we first derive the inequality (1), and then look for a relation between $\eta$ and the linear entropy or the entanglement entropy. Inserting the second into the first directly yields (3).

1.1 Relation Between $\eta$ and the Level of Classicality

Let’s keep the notations of §2, where we defined $\rho _{\mathcal {S}}^{(q)}(t) = \sum _{i \ne j} c_i \overline{c_j} {\langle {\mathcal {E}_j(t) \vert \mathcal {E}_i(t)}\rangle } {|{i}\rangle }{\langle {j}|}$. We have $|\hspace{-1.111pt}|\hspace{-1.111pt}| \rho _{\mathcal {S}}^{(q)}(t)|\hspace{-1.111pt}|\hspace{-1.111pt}| \leqslant \eta (t)$ because for all vectors ${|{\Psi }\rangle } = \sum _k \alpha _k {|{k}\rangle } \in \mathcal {H}_{\mathcal {S}}$ of norm 1,

$$\begin{aligned}&\rho _{\mathcal {S}}^{(q)}(t) {|{\Psi }\rangle } = \sum _{1 \leqslant i \ne j \leqslant d} c_i \overline{c_j} {\langle {\mathcal {E}_j(t) \vert \mathcal {E}_i(t)}\rangle } \alpha _j {|{i}\rangle } \\ \Rightarrow&\quad \Vert \rho _{\mathcal {S}}^{(q)}(t) {|{\Psi }\rangle } \Vert ^2 = \sum _{i=1}^d |c_i |^2 |\sum _{\begin{array}{c} j=1 \\ j \ne i \end{array}}^d \overline{c_j} {\langle {\mathcal {E}_j(t) \vert \mathcal {E}_i(t)}\rangle } \alpha _j |^2 \leqslant \eta (t)^2 \sum _{i=1}^d |c_i |^2 \sum _{j=1}^d |c_j |^2 \leqslant \eta (t)^2. \end{aligned}$$

Now, if F is a subspace of $\mathcal {H}_{\mathcal {S}}$ (i.e. a probabilistic event), let $(\varphi _k)_k$ be an orthonormal basis of F. We have:

$$\begin{aligned}&{\text {tr}}(\rho _{\mathcal {S}}(t) \Pi _F) - {\text {tr}}(\rho _{\mathcal {S}}^{(d)} \Pi _F) = {\text {tr}}(\rho _{\mathcal {S}}^{(q)}(t) \Pi _F) = \sum _{k=1}^{\dim (F)} {\langle {\varphi _k \vert \rho _{\mathcal {S}}^{(q)}(t) \varphi _k}\rangle } \\ \Rightarrow \quad&|{\text {tr}}(\rho _{\mathcal {S}}(t) \Pi _F) - {\text {tr}}(\rho _{\mathcal {S}}^{(d)} \Pi _F) |\leqslant \sum _{k=1}^{\dim (F)} |\hspace{-1.111pt}|\hspace{-1.111pt}|\rho _{\mathcal {S}}^{(q)}(t)|\hspace{-1.111pt}|\hspace{-1.111pt}| \leqslant \dim (F) \eta (t). \end{aligned}$$

In a nutshell: ${\mathbb {P}}_{\text {quantum}} = {\mathbb {P}}_{\text {classical}} + \mathcal {O}(\eta )$.

1.2 Relation Between $\eta$ and the Linear Entropy

We define the linear entropy (or purity defect) of a state $\rho$ to be $S_{\text {lin}}(\rho ) = 1 - {\text {tr}}(\rho ^2)$. Since $\mathcal {S}$ is initially in a pure state, the quantity $\frac{S_{\text {lin}}(\rho _{\mathcal {S}}(t))}{ S_{\text {lin}}(\rho _{\mathcal {S}}^{(d)})}$ goes from 0 at $t=0$ to almost 1 when $t \rightarrow +\infty$. It measures the ratio of purity that has already been lost compared to its final ideal value. Recall that $\rho _{\mathcal {S}}(t) = \sum _{i=1}^d |c_i |^2 {|{i}\rangle }{\langle {i}|} + \sum _{1 \leqslant i \ne j \leqslant d} c_i \overline{c_j} {\langle {\mathcal {E}_j(t) \vert \mathcal {E}_i(t)}\rangle } {|{i}\rangle }{\langle {j}|}$, so that:

$$\begin{aligned} \frac{S_{\text {lin}}(\rho _{\mathcal {S}}(t))}{S_{\text {lin}}(\rho _{\mathcal {S}}^{(d)})}&= \frac{1 - \sum _i |c_i |^4 - \sum _{i \ne j} |c_i |^2 |c_j |^2 |{\langle {\mathcal {E}_i(t) \vert \mathcal {E}_j(t)}\rangle } |^2 }{1 - \sum _i |c_i |^4} \\&\geqslant 1 - \eta ^2(t) \frac{ \sum _{i \ne j} |c_i |^2 |c_j |^2 }{1 - \sum _i |c_i |^4} \\&\geqslant 1 - \eta ^2(t), \end{aligned}$$

since the last fraction always equals 1 because $1 = (\sum _i |c_i |^2)(\sum _i |c_i |^2) = \sum _i |c_i |^4 + \sum _{i \ne j} |c_i |^2 |c_j |^2$. Note that, for any given time t, this inequality is actually an equality for the initial state ${|{\Psi _{\mathcal {S}}(0)}\rangle } = c_{i_0} {|{i_0}\rangle } + c_{j_0} {|{j_0}\rangle }$ where $i_0$ and $j_0$ denote two indices such that $\eta (t) = |{\langle {\mathcal {E}_{i_0}(t) \vert \mathcal {E}_{j_0}(t)}\rangle } |$. Thus:

$$\begin{aligned} \eta (t) = \sqrt{ 1 - \inf _{{|{\Psi _\mathcal {S}(0)}\rangle }} \frac{ S_{\text {lin}}(\rho _{\mathcal {S}}(t)) }{S_{\text {lin}}(\rho _{\mathcal {S}}^{(d)})} }. \end{aligned}$$

1.3 Relation Between $\eta$ and the Entanglement Entropy

The entanglement entropy is always much harder to manipulate. We were not able to prove in the general case a similar result when the linear entropy $S_{\text {lin}}$ is replaced by the entanglement entropy S, but numerical simulations tend to indicate that the same formula is still (almost) true and that there exists a deep link between the quantity $1 - \eta ^2(t)$ and the ratio $\frac{S(\rho _{\mathcal {S}}(t))}{S(\rho _{\mathcal {S}}^{(d)})}$. Here are some considerations to get convinced.

In dimension $d=2$, if one denotes $f(t) = {\langle {\mathcal {E}_2(t) \vert \mathcal {E}_1(t)}\rangle }$, one can write $\rho _{\mathcal {S}}(t) = \begin{pmatrix} |c_1 |^2 &{} c_1 \overline{c_2} f(t) \\ \overline{c_1} c_2 {\overline{f}}(t) &{} |c_1 |^2 \end{pmatrix}$, whose eigenvalues are $\text {ev}_{\pm } = \frac{1}{2}(1 \pm \sqrt{( |c_1 |^2 - |c_2 |^2)^2 + 4 f^2(t) |c_1 |^2 |c_2 |^2 })$. At large times, $f \ll 1$ and after some calculations we get at leading order:

$$\begin{aligned} \frac{S(\rho _{\mathcal {S}}(t))}{S(\rho _{\mathcal {S}}^{(d)})} = \frac{\text {ev}_{+}\ln (\text {ev}_{+}) + \text {ev}_{-}\ln (\text {ev}_{-}) }{ |c_1 |^2 \ln ( |c_1 |^2) + |c_2 |^2 \ln ( |c_2 |^2) } \simeq 1 + \frac{\frac{|c_1 |^2 |c_2 |^2}{ |c_1 |^2 - |c_2 |^2} (\ln ( |c_1 |^2) - \ln ( |c_2 |^2)) }{ |c_1 |^2 \ln ( |c_1 |^2) + |c_2 |^2 \ln ( |c_2 |^2)} \eta ^2(t). \end{aligned}$$

The ratio preceding $\eta ^2$ is a one-parameter real function in $|c_1 |^2$ (since $|c_2 |^2 = 1 - |c_1 |^2$) defined on [0, 1]; it turns out that is takes only values in $[-1, -0.7]$ and tends to $-1$ only when $|c_1 |^2$ tends to 0 or 1. Therefore, in dimension 2, we still have (at least at leading order):

$$\begin{aligned} \eta (t) = \sqrt{ 1 - \inf _{{|{\Psi _\mathcal {S}(0)}\rangle }} \frac{ S(\rho _{\mathcal {S}}(t)) }{S(\rho _{\mathcal {S}}^{(d)})} }. \end{aligned}$$

In higher dimension, if we suppose that one of the ${\langle {\mathcal {E}_j(t) \vert \mathcal {E}_i(t)}\rangle }$ decreases much slower than the others (assume without loss of generality that it is ${\langle {\mathcal {E}_2(t) \vert \mathcal {E}_1(t)}\rangle }$, still denoted f(t)), then after some time $\rho _{\mathcal {S}}(t)$ is not very different from:

$$\begin{aligned} \begin{pmatrix} |c_1 |^2 &{} c_1 \overline{c_2} f(t) &{} &{} &{} \\ \overline{c_1} c_2 {\overline{f}}(t) &{} |c_1 |^2 &{} &{} &{} \\ {} &{} &{} |c_3 |^2 &{} &{} \\ {} &{} &{} &{} \ddots &{} \\ {} &{} &{} &{} &{} |c_d |^2 \end{pmatrix}. \end{aligned}$$

Using the previous inequality in dimension 2:

$$\begin{aligned} \frac{ S(\rho _{\mathcal {S}}(t)) }{S(\rho _{\mathcal {S}}^{(d)})} \geqslant \frac{ (1 - \eta ^2(t))(|c_1 |^2 + |c_2 |^2 ) + |c_3 |^2 + \dots + |c_d |^2 }{|c_1 |^2 + \dots + |c_d |^2 } \geqslant 1 - \eta ^2(t), \end{aligned}$$

and, once again, this bound is attained for an appropriate choice of the $(c_i)_{1\leqslant i \leqslant d}$.