How bosonic is a pair of fermions?

Abstract

Composite particles made of two fermions can be treated as ideal elementary bosons as long as the constituent fermions are sufficiently entangled. In that case, the Pauli principle acting on the parts does not jeopardise the bosonic behaviour of the whole. An indicator for bosonic quality is the composite boson normalisation ratio $\chi _{N+1}/\chi _{N}$ of a state of $N$ composites. This quantity is prohibitively complicated to compute exactly for realistic two-fermion wavefunctions and large composite numbers $N$. Here, we provide an efficient characterisation in terms of the purity $P$ and the largest eigenvalue $\lambda _1$ of the reduced single-fermion state. We find the states that extremise $\chi _N$ for given $P$ and $\lambda _1$, and we provide easily evaluable, saturable upper and lower bounds for the normalisation ratio. Our results strengthen the relationship between the bosonic quality of a composite particle and the entanglement of its constituents.

Appendices

Appendix 1: Minimising distribution

For completeness, we reproduce the proofs from the Appendix of Ref. [24] in the Appendices 1 and 2, adapting the argument to our situation in which not only the purity $P$ is fixed, but also the largest Schmidt coefficient $\lambda_1$.

1.1 Uniforming operation

Following an analysis of the birthday problem with non-uniform birthday probabilities [32], we define a uniforming operation $\varGamma ^{u}$ on the distribution $\varvec{\Lambda }$ that can modify three selected $\lambda _j$ with indices ${2 \le j_1 < j_2 < j_3 \le S}$ (i.e. the operation never acts on the first Schmidt coefficient $\lambda _1$, since its value is fixed, by assumption). We will show that this operation always decreases $\chi _N$, and specify the distribution $\varvec{\Lambda }_{\text {min}}(P, \lambda _1)$ that remains invariant under the application of $\varGamma ^u$. This distribution thus minimises $\chi _N^{\varvec{\Lambda }}$ under the constraints $(P,\lambda _1)$.

The operation $\varGamma ^u$ modifies three coefficients in a distribution,

$$\begin{aligned} \varGamma ^u: ( \lambda _{j_1}, \lambda _{j_2},\lambda _{j_3} ) \rightarrow ( \lambda ^u_{j_1}, \lambda ^u_{j_2},\lambda ^u_{j_3} ) , \end{aligned}$$

(50)

such that it leaves

$$\begin{aligned} K_1&= \lambda _{j_1} + \lambda _{j_2} + \lambda _{j_3} , \nonumber \\ K_2&= \lambda _{j_1}^2 + \lambda _{j_2}^2 + \lambda _{j_3}^2 , \end{aligned}$$

(51)

invariant, and, consequently, also $\sum _{j} \lambda _j=1$ and $\sum _j \lambda _j^2=P$. The third power sum, $M(3)=\sum _j \lambda _j^3$, on the other hand, is changed by $\varGamma ^u$. Specifically,

$$\begin{aligned} \lambda _{j_1}^u = \lambda _{j_2}^u&= \frac{1}{6} \left( 2 K_1 + \sqrt{6 K_2-2 K_1^2} \right) \nonumber ,\\ \lambda _{j_3}^u&= \frac{1}{3} \left( K_1 - \sqrt{6 K_2-2 K_1^2} \right) . \end{aligned}$$

(52)

In the case $K_1^2 <2 K_2$, in order to avoid $\lambda _{j_3}^u<0$, we need to set

$$\begin{aligned} \lambda _{j_1/j_2}^u&= \frac{1}{2} \left( K_1 \pm \sqrt{2 K_2-K_1^2}\right) , \\ \lambda _{j_3}^u&= 0 \nonumber . \end{aligned}$$

(53)

1.1.1 The product $\lambda _{j_1} \lambda _{j_2}\lambda _{j_3}$ decreases under $\varGamma ^u$

It holds

$$\begin{aligned} \lambda _{j_1}^u \lambda _{j_2}^u \lambda _{j_3}^u \le \lambda _{j_1} \lambda _{j_2} \lambda _{j_3} . \end{aligned}$$

(54)

Proof

We write the left- and right-hand side of (54) in terms of $K_1, K_2$ and $\lambda _{j_1}$

$$\begin{aligned}&\lambda _{j_1}^u \lambda _{j_2}^u \lambda _{j_3}^u \nonumber \\&\quad = \left\{ \begin{array}{ll} \frac{1}{108} \left( K_1-\sqrt{6 K_2-2 K_1^2}\right) &{} \\ \times \left( 2 K_1+\sqrt{6 K_2-2 K_1^2}\right) ^2 &{}\hbox { for } K_1^2 > 2 K_2 \\ 0 &{}\hbox { for } K_1^2 \le 2 K_2 \end{array} \right. \\&\lambda _{j_1} \lambda _{j_2} \lambda _{j_3}= \frac{1}{2} \lambda _{j_1} \left( 2 \lambda _{j_1}^2-2 \lambda _{j_1} K_1+K_1^2-K_2 \right) \nonumber \end{aligned}$$

(55)

Given $K_1$ and $K_2$, the original $\lambda_{j_2/j_3}$ become functions of $\lambda _{_1},$

$$\begin{aligned} \lambda _{{j_2}/{j_3}} = \frac{1}{2} \left( K_1 - \lambda _{j_1} \pm \sqrt{2 \lambda _{j_1} K_1 - K_1^2 - 3 \lambda _{j_1}^2 + 2 K_2} \right) \nonumber \end{aligned}$$

(56)

The requirement $\lambda _{j_1} \ge \lambda _{{j_2}} \ge \lambda _{{j_3}} \ge 0$ imposes

$$\begin{aligned} \frac{ K_1}{3} + \frac{ \sqrt{3 K_2 - K_1^2}}{3\sqrt{2}}&\le \lambda _{j_1} \le \frac{ K_1}{3} +\frac{ \sqrt{6 K_2 -2 K_1^2} }{3}. \end{aligned}$$

(57)

The values of $\lambda _{j_1}$ constrained to this interval then always fulfil Eq. (54). $\square$

1.1.2 $\chi _N^{\varvec{\Lambda }}$ decreases upon application of $\varGamma ^u$

Upon application of $\varGamma ^u$, the normalisation constant $\chi _N$ and the normalisation ratio $\chi _{N+1}/\chi _N$ can only decrease:

$$\begin{aligned} \chi _{N}^{\varGamma ^{u}(\varvec{\Lambda }) }&\le \chi _{N}^{\varvec{\Lambda }} , \end{aligned}$$

(58)

$$\begin{aligned} \frac{\chi _{N+1}^{\varGamma ^{u}(\varvec{\Lambda }) } }{\chi _{N}^{\varGamma ^{u}(\varvec{\Lambda }) } }&\le \frac{ \chi _{N+1}^{\varvec{\Lambda }} }{ \chi _{N}^{\varvec{\Lambda }}} . \end{aligned}$$

(59)

To ease notation in the following, we exemplarily use $j_1=2,\, j_2=3,\, j_3=4$, and set

$$\begin{aligned} {\tilde{\chi }}_N&= \chi _N^{(\lambda _1, \lambda _5, \ldots , \lambda _S)}, \end{aligned}$$

(60)

which allows us to write $\chi _N$ as

$$\begin{aligned} \chi _N&= \lambda _2 \lambda _3 \lambda _4 \cdot {\tilde{\chi }}_{N-3} +(\lambda _2 \lambda _3 +\lambda _4 \lambda _3+\lambda _2 \lambda _4) {\tilde{\chi }}_{N-2} \nonumber \\&+(\lambda _2+\lambda _3+\lambda _4) {\tilde{\chi }}_{N-1}+ {\tilde{\chi }}_{N} . \end{aligned}$$

(61)

The terms

$$\begin{aligned} \lambda _2 \lambda _3 +\lambda _4 \lambda _3 +\lambda _2 \lambda _4&= \frac{1}{2} \left( K_1^2 - K_2 \right) , \\ \lambda _2+\lambda _3+\lambda _4&= K_1 , \end{aligned}$$

(62)

and ${{\tilde{\chi }}}_k \in \{ N-3, \dots , N \}$ remain invariant under the application of $\varGamma ^{u}$, whereas the product $\lambda _2 \lambda _3 \lambda _4$ decreases, due to Eq. (54). Consequently, also $\chi _N^{{\boldsymbol\Lambda }}$ decreases upon the application of $\varGamma ^u$.

Using $\chi _{N+1} \le \chi _{N}$, one can easily show in analogy to Ref. [24] that the inequality (58) is inherited by the normalisation ratio (59). $\square$

1.2 Properties of the minimising distribution

The distribution $\varvec{\Lambda }_{\text {min}}(P,\lambda _1)$ that minimises $\chi _N$ for fixed $\lambda _1$ and $P$ should remain invariant under the application of $\varGamma ^u$, for all choices of $2 \le j_1, j_2, j_3 \le S$. By the definition of $\varGamma ^u$, we see that any three coefficients with $\lambda _{j_1} > \lambda _{j_2} = \lambda _{j_3}$ never constitute a fixed point of $\varGamma ^u$. Therefore, the invariant distribution is of the form

$$\begin{aligned} \lambda _1 \ge \lambda _2=\ldots = \lambda _{S-1} \ge \lambda _S . \end{aligned}$$

(63)

It coincides with the distribution found in Ref. [23].

Appendix 2: Maximising distribution

1.1 Peaking operation

With $K_1$ and $K_2$ defined as in Eq. (51) above, we define the peaking operation $\varGamma ^p$ as follows [24]: For $K_1+\sqrt{6 K_2-2K_1^2} \le 3 \lambda _1$, we set

$$\begin{aligned} \lambda _{j_1}^{p}&= \frac{1}{3} \left( K_1 + \sqrt{6 K_2-2 K_1^2} \right) , \nonumber \\ \lambda _{j_2}^{p}=\lambda _{j_3}^{p}&= \frac{1}{6} \left( 2 K_1 - \sqrt{6 K_2-2 K_1^2} \right) . \end{aligned}$$

(64)

If $K_1+\sqrt{6 K_2-2K_1^2} > 3 \lambda _1$, the above definition leads to $\lambda _{j_1}^{p}> \lambda _1$, which we excluded by assumption. In this case, we define alternatively

$$\begin{aligned} \lambda _{j_1}^{p}&= \lambda _1, \\ \lambda _{j_2/j_3}^{p}&= \frac{ K_1 - \lambda _1 \pm \sqrt{2 (K_2 -\lambda _1^2) -(K_1-\lambda _1)^2 } }{{2}} , \nonumber \end{aligned}$$

(65)

for which $\lambda _1 = \lambda _{j_1}^p \ge \lambda _{j_2}^p \ge \lambda _{j_3}^p \ge 0$. In full analogy to the discussion in Appendix 1, one shows that

$$\begin{aligned} \chi _N^{\varvec{\Lambda }} \le \chi _N^{\varGamma ^p(\varvec{\Lambda })} , \qquad \frac{\chi _{N+1}^{\varvec{\Lambda }} }{\chi _N^{\varvec{\Lambda }} } \le \frac{\chi _{N+1}^{\varGamma ^p(\varvec{\Lambda })} }{\chi _N^{\varGamma ^p(\varvec{\Lambda })} } , \end{aligned}$$

(66)

i.e. the normalisation factor and ratio increase under the application of $\varGamma ^p$.

1.2 Properties of the maximising distribution

The distribution $\varvec{\Lambda }_{\text {max}}(P,\lambda _1)$ that maximises $\chi _N$ for fixed $\lambda _1$ and $P$ is obtained as follows: We maximise the multiplicity of $\lambda _1$ in $\varvec{\Lambda }$, i.e. $\lambda _1$ is repeated $L-1$ times, with $L=\lceil P/\lambda _1^2 \rceil$. The coefficients then need to fulfil

$$\begin{aligned} \lambda _1=\ldots = \lambda _{L-1} \ge \lambda _L \ge \lambda _{L+1}&= \ldots = \lambda _S , \end{aligned}$$

(67)

to ensure that $\varvec{\Lambda }_{\text {max}}(P,\lambda _1)$ be a fixed point of $\varGamma ^p$. Again, the distribution coincides with the one found in Ref. [23].