1. Introduction
Quantum entanglement is one of the key features of quantum mechanics [1–3]. As an essential resource, it plays a significant role in many tasks in quantum information and computation, such as quantum cryptography [4], quantum dense coding [5], quantum teleportation [6] and quantum secret sharing [7]. A fundamental issue in the theory of quantum entanglement is to determine whether a given quantum state is entangled or not. The issue is well understood for pure states. However, the situation is extremely complicated for mixed states and there are no general practical criteria until now.
For bipartite quantum systems, a number of criteria to detect entanglement have been proposed from different points of view, such as the positive partial transposition (PPT) criterion [8–10], the reduction criterion [11], the computable cross-norm or realignment (CCNR) criterion [12, 13], the covariance matrix criterion [14–16], the criterion based on Bloch representations [17], the criterion based on local uncertainty relations [18–20], and the criterion based on quantum Fisher information [21] etc. See, e.g., [2, 3] for comprehensive surveys. However, for multipartite quantum systems, the detection of entanglement becomes more difficult due to the complexity of multipartite quantum states. There are some separability criteria for multipartite quantum states, which are natural generalizations of those for bipartite cases, for instance, the multipartite reduction criterion [22], the multipartite realignment criterion [23], the multipartite covariance matrix criterion [24], the multipartite criterion based on Bloch representations [25], and the multipartite criterion based on local uncertainty relations [26].
Quantum measurements are important tools in quantum information processing for extracting information from quantum states. Recently, the separability criteria based on quantum measurements are attracting increasing interest, since they are easily implemented in experiments. There are two broadly useful and popular classes of measurements known as mutually unbiased measurements (MUMs) [27] and positive-operator-valued measures (GSIC-POVMs) [28], which are natural extensions of the familiar mutually unbiased bases (MUBs) [29–32] and symmetric informationally complete positive-operator-valued measures (SIC-POVMs) [33–35], respectively. Moreover, unlike the MUBs and SIC-POVMs, the complete set of MUMs and GSIC-POVMs can be constructed explicitly in all finite dimensions. These measurements are widely applied to the detection of entanglement, and thus a series of efficient separability criteria have been proposed both for bipartite and multipartite systems to date [36–45]. It has been shown that the existing criteria based on MUMs and GSIC-POVMs are very powerful in detecting the entanglement of bipartite quantum states, however, how to construct efficient criteria for multipartite quantum states is in general a hard issue. Recently, Siudzińska introduced a broad class of symmetric measurements [46], which includes the MUMs and GSIC-POVMs as special cases. Moreover, possible applications of these symmetric measurements are further discussed in [46–48], such as the entropic uncertainty relations and entanglement detection.
In this work, based on the correlation matrices constructed from the (informationally complete) symmetric measurements [46], we first propose a separability criterion for arbitrary dimensional bipartite systems, which is more powerful than the corresponding one in [46]. Interestingly, when choosing MUMs or GSIC-POVMs as local measurements for two subsystems, it reproduces several criteria known before. Employing these symmetric measurements, we further derive two effective separability criteria in terms of tripartite correlation matrices.
The work is arranged as follows. In section 2 , we review the notion of symmetric measurements and their main features. In section 3 , we derive several separability criteria based on (informationally complete) symmetric measurements for bipartite and tripartite systems, respectively. We further illustrate the effectiveness of our method through several examples in section 4 . Finally, we conclude with a summary in section 5 . Detailed proofs of the main results (theorems 1 –3 ) are relegated to the appendix .
2. Symmetric measurements
In [46], a broad class of symmetric measurements were introduced. Here we recall the definition and main properties.
A collection {Sμ: μ = 1, 2, …, s} consisting of s POVMs Sμ = {Sμ,i: i = 1, 2, …, t} on ${{\mathbb{C}}}^{d}$ is called a set of symmetric measurements if
with a free parameter χ satisfying
The set ${ \mathcal S }(s,t)$ is informationally complete if and only if
| i | (i) $\mathrm{tr}({S}_{\mu ,i})=\tfrac{d}{t}$, |
| ii | (ii) $\mathrm{tr}({S}_{\mu ,i}^{2})=\chi ,$ |
| iii | (iii) $\mathrm{tr}({S}_{\mu ,i}{S}_{\mu ,j})=\tfrac{d-t\chi }{t(t-1)},\quad i\ne j$, |
| iv | (iv) $\mathrm{tr}({S}_{\mu ,i}{S}_{\nu ,j})=\tfrac{d}{{t}^{2}},\quad \mu \ne \nu $ |
$\begin{eqnarray}\displaystyle \frac{d}{{t}^{2}}\lt \chi \leqslant \min \left\{\displaystyle \frac{{d}^{2}}{{t}^{2}},\displaystyle \frac{d}{t}\right\}.\end{eqnarray}$
The set ${ \mathcal S }(s,t)=\{{S}_{\mu ,i}:\mu =1,2,\,\ldots ,\,s;i\,=\,1,2,\,\ldots ,\,t\}$ is called an (s, t)-POVM [46]. If χ = d2/t2, then the constituent POVMs are projective measurements (t ≥ d). This notion encapsulates several important and special instances:| 1. | (1)a complete set of MUBs for s = d + 1, t = d and χ = 1, |
| 2. | (2)an equiangular tight frame for s = 1, χ = d2/t2, t ∈ {d, d + 1, …, d2}[49, 50], |
| 3. | (3)an SIC-POVM for s = 1, t = d2 and χ = 1/d2. |
$\begin{eqnarray}s=\displaystyle \frac{{d}^{2}-1}{t-1}.\end{eqnarray}$
It has been shown that for any finite dimension d (d > 2), there exist at least four informationally complete (s, t)-POVMs depending on the choice of s and t:| 1. | (1)s = 1 and t = d2 (GSIC-POVMs), |
| 2. | (2)s = d + 1 and t = d (MUMs), |
| 3. | (3)s = d2 − 1 and t = 2, |
| 4. | (4)s = d − 1 and t = d + 2. |
For the case d = 2, there are only two possible choices of the parameters s and t: s = 1, t = 4 (GSIC-POVMs) and s = 3, t = 2 (MUMs), which provide a unified method to describe GSIC-POVMs and MUMs.
In [46], a general construction of informationally complete (s, t)-POVMs for arbitrary dimensions has been presented by an orthonormal Hermitian operator basis. Let {Hμ,i: μ = 1, 2, …, s; i = 1, 2, …, t − 1} be a set of (st − 1) Hermitian and traceless operators on ${{\mathbb{C}}}^{d}$ satisfying $\mathrm{tr}({H}_{\mu ,i}{H}_{{\mu }^{{\prime} },{i}^{{\prime} }})={\delta }_{\mu {\mu }^{{\prime} }}{\delta }_{{{ii}}^{{\prime} }},$ then the operators
$\begin{eqnarray}{S}_{\mu ,i}=\displaystyle \frac{1}{t}{{\bf{1}}}_{d}+{{rR}}_{\mu ,i},\quad \mu =1,2,\quad \cdots ,\quad s;i=1,2,\quad \cdots ,\quad t\end{eqnarray}$
form an informationally complete (s, t)-POVM. Here $\begin{eqnarray}{R}_{\mu ,i}=\left\{\begin{array}{ll}{H}_{\mu }-\sqrt{t}(\sqrt{t}+1){H}_{\mu ,i}, & i=1,2,\,\ldots ,\,t-1,\\ (\sqrt{t}+1){H}_{\mu }, & i=t,\end{array}\right.\end{eqnarray}$
and ${H}_{\mu }={\sum }_{i=1}^{t-1}{H}_{\mu ,i}.$ The parameter r should be chosen such that Sμ,i ≥ 0, which implies that $\begin{eqnarray*}-\displaystyle \frac{1}{t{\lambda }_{\max }}\leqslant r\leqslant \displaystyle \frac{1}{t| {\lambda }_{\min }| },\end{eqnarray*}$
where ${\lambda }_{\max }$ and ${\lambda }_{\min }$ are the maximum and minimum eigenvalues among all eigenvalues of the operators Rμ,i, respectively. The parameter χ is related to r and t via $\begin{eqnarray}\chi =\displaystyle \frac{d}{{t}^{2}}+{r}^{2}(t-1){\left(\sqrt{t}+1\right)}^{2},\end{eqnarray}$
and the maximum value ${\chi }_{\max }$ is obtained when $\begin{eqnarray*}r=\max \left\{\displaystyle \frac{1}{t{\lambda }_{\max }},\displaystyle \frac{1}{t| {\lambda }_{\min }| }\right\}.\end{eqnarray*}$
For instance, when d = 3, we can construct four informationally complete (s, t)-POVMs from the generalized Gell-Mann (GM) operator basis [51]. We list the range of the parameter r and the maximum value ${\chi }_{\max }$ in table 1.Table 1. The range of the parameter r and the maximum value ${\chi }_{\max }$ for the four informationally complete (s, t)-POVMs on ${{\mathbb{C}}}^{3}$. These measurements are constructed from the generalized GM operators. |
| (s, t)-POVM | r | ${\chi }_{\max }$ |
|---|---|---|
| (1, 9) | −0.0121 ≤ r ≤ 0.0129 | 0.0583 |
| (4, 3) | −0.1093 ≤ r ≤ 0.1220 | 0.5555 |
| (8, 2) | −0.2536 ≤ r ≤ 0.2536 | 0.1250 |
| (2, 5) | −0.0383 ≤ r ≤ 0.0388 | 0.1831 |
Let {Sμ,i: μ = 1, 2, …, s; i = 1, 2, …, t} be any informationally complete (s, t)-POVM on ${{\mathbb{C}}}^{d}$. It has been shown that for any state ρ on ${{\mathbb{C}}}^{d}$, it holds that [46]
$\begin{eqnarray}\sum _{\mu =1}^{s}\sum _{i=1}^{t}{\left(\mathrm{tr}({S}_{\mu ,i}\rho )\right)}^{2}=\displaystyle \frac{d({t}^{2}\chi -d)\mathrm{tr}({\rho }^{2})+{d}^{3}-{t}^{2}\chi }{{dt}(t-1)}\end{eqnarray}$
$\begin{eqnarray}\leqslant \,\displaystyle \frac{(d-1)({d}^{2}+{t}^{2}\chi )}{{dt}(t-1)},\end{eqnarray}$
where the equality holds when ρ is a pure state.3. Detecting entanglement via symmetric measurements
In this section, based on the (informationally complete) symmetric measurements, we present some separability criteria for arbitrary dimensional bipartite and tripartite systems, respectively.
3.1. Separability criteria for bipartite systems
Let $\mathcal{S}^a=\{S_{\mu,m}^a;\mu=1,2,...,s_a;m=1,2,...,t_a\}$ and $\mathcal{S}^b=\left\{S_{\nu, n}^b: \nu=1,2, \ldots, s_b ; n=1,2, \ldots, t_b\right\}$ be two sets of informationally complete, symmetric measurements on ${{\mathbb{C}}}^{{d}_{a}}$ and ${{\mathbb{C}}}^{{d}_{b}}$ with parameters χa and χb, respectively. For a given state ρ on ${{\mathbb{C}}}^{{d}_{a}}\otimes {{\mathbb{C}}}^{{d}_{b}}$, the correlation matrix associated with these data is defined as A Space for the proof, see the appendix .
$\begin{eqnarray}C(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b})=\left(\Space{0ex}{1ex}{0ex}\mathrm{tr}\left(({S}_{i}^{a}\otimes {S}_{j}^{b})(\rho -{\rho }_{a}\times {\rho }_{b})\right)\Space{0ex}{1ex}{0ex}\right),\end{eqnarray}$
for i = 1, 2, …, sata, j = 1, 2, …, sbtb, where $\begin{eqnarray}\begin{array}{ll}{S}_{i}^{a}={S}_{\mu ,m}^{a}, & i=(\mu -1){t}_{a}+m,\\ & \mu =1,2,\,\ldots ,\,{s}_{a},m=1,2,\,\ldots ,\,{t}_{a},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{ll}{S}_{j}^{b}={S}_{\nu ,n}^{b}, & j=(\nu -1){t}_{b}+n,\\ & \nu =1,2,\,\ldots ,\,{s}_{b},n=1,2,\,\ldots ,\,{t}_{b},\end{array}\end{eqnarray}$
and ${\rho }_{a}={\mathrm{tr}}_{b}\rho $, ${\rho }_{b}={\mathrm{tr}}_{a}\rho $ are reduced states. Next we establish a separability criterion in terms of the above correlation matrix.With the above notation, if ρ is separable, then
$\begin{eqnarray}\parallel C(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b}){\parallel }_{\mathrm{tr}}\leqslant \sqrt{{\upsilon }_{a}(1-\mathrm{tr}({\rho }_{a}^{2})){\upsilon }_{b}(1-\mathrm{tr}({\rho }_{b}^{2}))},\end{eqnarray}$
where $\parallel \cdot {\parallel }_{\mathrm{tr}}$ stands for the trace norm (i.e., the sum of singular values of a matrix), and $\begin{eqnarray}{\upsilon }_{a}=\displaystyle \frac{{t}_{a}^{2}{\chi }_{a}-{d}_{a}}{{t}_{a}({t}_{a}-1)},\qquad {\upsilon }_{b}=\displaystyle \frac{{t}_{b}^{2}{\chi }_{b}-{d}_{b}}{{t}_{b}({t}_{b}-1)}.\end{eqnarray}$
If we define the matrix
$\begin{eqnarray*}D(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b})=\left(\Space{0ex}{1ex}{0ex}\mathrm{tr}\left(({S}_{i}^{a}\otimes {S}_{j}^{b})\rho \right)\Space{0ex}{1ex}{0ex}\right),\end{eqnarray*}$
then the separability criterion in [46] states that any separable state ρ on ${{\mathbb{C}}}^{{d}_{a}}\otimes {{\mathbb{C}}}^{{d}_{b}}$ must satisfy $\begin{eqnarray}\parallel D(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b}){\parallel }_{\mathrm{tr}}\leqslant \sqrt{{\widetilde{\upsilon }}_{a}{\widetilde{\upsilon }}_{b}},\end{eqnarray}$
with $\begin{eqnarray}{\widetilde{\upsilon }}_{a}=\displaystyle \frac{({d}_{a}-1)({d}_{a}^{2}+{t}_{a}^{2}{\chi }_{a})}{{d}_{a}{t}_{a}({t}_{a}-1)},{\widetilde{\upsilon }}_{b}=\displaystyle \frac{({d}_{b}-1)({d}_{b}^{2}+{t}_{b}^{2}{\chi }_{b})}{{d}_{b}{t}_{b}({t}_{b}-1)}.\end{eqnarray}$
Now we show that, for the same symmetric measurements, the separability criterion in theorem 1 is more efficient than the criterion determined by inequality (13 ).
Let11 ) holds, then
$\begin{eqnarray*}{\boldsymbol{\alpha }}=\left(\begin{array}{c}{\alpha }_{1}\\ {\alpha }_{2}\\ \vdots \\ {\alpha }_{{s}_{a}{t}_{a}}\end{array}\right),\qquad {\boldsymbol{\beta }}=\left(\begin{array}{c}{\beta }_{1}\\ {\beta }_{2}\\ \vdots \\ {\beta }_{{s}_{b}{t}_{b}}\end{array}\right)\end{eqnarray*}$
with $\begin{eqnarray*}\begin{array}{rcl}{\alpha }_{i} & = & \mathrm{tr}({S}_{i}^{a}{\rho }_{a}),\qquad i=1,2,\,\ldots ,\,{s}_{a}{t}_{a},\\ {\beta }_{j} & = & \mathrm{tr}({S}_{j}^{b}{\rho }_{b}),\qquad j=1,2,\,\ldots ,\,{s}_{b}{t}_{b},\end{array}\end{eqnarray*}$
then $C(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b})=D(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b})-{\boldsymbol{\alpha }}{{\boldsymbol{\beta }}}^{{\rm{T}}}$ (T stands for the transpose). Assuming that inequality ( $\begin{eqnarray*}\begin{array}{l}\parallel D(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b}){\parallel }_{\mathrm{tr}}\\ \,\leqslant \parallel C(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b}){\parallel }_{\mathrm{tr}}+\parallel {\boldsymbol{\alpha }}{{\boldsymbol{\beta }}}^{{\rm{T}}}{\parallel }_{\mathrm{tr}}\\ \,\leqslant \sqrt{{\upsilon }_{a}(1-\mathrm{tr}({\rho }_{a}^{2})){\upsilon }_{b}(1-\mathrm{tr}({\rho }_{b}^{2}))}+\parallel {\boldsymbol{\alpha }}\parallel \parallel {\boldsymbol{\beta }}\parallel \\ =\sqrt{{\widetilde{\upsilon }}_{a}-\sum _{i=1}^{{s}_{a}{t}_{a}}{\left(\mathrm{tr}{S}_{i}^{a}{\rho }_{a}\right)}^{2}}\sqrt{{\widetilde{\upsilon }}_{b}-\sum _{j=1}^{{s}_{b}{t}_{b}}{\left(\mathrm{tr}{S}_{j}^{b}{\rho }_{b}\right)}^{2}}\\ \qquad +\parallel {\boldsymbol{\alpha }}\parallel \parallel {\boldsymbol{\beta }}\parallel \\ =\sqrt{{\widetilde{\upsilon }}_{a}-\sum _{i=1}^{{s}_{a}{t}_{a}}{\left(\mathrm{tr}{S}_{i}^{a}{\rho }_{a}\right)}^{2}}\sqrt{{\widetilde{\upsilon }}_{b}-\sum _{j=1}^{{s}_{b}{t}_{b}}{\left(\mathrm{tr}{S}_{j}^{b}{\rho }_{b}\right)}^{2}}\\ \qquad +\sqrt{\sum _{i=1}^{{s}_{a}{t}_{a}}{\left(\mathrm{tr}{S}_{i}^{a}{\rho }_{a}\right)}^{2}}\sqrt{\sum _{j=1}^{{s}_{b}{t}_{b}}{\left(\mathrm{tr}{S}_{j}^{b}{\rho }_{b}\right)}^{2}}\\ \,\leqslant \sqrt{{\widetilde{\upsilon }}_{a}\widetilde{{\upsilon }_{b}}},\end{array}\end{eqnarray*}$
where ∥ · ∥ denotes the Euclidean norm of vectors and the third inequality is due to $\sqrt{c}\sqrt{d}+\sqrt{e}\sqrt{f}\leqslant \sqrt{c+e}\sqrt{d+f}$ for any non-negative c, d, e, f.Consider the simplest case da = db = d, sa = sb = s, ta = tb = t and χa = χb = χ, then from inequalities (11 ) and (13 ) we conclude that any separable state ρ satisfies
$\begin{eqnarray}\begin{array}{l}\sum _{\mu =1}^{s}\sum _{i=1}^{t}\left|\mathrm{tr}\left(({S}_{\mu ,i}^{a}\otimes {S}_{\mu ,i}^{b})(\rho -{\rho }_{a}\otimes {\rho }_{b})\right)\right|\\ \,\leqslant \displaystyle \frac{{t}^{2}\chi -d}{t(t-1)}\sqrt{(1-\mathrm{tr}({\rho }_{a}^{2}))(1-\mathrm{tr}({\rho }_{b}^{2}))},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\sum _{\mu =1}^{s}\sum _{i=1}^{t}\mathrm{tr}\left(({S}_{\mu ,i}^{a}\otimes {S}_{\mu ,i}^{b})\rho \right)\leqslant \displaystyle \frac{(d-1)({d}^{2}+{t}^{2}\chi )}{{dt}(t-1)},\end{eqnarray}$
by the inequality ${\sum }_{i=1}^{k}| {c}_{{ii}}| \leqslant \parallel C{\parallel }_{\mathrm{tr}}$ for any matrix $C\,=({c}_{{ij}})\in {{\mathbb{C}}}^{k\times k}$.3.2. Separability criteria for tripartite systems
Given a tripartite state ρ on ${{\mathbb{C}}}^{{d}_{a}}\otimes {{\mathbb{C}}}^{{d}_{b}}\otimes {{\mathbb{C}}}^{{d}_{c}}$, consider the informationally complete, symmetric measurements ${{ \mathcal S }}^{a}=\{{S}_{\mu ,m}^{a}:\mu =1,2,\,\ldots ,\,{s}_{a};m=1,2,\,\ldots ,\,{t}_{a}\}$ with the parameter χa, ${{ \mathcal S }}^{b}=\{{S}_{\nu ,n}^{b}:\nu =1,2,\,\ldots ,\,{s}_{b};n=1,2,\,\ldots ,\,{t}_{b}\}$ with the parameter χb, and ${{ \mathcal S }}^{c}=\{{S}_{\omega ,l}^{c}:\omega =1,2,\,\ldots ,\,{s}_{c};l=1,2,\,\ldots ,{t}_{c}\}$ with the parameter χc for the subsystems a, b and c, respectively. Let9 ) and (10 ), and 14 ), and For the proof, see the appendix .
$\begin{eqnarray*}{c}_{{ijk}}=\mathrm{tr}\left(({S}_{i}^{a}\otimes {S}_{j}^{b}\otimes {S}_{k}^{c})\rho \right)\end{eqnarray*}$
for i = 1, 2, …, sata, j = 1, 2, …, sbtb and k = 1, 2, …, sctc, where Sai, Sbj are defined by equations ( $\begin{eqnarray}{S}_{k}^{c}={S}_{\omega ,l}^{c}\end{eqnarray}$
with k = (&ohgr; − 1)tc + l, &ohgr; = 1, 2, …, sc, l = 1, 2, …, tc. We further define three correlation matrices ${E}^{\underline{a}{bc}}$, ${E}^{a\underline{b}c}$ and ${E}^{{ab}\underline{c}}$ among ${{ \mathcal S }}^{a}$, ${{ \mathcal S }}^{b}$ and ${{ \mathcal S }}^{c}$ by $\begin{eqnarray}{E}^{\underline{a}{bc}}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})={E}^{(1)}\oplus {E}^{(2)}\oplus \cdots \oplus \,{E}^{({s}_{a}{t}_{a})},\end{eqnarray}$
with ${E}^{(i)}=\left({e}_{{jk}}^{(i)}\right)$ and ${e}_{{jk}}^{(i)}={c}_{{ijk}}$ for i = 1, 2, …, sata, $\begin{eqnarray}{E}^{a\underline{b}c}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})={\widetilde{E}}^{(1)}\oplus {\widetilde{E}}^{(2)}\oplus \cdots \oplus \,{\widetilde{E}}^{({s}_{b}{t}_{b})},\end{eqnarray}$
with ${\widetilde{E}}^{(j)}=\left({\tilde{e}}_{{ik}}^{(j)}\right)$ and ${\tilde{e}}_{{ik}}^{(j)}={c}_{{ijk}}$ for j = 1, 2, …, sbtb, $\begin{eqnarray}{E}^{{ab}\underline{c}}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})={\hat{E}}^{(1)}\oplus {\hat{E}}^{(2)}\oplus \cdots \oplus \,{\hat{E}}^{({s}_{c}{t}_{c})},\end{eqnarray}$
with ${\hat{E}}^{(k)}=\left({\hat{e}}_{{ij}}^{(k)}\right)$ and ${\hat{e}}_{{ij}}^{(k)}={c}_{{ijk}}$ for k = 1, 2, …, sctc.With the above data, if a tripartite state ρ on ${{\mathbb{C}}}^{{d}_{a}}\otimes {{\mathbb{C}}}^{{d}_{b}}\otimes {{\mathbb{C}}}^{{d}_{c}}$ is fully separable, then
$\begin{eqnarray}{E}^{\underline{a}{bc}}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})\leqslant {s}_{a}\sqrt{{\widetilde{\upsilon }}_{b}{\widetilde{\upsilon }}_{c}},\end{eqnarray}$
$\begin{eqnarray}{E}^{a\underline{b}c}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})\leqslant {s}_{b}\sqrt{{\widetilde{\upsilon }}_{a}{\widetilde{\upsilon }}_{c}},\end{eqnarray}$
$\begin{eqnarray}{E}^{{ab}\underline{c}}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})\leqslant {s}_{c}\sqrt{{\widetilde{\upsilon }}_{a}{\widetilde{\upsilon }}_{b}},\end{eqnarray}$
where ${\widetilde{\upsilon }}_{a}$ and ${\widetilde{\upsilon }}_{b}$ are defined by equation ( $\begin{eqnarray}{\widetilde{\upsilon }}_{c}=\displaystyle \frac{({d}_{c}-1)({d}_{c}^{2}+{t}_{c}^{2}{\chi }_{c})}{{d}_{c}{t}_{c}({t}_{c}-1)}.\end{eqnarray}$
We can extend the bipartite separability criterion determined by equation (13 ) to the tripartite case: define three correlation matrices ${G}^{\underline{a}{bc}}$,${G}^{a\underline{b}c}$ and ${G}^{{ab}\underline{c}}$ among ${{ \mathcal S }}^{a}$, ${{ \mathcal S }}^{b}$ and ${{ \mathcal S }}^{c}$ by
$\begin{eqnarray}{G}^{\underline{a}{bc}}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})=\left(\Space{0ex}{1ex}{0ex}~{g}_{i,{s}_{c}~{t}_{c}~(j-1)~+k}~\Space{0ex}{1.6ex}{0ex}\right),\end{eqnarray}$
with ${g}_{i,~{s}_{c}~{t}_{c}~(j-1)+k}~=~{c}_{{~i~j~k}}$, $\begin{eqnarray}{G}^{a\underline{b}c}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})=\left(\Space{0ex}{1ex}{0ex}~{\tilde{g}}_{j,~{s}_{c}~{t}_{c}~(i-1)+k}\Space{0ex}{1ex}{0ex}~\right),\end{eqnarray}$
with ${\tilde{g}}_{j,~{s}_{c}~{t}_{c}~(i-1)+k}~=~{c}_{{i~j~k}}$, $\begin{eqnarray}{G}^{{ab}\underline{c}}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})=\left(\Space{0ex}{1ex}{0ex}~{\hat{g}}_{k,~{s}_{b}~{t}_{b}~(i-1)+j}~\Space{0ex}{1ex}{0ex}\right),\end{eqnarray}$
with ${\hat{g}}_{k,~{s}_{b}~{t}_{b}~(i-1)+j}~=~{c}_{{i~j~k}}$.In the following, we present another tripartite separability criterion based on the above correlation matrix. 2 . For the proof, see the appendix .
If a tripartite state ρ on ${{\mathbb{C}}}^{{d}_{a}}\otimes {{\mathbb{C}}}^{{d}_{b}}\otimes {{\mathbb{C}}}^{{d}_{c}}$ is fully separable, then
$\begin{eqnarray}{G}^{\underline{a}{bc}}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})\leqslant \sqrt{{\widetilde{\upsilon }}_{a}{\widetilde{\upsilon }}_{b}{\widetilde{\upsilon }}_{c}},\end{eqnarray}$
$\begin{eqnarray}{G}^{a\underline{b}c}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})\leqslant \sqrt{{\widetilde{\upsilon }}_{a}{\widetilde{\upsilon }}_{b}{\widetilde{\upsilon }}_{c}},\end{eqnarray}$
$\begin{eqnarray}{G}^{{ab}\underline{c}}(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b},{{ \mathcal S }}^{c})\leqslant \sqrt{{\widetilde{\upsilon }}_{a}{\widetilde{\upsilon }}_{b}{\widetilde{\upsilon }}_{c}},\end{eqnarray}$
where ${\widetilde{\upsilon }}_{a}$, ${\widetilde{\upsilon }}_{b}$ and ${\widetilde{\upsilon }}_{c}$ is given as in theorem 4. Illustration
In what follows, we provide some examples to illustrate the efficiency of the criteria in theorems 1 –3 .
Example 1. Consider the d × d-dimensional isotropic states
$\begin{eqnarray}{\rho }_{\mathrm{iso}}(p)=\displaystyle \frac{1-p}{{d}^{2}}{{\bf{1}}}_{{d}^{2}}+p| {\psi }_{d}^{+}\rangle \langle {\psi }_{d}^{+}| ,\qquad 0\leqslant p\leqslant 1,\end{eqnarray}$
where $| {\psi }_{d}^{+}\rangle ={\sum }_{i=0}^{d-1}| {ii}\rangle /\sqrt{d}$. The isotropic states ρiso(p) are entangled if and only if p > 1/(d + 1) [11].Let ${ \mathcal S }=\{{S}_{\mu ,m}:\mu =1,2,\,\ldots ,\,s;m=1\,2,\,\ldots ,\,t\}$ be a set of informationally complete, symmetric measurements on ${{\mathbb{C}}}^{d}$ with parameter χ, then the complex conjugate (relative to the computational basis of ${{\mathbb{C}}}^{d}$) ${{ \mathcal S }}^{* }=\{{S}_{\mu ,m}^{* }:\mu =1,2,\,\ldots ,\,s;m=1,2,\,\ldots ,\,t\}$ also is a set of informationally complete, symmetric measurements on ${{\mathbb{C}}}^{d}.$ We use them to detect the separability of ρiso(p).
After simple computations, we get the correlation matrix1 , we conclude that ρiso(p) are entangled for p > 1/(d + 1). Thus theorem 1 can detect all the entanglement of the isotropic states for any dimension d.
$\begin{eqnarray*}C({\rho }_{\mathrm{iso}}(p)| { \mathcal S },{{ \mathcal S }}^{* })=Z\oplus Z\oplus \cdots \oplus \,Z,\end{eqnarray*}$
where the number of summands is s, and Z = (zij) is a t × t matrix with the entries $\begin{eqnarray*}{z}_{{ij}}=\left\{\begin{array}{ll}\tfrac{p({t}^{2}\chi -d)}{{{dt}}^{2}}, & i=j\\ \tfrac{p(d-{t}^{2}\chi )}{{{dt}}^{2}(t-1)}, & i\ne j.\end{array}\right.\end{eqnarray*}$
Consequently, the matrix Z is real and non-negative definite, which has the spectrum $\begin{eqnarray*}\displaystyle \frac{p({t}^{2}\chi -d)}{{dt}(t-1)}\left\{1,1,\cdots ,1,0\right\}.\end{eqnarray*}$
Hence, $\begin{eqnarray*}\begin{array}{rcl}\parallel C({\rho }_{\mathrm{iso}}(p))| { \mathcal S },{{ \mathcal S }}^{* }{\parallel }_{\mathrm{tr}} & = & s(t-1)\displaystyle \frac{p({t}^{2}\chi -d)}{{dt}(t-1)}\\ & = & \displaystyle \frac{{ps}({t}^{2}\chi -d)}{{dt}}.\end{array}\end{eqnarray*}$
Moreover, we have $\mathrm{tr}({\rho }_{a}^{2})=\mathrm{tr}({\rho }_{b}^{2})=1/d$. Employing theorem Example 2. Consider the d-dimensional Werner states [52]11 ), implies that ρw(f) are entangled for $-1\leqslant f\lt \tfrac{2}{d}-1$. Thus theorem 1 can recognize the entanglement of ρw(f) for $-1\leqslant f\lt \tfrac{2}{d}-1$, which coincides with the result in [39].
$\begin{eqnarray}{\rho }_{{\rm{w}}}(f)=\displaystyle \frac{d-f}{{d}^{3}-d}{{\bf{1}}}_{{d}^{2}}+\displaystyle \frac{{df}-1}{{d}^{3}-d}V,\qquad -1\leqslant f\leqslant 1,\end{eqnarray}$
where $V={\sum }_{i,j=0}^{d-1}| {ij}\rangle \langle {ji}| $. It has been shown that ρw(f) are entangled if and only if −1 ≤ f ≤ 0. Let ${ \mathcal S }=\{{S}_{\mu ,m}:\mu =1,2,\,\ldots ,\,s;m=1,2,\,\ldots ,\,t\}$ be a set of informationally complete, symmetric measurements on ${{\mathbb{C}}}^{d}$ with parameter χ. By direct calculation, we have $\begin{eqnarray*}\parallel C({\rho }_{{\rm{w}}}(f)| { \mathcal S },{ \mathcal S }){\parallel }_{\mathrm{tr}}=\displaystyle \frac{s| {df}-1| ({t}^{2}\chi -d)}{t({d}^{3}-d)},\end{eqnarray*}$
which, via inequality (Example 3. Consider the family of 3 × 3 states
$\begin{eqnarray}\rho (\tau ,q)=q\rho (\tau )+(1-q)\displaystyle \frac{{{\bf{1}}}_{9}}{9},\end{eqnarray}$
where 0 < τ < 1 and 0 ≤ q ≤ 1, $\begin{eqnarray*}\rho (\tau )=\displaystyle \frac{1}{8\tau +1}\left(\begin{array}{ccccccccc}\tau & 0 & 0 & 0 & \tau & 0 & 0 & 0 & \tau \\ 0 & \tau & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \tau & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \tau & 0 & 0 & 0 & 0 & 0\\ \tau & 0 & 0 & 0 & \tau & 0 & 0 & 0 & \tau \\ 0 & 0 & 0 & 0 & 0 & \tau & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \tfrac{1+\tau }{2} & 0 & \tfrac{\sqrt{1-{\tau }^{2}}}{2}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \tau & 0\\ \tau & 0 & 0 & 0 & \tau & 0 & \tfrac{\sqrt{1-{\tau }^{2}}}{2} & 0 & \tfrac{1+\tau }{2}\end{array}\right)\end{eqnarray*}$
is a 3 × 3 bound entangled state given by Horodecki [10].We have shown that inequality (11 ) is stronger than inequality (13 ). Now we specifically compare the detection power of these two criteria using the informationally complete (8, 2)-POVM, which is constructed from the generalized GM matrices with the parameter χ = 0.125. In figure 1, we illustrate the ranges of the parameters q and τ for which the entanglement of ρ(τ, p) is detected by inequalities (11 ) and (13 ), respectively. It is clear that theorem 1 can detect more entangled states than that in [46] for the same measurement settings.
Figure 1. Comparison of the detection power between inequalities ( |
Example 4. Consider the 3 × 3 states given by1 with (2, 5)-POVM (the parameter χ = 0.1831), we get ρ(h) are entangled for 0 ≤ h < 1 and 5 < h ≤ 6. Moreover, we find that the same entangled conditions can be obtained with (4, 3)-POVM (MUMs with χ = 0.5555) and (1, 9)-POVM (GSIC-POVMs with χ = 0.0583) for these states. Therefore, in this case, the detection power of theorem 1 with three different symmetric measurements coincides and is more powerful than the PPT criterion.
$\begin{eqnarray}\rho (h)=\displaystyle \frac{1}{7}| {\psi }_{3}^{+}\rangle \langle {\psi }_{3}^{+}| +\displaystyle \frac{h}{7}{\rho }_{1}+\displaystyle \frac{6-h}{7}{\rho }_{2},\qquad 0\leqslant h\leqslant 1,\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}| {\psi }_{3}^{+}\rangle & = & \displaystyle \frac{| 00\rangle +| 11\rangle +| 22\rangle }{\sqrt{3}},\\ {\rho }_{1} & = & \displaystyle \frac{| 02\rangle \langle 02| +| 10\rangle \langle 10| +| 21\rangle \langle 21| }{3},\\ {\rho }_{2} & = & \displaystyle \frac{| 01\rangle \langle 01| +| 12\rangle \langle 12| +| 20\rangle \langle 20| }{3}.\end{array}\end{eqnarray*}$
From PPT criterion [8], ρ(h) are entangled for 0 ≤ h < 0.1716 and 5.8284 < h ≤ 6. Now employing theorem Example 5. Consider the three-qubit state $W=(| 100\rangle +| 010\rangle +| 001\rangle )/\sqrt{3}$ mixed with the identity operator [53]2 with three MUMs (i.e., s = 3, t = 2), we conclude that ρ(x) are entangled for 0 ≤ x < 4/7 ≈ 0.5714. In a similar way, theorem 2 with GSIC-POVMs (i.e., s = 1, t = 4) can also identify the entanglement of ρ(x) for 0 ≤ x < 4/7. Numerical computations show that the criterion in [44] based on MUMs cannot detect entanglement of ρ(x). Thus, our method in theorem 2 is more powerful than the criterion based on local uncertainty relations, and that in [44] for these states.
$\begin{eqnarray}\rho (x)=\displaystyle \frac{x}{8}{{\bf{1}}}_{8}+(1-x)| {\rm{W}}\rangle \langle {\rm{W}}| ,\qquad 0\leqslant x\leqslant 1.\end{eqnarray}$
It has been shown that ρ(x) are entangled for 0 ≤ x < 0.566 from the local uncertainty relation criterion in [26]. Now employing theorem Example 6. Consider the three-qutrit states
$\begin{eqnarray}\rho (y)=\displaystyle \frac{1-y}{27}{{\bf{1}}}_{27}+y| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}| ,\qquad 0\leqslant y\leqslant 1\end{eqnarray}$
with ∣Φ⟩ = $\tfrac{1}{\sqrt{6}}(| 002\rangle $ + ∣020⟩ + ∣200⟩ + ∣112⟩ + ∣121⟩ + ∣211⟩).Now we compare the detection power between theorems 2 and 3 . Using theorem 2 with GSIC-POVMs, we conclude that ρ(y) are entangled for 0.3208 ≤ y ≤ 1. Moreover, theorem 3 can detect the entanglement of ρ(y) for 0.3902 ≤ y ≤ 1. Thus theorems 2 and 3 are efficient in detecting the entanglement for certain multipartite systems, and theorem 2 is more efficient than theorem 3 in this case.
5. Summary
In this paper, a class of symmetric measurements introduced in [46], which include MUMs and GSIC-POVMs as special cases, have been used to study the problem of quantum separability. Based on the correlation matrices constructed from these (informationally complete) symmetric measurements, we have derived a separability criterion for bipartite systems, which enhances a result in [46]. Furthermore, using tripartite correlation matrices defined via these symmetric measurements, two effective separability criteria have been proposed to detect the entanglement of tripartite quantum states. We have illustrated the detection power of these separability criteria via several examples, and in particular, we have shown that these criteria can detect some entangled states which cannot be detected by some other methods.
We emphasize that theorems 1 –3 depend on the choice of (informationally complete) symmetric measurements. Thus it would be interesting to analyze how to choose and construct suitable symmetric measurements such that these criteria can detect more entangled states. Moreover, it should be noted that theorem 3 can be viewed as a natural generalization of the separability criterion for the bipartite scenario presented in [46]. Nevertheless, the problem of how to extend our result (i.e., theorem 1 ) to multipartite systems needs to be further studied. Also, the phenomenon of genuine multipartite entanglement is attracting increasing interest due to its physical significance and applications. It is desirable to explore this topic via general symmetric measurements.
Acknowledgments
This work was supported by the National Key R&D Program of China, Grant No. 2020YFA0712700, and the National Natural Science Foundation of China, Grant Nos. 11875317 and 61833010.
Appendix
Here we present the detailed proofs of theorems 1 –3 .
A.1. Proof of theorem 1
Since ρ is separable, then it can be written asA1 ), we get6 ) and (7 ) are used in the last inequality.
$\begin{eqnarray*}\rho =\sum _{\xi }{p}_{\xi }{\rho }_{\xi }^{a}\otimes {\rho }_{\xi }^{b}\end{eqnarray*}$
with 0 ≤ pξ ≤ 1 and ∑ξpξ = 1, and ${\rho }_{a}={\sum }_{\xi }{p}_{\xi }{\rho }_{\xi }^{a}$, ${\rho }_{b}={\sum }_{\xi }{p}_{\xi }{\rho }_{\xi }^{b},$ $\begin{eqnarray*}\rho -{\rho }_{a}\otimes {\rho }_{b}=\sum _{\xi }{p}_{\xi }({\rho }_{\xi }^{a}-{\rho }_{a})\otimes ({\rho }_{\xi }^{b}-{\rho }_{b}).\end{eqnarray*}$
Consequently, for the symmetric measurements $\begin{eqnarray*}\begin{array}{rcl}{{ \mathcal S }}^{a} & = & \{{S}_{i}^{a}:i=1,2,\,\ldots ,\,{s}_{a}{t}_{a}\},\\ {{ \mathcal S }}^{b} & = & \{{S}_{j}^{b}:j=1,2,\,\ldots ,\,{s}_{b}{t}_{b}\},\end{array}\end{eqnarray*}$
we have $\begin{eqnarray}C(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b})=\sum _{\xi }{p}_{\xi }{{\boldsymbol{\eta }}}_{\xi }{{\boldsymbol{\gamma }}}_{\xi }^{{\rm{T}}}\end{eqnarray}$
with $\begin{eqnarray*}{{\boldsymbol{\eta }}}_{\xi }=\left(\begin{array}{c}{\eta }_{1}\\ {\eta }_{2}\\ \vdots \\ {\eta }_{{s}_{a}{t}_{a}}\end{array}\right),\qquad {{\boldsymbol{\gamma }}}_{\xi }=\left(\begin{array}{c}{\gamma }_{1}\\ {\gamma }_{2}\\ \vdots \\ {\gamma }_{{s}_{b}{t}_{b}}\end{array}\right)\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{rcl}{\eta }_{i} & = & \mathrm{tr}({S}_{i}^{a}{\rho }_{\xi }^{a})-\mathrm{tr}({S}_{i}^{a}{\rho }_{a}),\qquad i=1,2,\,\ldots ,\,{s}_{a}{t}_{a},\\ {\gamma }_{j} & = & \mathrm{tr}({S}_{j}^{b}{\rho }_{\xi }^{b})-\mathrm{tr}({S}_{j}^{b}{\rho }_{b}),\qquad j=1,2,\,\ldots ,\,{s}_{b}{t}_{b}.\end{array}\end{eqnarray*}$
From equation ( $\begin{eqnarray*}\begin{array}{l}{\parallel C\left(\rho | {{ \mathcal S }}^{a},{{ \mathcal S }}^{b}\right)\parallel }_{\mathrm{tr}}\\ \,\leqslant \sum _{\xi }{p}_{\xi }{\parallel {{\boldsymbol{\eta }}}_{\xi }{{\boldsymbol{\gamma }}}_{\xi }^{{\rm{T}}}\parallel }_{\mathrm{tr}}\\ =\sum _{\xi }{p}_{\xi }\parallel {{\boldsymbol{\eta }}}_{\xi }\parallel \parallel {{\boldsymbol{\gamma }}}_{\xi }\parallel \\ \,\leqslant \sqrt{\sum _{\xi }{p}_{\xi }\parallel {{\boldsymbol{\eta }}}_{\xi }{\parallel }^{2}}\sqrt{\sum _{\xi }{p}_{\xi }\parallel {{\boldsymbol{\gamma }}}_{\xi }{\parallel }^{2}}\\ =\sqrt{\sum _{\xi }{p}_{\xi }\sum _{i=1}^{{s}_{a}{t}_{a}}{\left(\mathrm{tr}({S}_{i}^{a}{\rho }_{\xi }^{a})-\mathrm{tr}({S}_{i}^{a}{\rho }_{a})\right)}^{2}}\\ \,\times \sqrt{\sum _{\xi }{p}_{\xi }\sum _{j=1}^{{s}_{b}{t}_{b}}{\left(\mathrm{tr}({S}_{j}^{b}{\rho }_{\xi }^{b})-\mathrm{tr}({S}_{j}^{b}{\rho }_{b})\right)}^{2}}\\ =\sqrt{\sum _{\xi }\sum _{i=1}^{{s}_{a}{t}_{a}}{p}_{\xi }{\left(\mathrm{tr}{S}_{i}^{a}{\rho }_{\xi }^{a}\right)}^{2}-\sum _{i=1}^{{s}_{a}{t}_{a}}{\left(\mathrm{tr}{S}_{i}^{a}{\rho }_{a}\right)}^{2}}\\ \ \ \ \times \sqrt{\sum _{\xi }\sum _{j=1}^{{s}_{b}{t}_{b}}{p}_{\xi }{\left(\mathrm{tr}{S}_{j}^{b}{\rho }_{\xi }^{b}\right)}^{2}-\sum _{j=1}^{{s}_{b}{t}_{b}}{\left(\mathrm{tr}{S}_{j}^{b}{\rho }_{b}\right)}^{2}}\\ \,\leqslant \sqrt{\displaystyle \frac{({d}_{a}-1)({d}_{a}^{2}+{t}_{a}^{2}{\chi }_{a})}{{d}_{a}{t}_{a}({t}_{a}-1)}-\displaystyle \frac{{d}_{a}({t}_{a}^{2}{\chi }_{a}-{d}_{a})\mathrm{tr}({\rho }_{a}^{2})+{d}_{a}^{3}-{t}_{a}^{2}{\chi }_{a}}{{d}_{a}{t}_{a}({t}_{a}-1)}}\\ \ \ \ \times \sqrt{\displaystyle \frac{({d}_{b}-1)({d}_{b}^{2}+{t}_{b}^{2}{\chi }_{b})}{{d}_{b}{t}_{b}({t}_{b}-1)}-\displaystyle \frac{{d}_{b}({t}_{b}^{2}{\chi }_{b}-{d}_{b})\mathrm{tr}({\rho }_{b}^{2})+{d}_{b}^{3}-{t}_{b}^{2}{\chi }_{b}}{{d}_{b}{t}_{b}({t}_{b}-1)}}\\ =\sqrt{{\upsilon }_{a}(1-\mathrm{tr}({\rho }_{a}^{2})){\upsilon }_{b}(1-\mathrm{tr}({\rho }_{b}^{2}))},\end{array}\end{eqnarray*}$
where the first equality is due to $\parallel | \psi \rangle \langle \varphi | {\parallel }_{\mathrm{tr}}=\parallel | \psi \rangle \parallel \parallel | \varphi \rangle \parallel $ for any vectors ∣ψ⟩ and ∣φ⟩, the second inequality follows from the Cauchy-Schwarz inequality, and equations (A.2. Proof of theorem 2
First, any tripartite separable state ρ can be decomposed as7 ) is used in the last inequality.
$\begin{eqnarray}\rho =\sum _{\zeta }{p}_{\zeta }{\rho }_{\zeta }^{a}\otimes {\rho }_{\zeta }^{b}\otimes {\rho }_{\zeta }^{c}\end{eqnarray}$
with 0 ≤ pζ ≤ 1 and ∑ζpζ = 1. Thus, we can derive $\begin{eqnarray*}{E}^{\underline{a}{bc}}({\rho }_{\zeta }^{a}\otimes {\rho }_{\zeta }^{b}\otimes {\rho }_{\zeta }^{c})={E}_{\zeta }^{a}\otimes {{\boldsymbol{e}}}_{\zeta }^{b}{\left({{\boldsymbol{e}}}_{\zeta }^{c}\right)}^{{\rm{T}}},\end{eqnarray*}$
where $\begin{eqnarray*}{E}_{\zeta }^{a}=\left(\begin{array}{cccc}{e}_{1}^{a} & 0 & \cdots & 0\\ 0 & {e}_{2}^{a} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & {e}_{{s}_{a}{t}_{a}}^{a}\end{array}\right),\end{eqnarray*}$
$\begin{eqnarray*}{{\boldsymbol{e}}}_{\zeta }^{b}=\left(\begin{array}{c}{e}_{1}^{b}\\ {e}_{2}^{b}\\ \vdots \\ {e}_{{s}_{b}{t}_{b}}^{b}\end{array}\right),\qquad {{\boldsymbol{e}}}_{\zeta }^{c}=\left(\begin{array}{c}{e}_{1}^{c}\\ {e}_{2}^{c}\\ \vdots \\ {e}_{{s}_{c}{t}_{c}}^{c}\end{array}\right)\end{eqnarray*}$
with $\begin{eqnarray*}\begin{array}{rcl}{e}_{i}^{a} & = & \mathrm{tr}({S}_{i}^{a}{\rho }_{\zeta }^{a}),\qquad i=1,2,\,\ldots ,\,{s}_{a}{t}_{a},\\ {e}_{j}^{b} & = & \mathrm{tr}({S}_{j}^{b}{\rho }_{\zeta }^{b}),\qquad j=1,2,\,\ldots ,\,{s}_{b}{t}_{b},\\ {e}_{k}^{c} & = & \mathrm{tr}({S}_{k}^{c}{\rho }_{\zeta }^{c}),\qquad k=1,2,\,\ldots ,\,{s}_{c}{t}_{c}.\end{array}\end{eqnarray*}$
Then we get $\begin{eqnarray*}\begin{array}{rcl}\parallel {E}^{\underline{a}{bc}}(\rho ){\parallel }_{\mathrm{tr}} & \leqslant & \sum _{\zeta }{p}_{\zeta }\parallel {E}^{\underline{a}{bc}}({\rho }_{\zeta }^{a}\otimes {\rho }_{\zeta }^{b}\otimes {\rho }_{\zeta }^{c}){\parallel }_{\mathrm{tr}}\\ & = & \sum _{\zeta }{p}_{\zeta }\parallel {E}_{\zeta }^{a}\otimes {{\boldsymbol{e}}}_{\zeta }^{b}{\left({{\boldsymbol{e}}}_{\zeta }^{c}\right)}^{{\rm{T}}}{\parallel }_{\mathrm{tr}}\\ & = & \sum _{\zeta }{p}_{\zeta }\parallel {E}_{\zeta }^{a}{\parallel }_{\mathrm{tr}}\parallel {{\boldsymbol{e}}}_{\zeta }^{b}\parallel \parallel {{\boldsymbol{e}}}_{\zeta }^{c}\parallel \\ & = & \sum _{\zeta }{p}_{\zeta }\mathrm{tr}({E}_{\zeta }^{a})\parallel {{\boldsymbol{e}}}_{\zeta }^{b}\parallel \parallel {{\boldsymbol{e}}}_{\zeta }^{c}\parallel \\ & = & \sum _{\zeta }{p}_{\zeta }\mathrm{tr}\left(\left(\sum _{i=1}^{{s}_{a}{t}_{a}}{S}_{i}^{a}\right){\rho }_{\zeta }^{a}\right)\parallel {{\boldsymbol{e}}}_{\zeta }^{b}\parallel \parallel {{\boldsymbol{e}}}_{\zeta }^{c}\parallel \\ & = & \sum _{\zeta }{p}_{\zeta }\mathrm{tr}\left({s}_{a}{{\bf{1}}}_{{d}_{a}}{\rho }_{\zeta }^{a}\right)\parallel {{\boldsymbol{e}}}_{\zeta }^{b}\parallel \parallel {{\boldsymbol{e}}}_{\zeta }^{c}\parallel \\ & = & \sum _{\zeta }{p}_{\zeta }{s}_{a}\parallel {{\boldsymbol{e}}}_{\zeta }^{b}\parallel \parallel {{\boldsymbol{e}}}_{\zeta }^{c}\parallel \\ & \leqslant & {s}_{a}\sqrt{{\widetilde{\upsilon }}_{b}{\widetilde{\upsilon }}_{c}},\end{array}\end{eqnarray*}$
where the fifth equality follows that $\{{S}_{i}^{a}:i=1,2,\,\ldots ,\,{s}_{a}{t}_{a}\}$ consists of sa POVMs, and the equation (Similarly, we have21 )–(23 ) hold.
$\begin{eqnarray*}\parallel {E}^{a\underline{b}c}(\rho ){\parallel }_{\mathrm{tr}}\leqslant {s}_{b}\sqrt{{\widetilde{\upsilon }}_{a}{\widetilde{\upsilon }}_{c}},\quad \parallel {E}^{{ab}\underline{c}}(\rho ){\parallel }_{\mathrm{tr}}\leqslant {s}_{c}\sqrt{{\widetilde{\upsilon }}_{a}{\widetilde{\upsilon }}_{b}}.\end{eqnarray*}$
Therefore, inequalities (A.3. Proof of theorem 3
Assume that ρ can be written as in equation (A2 ), then
$\begin{eqnarray*}{G}^{\underline{a}{bc}}({\rho }_{\zeta }^{a}\otimes {\rho }_{\zeta }^{b}\otimes {\rho }_{\zeta }^{c})={{\boldsymbol{g}}}_{\zeta }^{a}{\left({{\boldsymbol{g}}}_{\zeta }^{b}\otimes {{\boldsymbol{g}}}_{\zeta }^{c}\right)}^{{\rm{T}}},\end{eqnarray*}$
where $\begin{eqnarray*}{{\boldsymbol{g}}}_{\zeta }^{a}=\left(\begin{array}{c}{g}_{1}^{a}\\ {g}_{2}^{a}\\ \vdots \\ {g}_{{s}_{a}{t}_{a}}^{a}\end{array}\right),\qquad {{\boldsymbol{g}}}_{\zeta }^{b}=\left(\begin{array}{c}{g}_{1}^{b}\\ {g}_{2}^{b}\\ \vdots \\ {g}_{{s}_{b}{t}_{b}}^{b}\end{array}\right),\qquad {{\boldsymbol{g}}}_{\zeta }^{c}=\left(\begin{array}{c}{g}_{1}^{c}\\ {g}_{2}^{c}\\ \vdots \\ {g}_{{s}_{c}{t}_{c}}^{c}\end{array}\right)\end{eqnarray*}$
with ${g}_{i}^{a}=\mathrm{tr}({S}_{i}^{a}{\rho }_{\zeta }^{a}),$ i = 1, 2, …, sata, ${g}_{j}^{b}=\mathrm{tr}({S}_{j}^{b}{\rho }_{\zeta }^{b}),$ j = 1, 2, …, sbtb, and ${g}_{k}^{c}=\mathrm{tr}({S}_{k}^{c}{\rho }_{\zeta }^{c}),$ k = 1, 2, …, sctc. Therefore, we have $\begin{eqnarray*}\begin{array}{rcl}\parallel {G}^{\underline{a}{bc}}(\rho ){\parallel }_{\mathrm{tr}} & \leqslant & \sum _{\xi }{p}_{\zeta }\parallel {G}^{\underline{a}{bc}}({\rho }_{\zeta }^{a}\otimes {\rho }_{\zeta }^{b}\otimes {\rho }_{\zeta }^{c}){\parallel }_{\mathrm{tr}}\\ & = & \sum _{\zeta }{p}_{\zeta }\parallel {{\boldsymbol{g}}}_{\zeta }^{a}{\left({{\boldsymbol{g}}}_{\zeta }^{b}\otimes {{\boldsymbol{g}}}_{\zeta }^{c}\right)}^{{\rm{T}}}{\parallel }_{\mathrm{tr}}\\ & = & \sum _{\zeta }{p}_{\zeta }\parallel {{\boldsymbol{g}}}_{\zeta }^{a}\parallel \parallel {{\boldsymbol{g}}}_{\zeta }^{b}\parallel \parallel {{\boldsymbol{g}}}_{\zeta }^{c}\parallel \\ & \leqslant & \sqrt{{\widetilde{\upsilon }}_{a}{\widetilde{\upsilon }}_{b}{\widetilde{\upsilon }}_{c}}.\end{array}\end{eqnarray*}$
With the same method we get $\begin{eqnarray*}\parallel {G}^{a\underline{b}c}(\rho ){\parallel }_{\mathrm{tr}}\leqslant \sqrt{{\widetilde{\upsilon }}_{a}{\widetilde{\upsilon }}_{b}{\widetilde{\upsilon }}_{c}},\quad \parallel {G}^{{ab}\underline{c}}(\rho ){\parallel }_{\mathrm{tr}}\leqslant \sqrt{{\widetilde{\upsilon }}_{a}{\widetilde{\upsilon }}_{b}{\widetilde{\upsilon }}_{c}}.\end{eqnarray*}$