Ordering states with coherence measures

Abstract

The quantification of quantum coherence has attracted a growing attention, and based on various physical contexts, several coherence measures have been put forward. An interesting question is whether these coherence measures give the same ordering when they are used to quantify the coherence of quantum states. In this paper, we consider the two well-known coherence measures, the $l_1$ norm of coherence and the relative entropy of coherence, to show that there are the states for which the two measures give a different ordering. Our analysis can be extended to other coherence measures, and as an illustration of the extension we further consider the formation of coherence to show that the $l_1$ norm of coherence and the formation of coherence, as well as the relative entropy of coherence and the coherence of formation, do not give the same ordering too.

Appendix

We show that any two d($\ge$4)-dimensional pure states defined by Eq. (15) necessarily satisfy $\mathcal {C}_{l_1}(|\varphi ^{(d)}_1\rangle )<\mathcal {C}_{l_1}(|\varphi ^{(d)}_2\rangle )$ and $\mathcal {C}_r(|\varphi ^{(d)}_1\rangle )>\mathcal {C}_r(|\varphi ^{(d)}_2\rangle )$ as long as $|\varphi _1\rangle$ and $|\varphi _2\rangle$ do. To this end, we only need to prove the following theorem: If the two $(d-1)$-dimensional states expressed as

$$\begin{aligned} |\varphi ^{(d-1)}_1\rangle =\sum _{i=1}^{d-1}a_i|i\rangle ,\quad |\varphi ^{(d-1)}_2\rangle&=\sum _{i=1}^{d-1}b_i|i\rangle , \end{aligned}$$

(16)

satisfy $\mathcal {C}_{l_1}(|\varphi ^{(d-1)}_1\rangle )<\mathcal {C}_{l_1}(|\varphi ^{(d-1)}_2\rangle )$ and $\mathcal {C}_r(|\varphi ^{(d-1)}_1\rangle )>\mathcal {C}_r(|\phi ^{(d-1)}_2\rangle )$, then the two d-dimensional states defined by

$$\begin{aligned} |\varphi ^{(d)}_1\rangle =\alpha _d|\varphi ^{(d-1)}_1\rangle +\beta _d|d\rangle ,~~ |\varphi ^{(d)}_2\rangle =\alpha _d|\varphi ^{(d-1)}_1\rangle +\beta _d|d\rangle \end{aligned}$$

(17)

with $|\alpha _d|^2+|\beta _d|^2=1$ and $0<|\alpha _d|<1$, satisfy $\mathcal {C}_{l_1}(|\varphi ^{(d)}_1\rangle )<\mathcal {C}_{l_1}(|\varphi ^{(d)}_2\rangle )$ and $\mathcal {C}_r(|\varphi ^{(d)}_1\rangle )>\mathcal {C}_r(|\varphi ^{(d)}_2\rangle )$.

We now prove the theorem.

By the definition of the $l_1$-norm of coherence, we have

$$\begin{aligned} \begin{aligned} \mathcal {C}_{l_1}(|\varphi ^{(d-1)}_1\rangle )=&2\sum _{1\le i<j\le d-1}|a_ia_j|, \end{aligned} \end{aligned}$$

(18)

and

$$\begin{aligned} \begin{aligned} \mathcal {C}_{l_1}(|\varphi ^{(d)}_1\rangle )=&2|\alpha _d|^2\sum _{1\le i<j\le d-1}|a_ia_j|+2|\alpha _d\beta _d|\sum _{i=1}^{d-1}|a_i|,\\ =&2|\alpha _d|^2\sum _{1\le i<j\le d-1}|a_ia_j|+2|\alpha _d\beta _d|\sqrt{\left( \sum _{i=1}^{d-1}|a_i|\right) ^2}\\ =&2|\alpha _d|^2\sum _{1\le i<j\le d-1}|a_ia_j|+2|\alpha _d\beta _d|\sqrt{1+2\sum _{1\le i<j\le d-1}|a_ia_j|},\\ =&|\alpha _d|^2C_{l_1}(|\varphi ^{(d-1)}_1\rangle )+2|\alpha _d\beta _d|\sqrt{1+C_{l_1}(|\varphi ^{(d-1)}_1\rangle )}, \end{aligned} \end{aligned}$$

(19)

and similarly,

$$\begin{aligned} \begin{aligned} \mathcal {C}_{l_1}(|\varphi ^{(d-1)}_2\rangle )=&2\sum _{1\le i<j\le d-1}|b_ib_j|, \end{aligned} \end{aligned}$$

(20)

and

$$\begin{aligned} \begin{aligned} \mathcal {C}_{l_1}(|\varphi ^{(d)}_2\rangle )=&|\beta _d|^2C_{l_1}(|\varphi ^{(d-1)}_2\rangle )+2|\alpha _d\beta _d|\sqrt{1+C_{l_1}(|\varphi ^{(d-1)}_2\rangle }). \end{aligned}\end{aligned}$$

(21)

Eqs. (19) and (21) show that $\mathcal {C}_{l_1}(|\phi ^{(d)}_1\rangle )<\mathcal {C}_{l_1}(|\phi ^{(d)}_2\rangle )$ if $\mathcal {C}_{l_1}(|\varphi ^{(d-1)}_1\rangle )<\mathcal {C}_{l_1}(|\varphi ^{(d-1)}_2\rangle )$.

By the definition of the relative entropy of coherence, we have

$$\begin{aligned} \begin{aligned} \mathcal {C}_r(|\varphi ^{(d-1)}_1\rangle )=&-\sum _{i=1}^{d-1}|a_i|^2\log _2|a_i|^2, \end{aligned} \end{aligned}$$

(22)

and

$$\begin{aligned} \begin{aligned} \mathcal {C}_r({|\varphi ^{(d)}_1\rangle })=&-|\beta _d|^2\log _2|\beta _d|^2 -\sum _{i=1}^{d-1}|\alpha _d|^2|a_i|^2\log _2(|\alpha _d|^2|a_i|^2)\\ =&-|\beta _d|^2\log _2|\beta _d|^2-|\alpha _d|^2\log _2|\alpha _d|^2 -|\alpha _d|^2\sum _{i=1}^{d-1}|a_i|^2\log _2|a_i|^2\\ =&|\alpha _d|^2C_r(|\varphi ^{(d-1)}_1\rangle )+H(|\alpha _d|^2), \end{aligned} \end{aligned}$$

(23)

where $H(x)=-x\log _2x-(1-x)\log _2(1-x)$ is the binary Shannon entropy function, and similarly,

$$\begin{aligned} \begin{aligned} \mathcal {C}_r(|\varphi ^{(d-1)}_2\rangle )=&-\sum _{i=1}^{d-1}|b_i|^2\log _2|b_i|^2, \end{aligned} \end{aligned}$$

(24)

and

$$\begin{aligned} \mathcal {C}_r({|\varphi ^{(d)}_2\rangle })=|\alpha _d|^2C_r(|\varphi ^{(d-1)}_2\rangle )+H(|\alpha _d|^2). \end{aligned}$$

(25)

Eqs. (24) and (25) show that $\mathcal {C}_r(|\varphi ^{(d)}_1\rangle )>\mathcal {C}_r(|\varphi ^{(d)}_2\rangle )$ if $\mathcal {C}_r(|\varphi ^{(d-1)}_1\rangle )>\mathcal {C}_r(|\varphi ^{(d-1)}_2\rangle )$. This completes the proof of the theorem. With this theorem, it is easy to obtain the conclusion related to Eq. (15).