1. Introduction
Quantum entanglement is a strange phenomenon in quantum mechanics, where two quantum states in an entangled state have a correlation no matter how far apart they are. One quantum state changes, and the other quantum state changes instantaneously. It is an accepted fact that the entanglement phenomenon is unique to the core resource of modern quantum information science [1, 2], such as quantum computation, quantum information and quantum communication protocols [3–6].
Obviously, judging whether a quantum state is entangled is a core problem in quantum information processing, and it is described from different mathematical and physical angles. Until now, various forms of entanglement criteria have been obtained successively, mainly including positive partial transpose criteria [7–9], realignment criteria [10–16], correlation matrix criteria [17] and local uncertainty relation criteria [18], as well as quantum Fisher information criteria [19, 20], and more [21, 22]. In particular, a comprehensive investigation has been carried out in [23]. While considerable results have been achieved, there is still no universal method to check for entanglement.
In fact, observing and verifying quantum entanglement requires one to conduct experiments and use various quantum measurement techniques. Hitherto, various kinds of quantum measurements are constructed via projection measurement and positive operator-valued measurements. A useful entanglement detection scheme for entanglement detection may distinguish the largest collection of entangled states with as few measurements as possible. In [24], using mutually unbiased bases (MUBs) [25–36], entanglement criteria have been presented in different quantum systems. However, a complete set of MUBs is known to exist in prime power dimensions, but has not been found in any other complex dimension. As a result, a mutual unbiased measurement (MUM) arises at the historic moment and is subsequently applied to detect quantum entanglement as well [37–40]. The good news is that, unlike MUBs, a complete set of MUMs has been explicitly established in any finite dimension.
Besides, symmetry has attracted great attention in the quantum physics world because of its beautiful characteristics. It is noteworthy that symmetric informationally complete positive operator-valued measurements (SIC-POVMs), in quantum measurement theory, is a bright spot, whose arbitrary elements is rank one. SIC-POVMs are intimately related to MUBs [41–43], the study of which constitutes another outstanding issue in quantum measurement theory [44]. The authors of [39, 41, 45, 46] have obtained a series of powerful entanglement criteria from different perspectives for many-body quantum systems, and some well-known examples have been provided to demonstrate its superiority. It is a pity that the existence of SIC-POVMs in arbitrary dimensions remains a challenging problem, and analytical solutions have only been found in a particular dimensional space [42, 43]. To harness this problem, a general SIC-POVMs (GSIC-POVMs) is introduced [44]; that is, there is no necessity for each measurement operator to be rank one, and GSIC-POVMs can be constructed in all finite dimensions via generalized Gell-Mann matrices [47]. In [39, 48, 49], the separability criteria of bipartite and many-body systems have been given, respectively.
On the other hand, from the perspective of mathematical structure, although the definition constraints of the above measurement are beautiful and interesting, they are still harsh. For example, the definitions of SIC-POVMs and MUBs are closely related to dimensions, and the operators of SIC-POVMs are related to symmetry; also, the efficiency parameters of MUMs and GSIC-POVMs are very important and pave the way for us to further optimize quantum measurements. These properties may bring convenience to quantum tasks, such as the fact that symmetry is closely related to the geometry of quantum state space, which also has a far-reaching impact in quantum measurement theory. However, there is no universal method that enables one to build SIC-POVMs for all d, although many explicit constructions have been given.
Based on these concerns, in turn, it provides a reasonable and important reason to study the arbitrary equiangular tight frames (ETFs) in more detail, which are a natural generalization of orthonormal bases, and the maximal sets of complex equiangular vectors provide SIC-POVMs [50–53]. Besides, the angles between these vectors of ETFs are equal and satisfy the definition of tight frames. Because of these inherently unique mathematical properties, ETFs have been shown to be useful for coding, error correction and robust data transmission [4, 50] and more. In particular, [54] proposed the concept of equioverlapping measurements and studied their properties. In this paper, based on ETFs and the index of coincidence from [55], a series of separability criteria are proposed for different quantum systems. The rigorous proofs show that the criteria presented in this paper are more powerful and comprehensive than their counterparts in [38–40, 46, 48] for entanglement detection via some examples. Moreover, our derived criteria may also provide theoretical experimental methods for detecting entanglement of unknown quantum states [56].
The remainder of the paper is arranged as follows. In section 2 , some basic notions and properties of ETFs are recalled. Section 3 provides some entanglement criteria based on ETFs for different quantum systems via the mean inequality and Cauchy–Schwarz inequality, respectively. Then, these criteria are tested using various entangled states in section 4 . Finally, we also generalize the results to obtain some necessary criteria to detect entanglement for many-body quantum states in section 5 . The paper is concluded in section 6 .
2. Equiangular tight frames
As a general rule, quantum measurement is usually depicted by a POVM, such that
$\begin{eqnarray*}{\mathscr{E}}=\left\{{{\mathscr{E}}}_{\alpha }:\alpha =1,\,2,\,\cdots ,\,m\right\}\end{eqnarray*}$
composing a set of non-negative operators ${{\mathscr{E}}}_{\alpha }$ , and satisfying ${\sum }_{\alpha =1}^{m}{{\mathscr{E}}}_{\alpha }={\bf{1}}$ . Note that the measuring index operator m need not be equal to d, and it can be one or infinite. In addition, ${{\mathscr{E}}}_{\alpha }$ does not necessarily satisfy the properties of being projective or having rank one. For any density operator ρ, the probability outcome associated with ${{\mathscr{E}}}_{\alpha }$ can be expressed as ${p}_{\alpha }\,=\mathrm{Tr}({{\mathscr{E}}}_{\alpha }\rho )$ [4].To further characterize the internal structure of quantum measurement, a series of measurement operators have been proposed successively. Of particular concern to us is that SIC-POVM has a highly symmetrical structure, where symmetric means that the measurement operator is both equiangular and equidistant in the Hilbert space, and the rank of any measurement operator is one. However, an outstanding open problem is the conjecture of the existence of SIC-POVMs in any dimension (Zauners conjecture) [57, 58], which is widely believed to be true and supported by vast evidence [59, 60]. Despite many effects, the issue remains unsolved, and in recent years, its connections with algebraic number theory have been revealed.
Recently, in [49–52], the existence of such frames in both real and complex forms have been discussed, respectively. It is worth noting that all the frames are assumed to be complex in our works. Let ${{\mathscr{H}}}_{d}$ be d-dimensional Hilbert space, and a set of n ≥ d unit vectors ${\mathscr{F}}=\left\{\left|{\phi }_{j}\right\rangle \right\}$ is called a frame if 0 < S0 < S1 < ∞ exists, such that
$\begin{eqnarray}{S}_{0}\leqslant \displaystyle \sum _{j=1}^{n}{\left|\,,\left\langle {\phi }_{j}| \psi \right\rangle ,\,\right|}^{2}\leqslant {S}_{1},\end{eqnarray}$
for all units $| \psi \rangle \in {{\mathscr{H}}}_{d}$ , where S0 and S1 are the minimal and maximal eigenvalues of the frame operator $\begin{eqnarray}S=\displaystyle \sum _{j=1}^{n}\left|{\phi }_{j}\right\rangle \left\langle {\phi }_{j}\right|.\end{eqnarray}$
If taking S0 = S1 = n/d, a tight frame is given, where the frame operator is scalar with the eigenvalue S = n/d of multiplicity d. Every Parseval tight frame corresponds to a rank-one POVM when S0 = S1 = 1. In particular, an important tight frame is known as equiangular. Without loss of generality, a tight frame ${\mathscr{F}}$ is called equiangular, such that $\begin{eqnarray}{\left|\,\left\langle {\phi }_{i}| {\phi }_{j}\right\rangle \,\right|}^{2}=c,\end{eqnarray}$
for each pair i ≠ j and c > 0. It is easy to derive $\begin{eqnarray}S={nc}+1-c=\displaystyle \frac{n}{d},\quad c=\displaystyle \frac{S-1}{n-1}=\displaystyle \frac{n-d}{(n-1)d}.\end{eqnarray}$
If an ETF with n elements exists in d dimensions, then n ≤ d2 and an ETF with n elements also exists in n − d dimensions [52, 55]. The ETF is constructed in quantum measurement to optimize quantum measurement performance, distinguish quantum states, encode and decode quantum information, and optimize quantum computing resources [61, 62]; thus, this framework has important theoretical and practical significance in quantum information processing. On the other hand, an important generalized result is presented in [51].Fix $d\geqslant 2$ , and suppose ${\boldsymbol{\phi }}$ is an n, d matrix with unit-norm columns. Then, any two quantum states satisfy the following inequality, that is
$\begin{eqnarray}\mathop{\max }\limits_{i\ne j}\left|\,,\left\langle {\phi }_{i}| {\phi }_{j}\right\rangle ,\,\right|\geqslant \sqrt{\displaystyle \frac{n-d}{d(n-1)}}.\end{eqnarray}$
Notably, this bound is reached if and only if φ is an ETF. The inequality is originally due to Welch [63], and the insightful proof has been given in [64, 65]. Furthermore, taking $n={d}^{2}$ in equation ( $\begin{eqnarray}{\left|\,,\left\langle {\phi }_{i}| {\phi }_{j}\right\rangle ,\,\right|}^{2}=\displaystyle \frac{1}{d+1}\quad (i\ne j).\end{eqnarray}$
That is a special case of SIC-POVMs. In short, equation (In the following, generally speaking, any POVM elements of ETFs can be expressed as
$\begin{eqnarray}{E}_{j}=\displaystyle \frac{d}{n}\left|{\phi }_{j}\right\rangle \left\langle {\phi }_{j}\right|,\end{eqnarray}$
then, the states of any ETF result from the resolution ${\mathscr{E}}=\left\{{E}_{j}\right\}$ of the identity and satisfy $\begin{eqnarray}\displaystyle \frac{d}{n}S=\displaystyle \sum _{j=1}^{n}{E}_{j}={I}_{d}.\end{eqnarray}$
When the pre-measurement state is described by the density matrix ρ, the probability of the jth outcome is equal to $\begin{eqnarray}{p}_{j}({\mathscr{E}};{\boldsymbol{\rho }})=\displaystyle \frac{d}{n}\left\langle {\phi }_{j}| {\boldsymbol{\rho }}| {\phi }_{j}\right\rangle .\end{eqnarray}$
To formulate the quantum task of ETF-based measurement, conforming to the index of coincidence is an important factor in it. Let n unit kets $\left|{\phi }_{j}\right\rangle $ form an ETF in ${{\mathscr{H}}}_{d}$ , and let the POVM ${\mathscr{E}}$ be assigned to the ETF. For a given density matrix ρ, it holds that [55]4 ).
$\begin{eqnarray}I({\mathscr{E}};{\boldsymbol{\rho }})=\displaystyle \sum _{j=1}^{n}{p}_{j}{\left({\mathscr{E}};{\boldsymbol{\rho }}\right)}^{2}\leqslant \displaystyle \frac{{Sc}+(1-c)\mathrm{tr}\left({{\boldsymbol{\rho }}}^{2}\right)}{{S}^{2}},\end{eqnarray}$
where S and c obey equation (Some special cases are worth noting when the inequality, equation (10 ), is saturated. Consider the maximally mixed state ρ* = Id/d; one has ${p}_{j}\left({\mathscr{E}};{{\boldsymbol{\rho }}}_{* }\right)=1/n$ for all j and, herewith, $I\left({\mathscr{E}};{{\boldsymbol{\rho }}}_{* }\right)=1/n$ . In fact, due to 1 − c = S − nc and Sd = n, the right-hand side of equation (10 ) reads as
$\begin{eqnarray}\displaystyle \frac{{Scd}+1-c}{{S}^{2}d}=\displaystyle \frac{{nc}+S-{nc}}{{S}^{2}d}=\displaystyle \frac{1}{n}.\end{eqnarray}$
Another important case is deduced for any pure state taken from the ETF, and then the index of coincidence is given by10 ) coincides with equation (12 ) for a pure state. Let us trace the case of SIC-POVMs. For n = d2, S = d and c = (d + 1)−1, the result in equation (10 ) is turned into13 ), has been calculated in [44].
$\begin{eqnarray}{\left(\displaystyle \frac{d}{n}\right)}^{2}+(n-1){\left(\displaystyle \frac{{cd}}{n}\right)}^{2}-\displaystyle \frac{1+(n-1){c}^{2}}{{S}^{2}}=\displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n},\end{eqnarray}$
where (n − 1)c = S − 1. It is easy to see that the right-hand side of equation ( $\begin{eqnarray}I({\mathscr{E}};{\boldsymbol{\rho }})=\displaystyle \frac{1+\mathrm{tr}\left({{\boldsymbol{\rho }}}^{2}\right)}{d(d+1)},\end{eqnarray}$
and the corresponding probabilities can also be given by $\begin{eqnarray}{p}_{j}({\mathscr{E}};{\boldsymbol{\rho }})=\displaystyle \frac{1}{d}\left\langle {\phi }_{j}| {\boldsymbol{\rho }}| {\phi }_{j}\right\rangle .\end{eqnarray}$
As was mentioned, the index of coincidence, equation (3. Entanglement criteria based on equiangular tight frames
As we all know, the detection of entangled states is usually closely related to the quantum measurements. In fact, implementing measurements using a unitary operator whose rank is not one may incur excessive quantum costs. Such resource consumption may be incompatible with the trend towards achieving quantum information tasks. Thus, ETF-based measurements are still of interest as having rank-one elements. Based on this consideration, due to the close relationship between tight frames and POVMs, a series of effective entanglement criteria that are applicable to different quantum systems can be obtained.
Based on equation (9 ), for a density matrix ρAB of bipartite quantum systems and the ETF ${\mathscr{E}}$ , the corresponding (j, k)-probability can be expressed as
$\begin{eqnarray}{p}_{j,k}({\mathscr{E}};{{\boldsymbol{\rho }}}_{{AB}})=\displaystyle \frac{{d}^{2}}{{n}^{2}}\left\langle {\phi }_{j}{\phi }_{k}^{* }\left|{{\boldsymbol{\rho }}}_{{AB}}\right|{\phi }_{j}{\phi }_{k}^{* }\right\rangle .\end{eqnarray}$
It is worth noting that if we sum these probabilities over all j = k, by using the corresponding index of coincidence in equation (10 ) and the well-known inequality, it is not hard to obtain the following important result from [55].
Suppose that ${{\boldsymbol{\rho }}}_{{AB}}$ is a density matrix in bipartite quantum systems ${{\mathscr{C}}}^{d}\otimes {{\mathscr{C}}}^{d}$ with the same dimension. Let ${{\mathscr{E}}}_{i}$ and ${{\mathscr{Q}}}_{i}$ be any two sets of ETF in ${{\mathscr{C}}}^{d}$ , and then one defines
$\begin{eqnarray}{\mathscr{S}}\left({{\boldsymbol{\rho }}}_{{AB}}\right)=\displaystyle \sum _{i=1}^{n}\mathrm{Tr}\left({{\mathscr{E}}}_{i}\otimes {{\mathscr{Q}}}_{i}{{\boldsymbol{\rho }}}_{{AB}}\right).\end{eqnarray}$
Thus, ${{\boldsymbol{\rho }}}_{{AB}}$ is a separable state if $\begin{eqnarray}{\mathscr{S}}\left({{\boldsymbol{\rho }}}_{{AB}}\right)\leqslant \displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n}.\end{eqnarray}$
Without loss of generality, any separable pure state ${{\boldsymbol{\rho }}}_{{AB}}$ is read as
$\begin{eqnarray}{{\boldsymbol{\rho }}}_{{AB}}=| \phi \rangle \langle \phi | \otimes | \psi \rangle \langle \psi | .\end{eqnarray}$
It is obvious that equation (16 ) is a linear relation of the density matrix ρAB. Thus, based on mean inequality and the relation of matrix traces for tensor products, it is easy to derive
$\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{i=1}^{n}\mathrm{Tr}({{\mathscr{E}}}_{i}\otimes {{\mathscr{Q}}}_{i}{{\boldsymbol{\rho }}}_{{AB}})=\displaystyle \sum _{i=1}^{n}\mathrm{Tr}[({{\mathscr{E}}}_{i}\otimes {{\mathscr{Q}}}_{i})(| \phi \rangle \langle \phi | \otimes | \psi \rangle \langle \psi | )]\\ \,=\displaystyle \sum _{i=1}^{n}\mathrm{Tr}({{\mathscr{E}}}_{i}| \phi \rangle \langle \phi | )\mathrm{Tr}({{\mathscr{Q}}}_{i}| \psi \rangle \langle \psi | )\\ \,\leqslant \displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{n}\left\{{\left[\mathrm{Tr}({{\mathscr{E}}}_{i}| \phi \rangle \langle \phi | )\right]}^{2}+{\left[\mathrm{Tr}({{\mathscr{Q}}}_{i}| \psi \rangle \langle \psi | )\right]}^{2}\right\}.\end{array}\end{eqnarray}$
Besides, based on equation (10), the previous inequality, equation (19), becomes1 , a well-known entanglement criterion is readily available, such that1 can detect more entanglement than MUMs.
$\begin{eqnarray}\displaystyle \sum _{i=1}^{n}\mathrm{Tr}({{\mathscr{E}}}_{i}\otimes {{\mathscr{Q}}}_{i}{{\boldsymbol{\rho }}}_{{AB}})\leqslant \displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n},\end{eqnarray}$
if ρAB is a pure state. For convenience, this theorem is named the ETF criterion. Besides, if taking n = d in equation (20), one has ${\sum }_{i=1}^{d}\mathrm{Tr}({{\mathscr{E}}}_{i}\otimes {{\mathscr{Q}}}_{i}{{\boldsymbol{\rho }}}_{{AB}})\leqslant 1$ . It is worth noting that when taking n = d2 in theorem $\begin{eqnarray}\displaystyle \sum _{i=1}^{{d}^{2}}\mathrm{Tr}({{\mathscr{E}}}_{i}\otimes {{\mathscr{Q}}}_{i}{\boldsymbol{\rho }})\leqslant \displaystyle \frac{2}{d(d+1)}.\end{eqnarray}$
It is easy to find that the criterion is consistent with the criterion in [39]. In [38], the corresponding entanglement criterion based on MUMs is proposed, and showed that if ρ is separable, then J(ρ) ≤ 1 + κ, where 1/d < κ ≤ 1. And when κ = 1, MUMs is MUBs, i.e. J(ρ) ≤ 2. Obviously, theorem In addition, for different quantum subsystems, a more effective entanglement criterion than the ETF criterion is given.
If ${\boldsymbol{\rho }}$ is a density matrix in ${{\mathscr{C}}}^{{d}_{1}}\otimes {{\mathscr{C}}}^{{d}_{2}}$ with unequal dimensions, then let ${{\mathscr{E}}}_{i}$ and ${{\mathscr{Q}}}_{j}$ be any of a different set of ETFs in ${{\mathscr{C}}}^{{d}_{1}}$ and ${{\mathscr{C}}}^{{d}_{2}}$ , respectively. Thus, one can be defined directly
$\begin{eqnarray}{{\mathscr{S}}}_{1}({\boldsymbol{\rho }})=\displaystyle \sum _{i=1}^{n}{\rm{Tr}}({{\mathscr{E}}}_{i}\otimes {{\mathscr{Q}}}_{i}{\boldsymbol{\rho }}),\end{eqnarray}$
where $n=\min \{{n}_{1},{n}_{2}\}$ .If the density matrix ρ is separable, then
$\begin{eqnarray}{{\mathscr{S}}}_{1}({\boldsymbol{\rho }})\leqslant \displaystyle \frac{1}{2}\left(\displaystyle \frac{{d}_{1}^{2}-2{d}_{1}+{n}_{1}}{{n}_{1}^{2}-{n}_{1}}+\displaystyle \frac{{d}_{2}^{2}-2{d}_{2}+{n}_{2}}{{n}_{2}^{2}-{n}_{2}}\right).\end{eqnarray}$
Similarly, equation (
$\begin{eqnarray}{\boldsymbol{\rho }}=| \phi \rangle \langle \phi | \otimes | \psi \rangle \langle \psi | ,\end{eqnarray}$
and it is easy to prove that $\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{i=1}^{n}\mathrm{Tr}({{\mathscr{E}}}_{i}\otimes {{\mathscr{Q}}}_{i}{\boldsymbol{\rho }}) & = & \displaystyle \sum _{i=1}^{n}\mathrm{Tr}({{\mathscr{E}}}_{i}| \phi \rangle \langle \phi | )\displaystyle \sum _{i=1}^{n}\mathrm{Tr}({{\mathscr{Q}}}_{i}| \psi \rangle \langle \psi | )\\ & \leqslant & \displaystyle \frac{1}{2}\left\{\displaystyle \sum _{i=1}^{{n}_{1}}{\left[\mathrm{Tr}({{\mathscr{E}}}_{i}| \phi \rangle \langle \phi | )\right]}^{2}\right.\\ & & \left.+\,\displaystyle \sum _{j=1}^{{n}_{2}}{\left[\mathrm{Tr}({{\mathscr{Q}}}_{j}| \psi \rangle \langle \psi | )\right]}^{2}\right\}\\ & \leqslant & \displaystyle \frac{1}{2}\left[\displaystyle \frac{{d}_{1}^{2}-2{d}_{1}+{n}_{1}}{{n}_{1}^{2}-{n}_{1}}+\displaystyle \frac{{d}_{2}^{2}-2{d}_{2}+{n}_{2}}{{n}_{2}^{2}-{n}_{2}}\right].\end{array}\end{eqnarray}$
Therefore, one can obtain $\begin{eqnarray}{{\mathscr{S}}}_{1}({\boldsymbol{\rho }})\leqslant \displaystyle \frac{1}{2}\left(\displaystyle \frac{{d}_{1}^{2}-2{d}_{1}+{n}_{1}}{{n}_{1}^{2}-{n}_{1}}+\displaystyle \frac{{d}_{2}^{2}-2{d}_{2}+{n}_{2}}{{n}_{2}^{2}-{n}_{2}}\right).\end{eqnarray}$
According to theorem 2 , a special entanglement criterion can be obtained, i.e. taking ${n}_{1}={d}_{1}^{2}$ , ${n}_{2}={d}_{2}^{2}$ , respectively, and one has
$\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{j=1}^{{d}_{1}^{2}}\displaystyle \sum _{j=1}^{{d}_{2}^{2}}\mathrm{Tr}({{\mathscr{E}}}_{j}\otimes {{\mathscr{Q}}}_{j}{\boldsymbol{\rho }})\\ \quad \leqslant \,\displaystyle \frac{1}{2}\left[\displaystyle \frac{2}{{d}_{1}({d}_{1}+1)}+\displaystyle \frac{2}{{d}_{2}({d}_{2}+1)}\right]\leqslant \displaystyle \frac{1}{6}.\end{array}\end{eqnarray}$
Compared with theorem 1 , theorem 2 is more efficient and has a wider application range, i.e. more entangled states can be detected. Moreover, the result of the above theorem can be further improved using the Cauchy–Schwarz inequality, and the following theorem can be obtained.
If ${\boldsymbol{\rho }}$ is separable, then
$\begin{eqnarray}{{\mathscr{S}}}_{2}({\boldsymbol{\rho }})\leqslant \sqrt{\displaystyle \frac{{d}_{1}^{2}-2{d}_{1}+{n}_{1}}{{n}_{1}^{2}-{n}_{1}}}\sqrt{\displaystyle \frac{{d}_{2}^{2}-2{d}_{2}+{n}_{2}}{{n}_{2}^{2}-{n}_{2}}},\end{eqnarray}$
where ${{\mathscr{S}}}_{2}({\boldsymbol{\rho }})={\sum }_{i\,=\,1}^{n}\mathrm{Tr}({{\mathscr{E}}}_{i}\otimes {{\mathscr{Q}}}_{i}{\boldsymbol{\rho }})$ with $n=\min \{{n}_{1},{n}_{2}\}$ , and ${\boldsymbol{\rho }}=| \phi \rangle \langle \phi | \otimes | \psi \rangle \langle \psi | $ .Obviously, by using the Cauchy–Schwarz inequality, one has
$\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{i=1}^{n}\mathrm{Tr}({{\mathscr{E}}}_{i}\otimes {{\mathscr{Q}}}_{i}{\boldsymbol{\rho }})=\displaystyle \sum _{i=1}^{n}\mathrm{Tr}({{\mathscr{E}}}_{i}| \phi \rangle \langle \phi | )\displaystyle \sum _{i=1}^{n}\mathrm{Tr}({{\mathscr{Q}}}_{i}| \psi \rangle \langle \psi | )\leqslant \sqrt{\displaystyle \sum _{i=1}^{{n}_{1}}\left[\right.\mathrm{Tr}({{\mathscr{E}}}_{i}| \phi \rangle \langle \phi | )}{\left.\right]}^{2}\sqrt{\displaystyle \sum _{j=1}^{{n}_{2}}\left[\right.\mathrm{Tr}({{\mathscr{Q}}}_{j}| \psi \rangle \langle \psi | ){\left.\right]}^{2}}\\ \,\quad \leqslant \,\sqrt{\displaystyle \frac{{d}_{1}^{2}-2{d}_{1}+{n}_{1}}{{n}_{1}^{2}-{n}_{1}}}\sqrt{\displaystyle \frac{{d}_{2}^{2}-2{d}_{2}+{n}_{2}}{{n}_{2}^{2}-{n}_{2}}},\end{array}\end{eqnarray}$
and the proof is completed. It is not difficult to prove that the following result can be easily reached.Theorem
4. Tighter entanglement criteria
For any quantum state, to detect as much entanglement as possible, it is important to construct stronger entanglement criteria. In the following, a tighter separability criterion is put forward based on the Cauchy–Schwarz inequality and ρ − ρA ⨂ ρB, where ρ is a density matrix in bipartite quantum systems ${{\mathscr{C}}}^{d}\otimes {{\mathscr{C}}}^{d^{\prime} }$ , and ${\rho }^{A}\left({\rho }^{B}\right)$ stands for the reduced density matrix of the first (second) subsystem. These derived criteria also provide a simple experimental method for detecting entanglement of unknown quantum states.
Given that ${\{{{\mathscr{E}}}_{\alpha }\}}_{\alpha =1}^{n}$ and ${\{{{\mathscr{Q}}}_{\alpha }\}}_{\alpha =1}^{n}$ are any two sets of n ETF on ${{\mathscr{C}}}^{d}$ . For ρ is a density matrix in ${{\mathscr{C}}}^{d}\otimes {{\mathscr{C}}}^{d}$ and a separable state, one has
$\begin{eqnarray}\begin{array}{l}{{\mathscr{S}}}_{3}(\rho )=\displaystyle \sum _{\alpha =1}^{n}| \mathrm{Tr}({{\mathscr{E}}}_{\alpha }\otimes {{\mathscr{Q}}}_{\alpha }(\rho -{\rho }^{A}\otimes {\rho }^{B}))| \\ \quad \leqslant \sqrt{\left(\displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n}-\displaystyle \sum _{\alpha =1}^{n}{\left[\mathrm{Tr}\left({{\mathscr{E}}}_{\alpha }{\rho }^{A}\right)\right]}^{2}\right)\left(\displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n}-\displaystyle \sum _{\alpha =1}^{n}{\left[\mathrm{Tr}\left({{\mathscr{Q}}}_{\alpha }{\rho }^{B}\right)\right]}^{2}\right)}.\end{array}\end{eqnarray}$
Without loss of generality, any separable state ρ is expressed as
$\begin{eqnarray}\rho =\displaystyle \sum _{i=1}^{r}{p}_{i}{\rho }_{i}^{A}\otimes {\rho }_{i}^{B}.\end{eqnarray}$
Here, $0\leqslant {p}_{i}\leqslant 1$ and ${\sum }_{i=1}^{r}{p}_{i}=1$ , ${\rho }_{i}^{A}\left({\rho }_{i}^{B}\right)$ stands for the density matrix acting on the first (second) subsystem, respectively. Hence, one has ${\rho }^{A}={\sum }_{i=1}^{r}{p}_{i}{\rho }_{i}^{A},\,{\rho }^{B}={\sum }_{i=1}^{r}{p}_{i}{\rho }_{i}^{B}$ . Thus, in [66], $\rho -{\rho }^{A}\otimes {\rho }^{B}$ is given by $\begin{eqnarray}\rho -{\rho }^{A}\otimes {\rho }^{B}=\displaystyle \frac{1}{2}\left(\displaystyle \sum _{s,\,t=1}^{r}{p}_{s}{p}_{t}\left({\rho }_{s}^{A}-{\rho }_{t}^{A}\right)\otimes \left({\rho }_{s}^{B}-{\rho }_{t}^{B}\right)\right).\end{eqnarray}$
Using the Cauchy–Schwarz inequality with equation ( $\begin{eqnarray}\begin{array}{rcl} & & {{\mathscr{S}}}_{3}(\rho )\\ & \leqslant & \displaystyle \frac{1}{2}\displaystyle \sum _{\alpha =1}^{n}\displaystyle \sum _{s,t=1}^{r}| \sqrt{{p}_{s}{p}_{t}}\mathrm{Tr}\left({{\mathscr{E}}}_{\alpha }\left({\rho }_{s}^{A}-{\rho }_{t}^{A}\right)\right)| | \sqrt{{p}_{s}{p}_{t}}\mathrm{Tr}\left({{\mathscr{Q}}}_{\alpha }\left({\rho }_{s}^{B}-{\rho }_{t}^{B}\right)\right)| \\ & \leqslant & \displaystyle \frac{1}{2}\sqrt{\displaystyle \sum _{\alpha =1}^{n}\displaystyle \sum _{s,t=1}^{r}{p}_{s}{p}_{t}{\left[\mathrm{Tr}\left({{\mathscr{E}}}_{\alpha }\left({\rho }_{s}^{A}-{\rho }_{t}^{A}\right)\right)\right]}^{2}}\sqrt{\displaystyle \sum _{\alpha =1}^{n}\displaystyle \sum _{s,t=1}^{r}{p}_{s}{p}_{t}{\left[\mathrm{Tr}\left({{\mathscr{Q}}}_{\alpha }\left({\rho }_{s}^{B}-{\rho }_{t}^{B}\right)\right)\right]}^{2}}\\ & = & \sqrt{\left(\displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n}-\displaystyle \sum _{\alpha =1}^{n}{\left[\mathrm{Tr}\left({{\mathscr{E}}}_{\alpha }{\rho }^{A}\right)\right]}^{2}\right)\left(\displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n}-\displaystyle \sum _{\alpha =1}^{n}{\left[\mathrm{Tr}\left({{\mathscr{Q}}}_{\alpha }{\rho }^{B}\right)\right]}^{2}\right)}.\end{array}\end{eqnarray}$
Compared with the ETF criterion, theorem 4 has a tighter lower bound; therefore, more entangled states can be detected, and we have the following results.
Theorem
On the other hand, Shen et al proposed an entanglement criterion for ρ − ρA ⨂ ρB based on MUMs [40]. Let ${\{{{\mathscr{P}}}^{\left(b\right)}\}}_{b=1}^{d+1}$ and ${\{{{\mathscr{Q}}}^{\left(b\right)}\}}_{b=1}^{d+1}$ be two sets of d + 1 MUMs on ${{\mathbb{C}}}^{d}$ with the same parameter κ, where ${{\mathscr{P}}}^{\left(b\right)}=\{{P}_{n}^{\left(b\right)}\}{}_{n=1}^{d}$ , ${{\mathscr{Q}}}^{\left(b\right)}=\{{Q}_{n}^{\left(b\right)}\}{}_{n=1}^{d},b\,=\,1,\,2,\,\cdots ,\,d+1$ . For any separable state ρ in ${{\mathscr{C}}}^{d}\otimes {{\mathscr{C}}}^{d}$ ,4 can detect more entanglement.
$\begin{eqnarray*}\begin{array}{l}S(\rho )=\displaystyle \sum _{b=1}^{d+1}\displaystyle \sum _{n=1}^{d}\left|\mathrm{Tr}\left({P}_{n}^{\left(b\right)}\otimes {Q}_{n}^{\left(b\right)}\left(\rho -{\rho }^{A}\otimes {\rho }^{B}\right)\right)\right|\\ \quad \leqslant \sqrt{\left(1+\kappa -{\sum }_{b=1}^{d+1}{\sum }_{n=1}^{d}{\left(\mathrm{Tr}\left({P}_{n}^{\left(b\right)}{\rho }^{A}\right)\right)}^{2}\right)\left(1+\kappa -{\sum }_{b=1}^{d+1}{\sum }_{n=1}^{d}{\left(\mathrm{Tr}\left({Q}_{n}^{\left(b\right)}{\rho }^{B}\right)\right)}^{2}\right)}.\end{array}\end{eqnarray*}$
By simple calculation, it is easy to show that theorem Obviously, it is easy to obtain the particular case according to theorem 4 .
If $n={d}^{2}$ in the ETF, theorem
$\begin{eqnarray}\begin{array}{rcl}{{\mathscr{S}}}_{4}(\rho ) & = & \displaystyle \sum _{i=1}^{{d}^{2}}\left|\mathrm{Tr}\left({{\mathscr{P}}}_{i}\otimes {{\mathscr{Q}}}_{i}\left(\rho -{\rho }^{A}\otimes {\rho }^{B}\right)\right)\right|\\ & \leqslant & \sqrt{\displaystyle \frac{2}{d(d+1)}-\displaystyle \sum _{i=1}^{{d}^{2}}{\left[\mathrm{Tr}\left({{\mathscr{P}}}_{i}{\rho }^{A}\right)\right]}^{2}}\\ & & \times \sqrt{\displaystyle \frac{2}{d(d+1)}-\displaystyle \sum _{i=1}^{{d}^{2}}{\left[\mathrm{Tr}\left({{\mathscr{Q}}}_{i}{\rho }^{B}\right)\right]}^{2}}.\end{array}\end{eqnarray}$
Notice that for two subsystems with different dimensions, the corresponding result can also be obtained.
Let ${\{{{\mathscr{E}}}_{\alpha }\}}_{\alpha =1}^{{n}_{1}}$ and ${\{{{\mathscr{Q}}}_{\beta }\}}_{\beta =1}^{{n}_{2}}$ be any two sets of ${n}_{1}({n}_{2})$ ETF on ${{\mathscr{C}}}^{d}$ . Let the separable state ρ be a density matrix in ${{\mathscr{C}}}^{d}\otimes {{\mathscr{C}}}^{d^{\prime} }$ , and one has
$\begin{eqnarray}\begin{array}{l}{{\mathscr{S}}}_{5}(\rho )=\displaystyle \sum _{\alpha =1}^{{n}_{1}}\displaystyle \sum _{\beta =1}^{{n}_{2}}| \mathrm{Tr}({{\mathscr{E}}}_{\alpha }\otimes {{\mathscr{Q}}}_{\beta }(\rho -{\rho }^{A}\otimes {\rho }^{B}))| \\ \quad \leqslant \sqrt{\left(\displaystyle \frac{{d}^{2}-2d+{n}_{1}}{{n}_{1}^{2}-{n}_{1}}-\displaystyle \sum _{\alpha =1}^{{n}_{1}}{\left[\mathrm{Tr}\left({{\mathscr{E}}}_{\alpha }{\rho }^{A}\right)\right]}^{2}\right)\left(\displaystyle \frac{d{{\prime} }^{2}-2d^{\prime} +{n}_{2}}{{n}_{2}^{2}-{n}_{2}}-\displaystyle \sum _{\beta =1}^{{n}_{2}}{\left[\mathrm{Tr}\left({{\mathscr{Q}}}_{\beta }{\rho }^{B}\right)\right]}^{2}\right)}.\end{array}\end{eqnarray}$
Theorem
5. Examples
To prove the superiority and practicability of the criteria, suppose that ${{\mathscr{P}}}_{\alpha }={\sum }_{\alpha =1}^{n}{{\mathscr{E}}}_{\alpha }$ is a set of ETF in ${{\mathscr{C}}}^{d}$ . Let $\overline{{{\mathscr{P}}}_{\alpha }}$ denote the conjugation of ${{\mathscr{P}}}_{\alpha }$ , where the $\overline{{{\mathscr{P}}}_{\alpha }}$ is another set of ETFs. Furthermore, some well-known examples are given in the following.
In bipartite quantum systems ${{\mathscr{C}}}^{d}\otimes {{\mathscr{C}}}^{d}$ , any maximally entangled state is read as
$\begin{eqnarray}| {\phi }^{+}\rangle =\displaystyle \frac{1}{\sqrt{d}}\displaystyle \sum _{i=0}^{d-1}| {ii}\rangle .\end{eqnarray}$
Furthermore, based on theorem $\begin{eqnarray}\begin{array}{rcl}J\left(| {\phi }^{+}\rangle \right) & = & \displaystyle \sum _{\alpha =1}^{n}\mathrm{Tr}\left({{\mathscr{P}}}_{\alpha }\otimes \overline{{{\mathscr{P}}}_{\alpha }}\left|{\phi }^{+}\right\rangle \left\langle {\phi }^{+}\right|\right)\\ & = & \displaystyle \frac{d}{n}\gt \displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n}.\end{array}\end{eqnarray}$
Thus, this criterion can detect all the maximally entangled pure states, if and only if, $n\gt d$ . Clearly, our detection scheme makes sense.On the one hand, the most efficient scheme takes place for n = d2, then the ETF can turn into SIC-POVMs. Hence, we have
$\begin{eqnarray}{J}_{a}\left(| {{\rm{\Phi }}}^{+}\rangle \right)=\displaystyle \frac{1}{d}\gt \displaystyle \frac{2}{d(d+1)}.\end{eqnarray}$
Obviously, the criterion can detect all the maximally entangled pure states.We now consider the Bell diagonal states,
$\begin{eqnarray}{\rho }_{\mathrm{Bell}}=\displaystyle \sum _{s,\,t=0}^{d-1}{p}_{s,\,t}\left|{{\rm{\Phi }}}_{s,\,t}^{+}\right\rangle \left\langle {{\rm{\Phi }}}_{s,\,t}^{+}\right|,\end{eqnarray}$
where ${p}_{s,\,t}\geqslant 0,{\sum }_{s,\,t=0}^{d-1}{p}_{s,\,t}=1,\left|{{\rm{\Phi }}}_{s,\,t}^{+}\right\rangle =\left({U}_{s,\,t}\otimes I\right)\left|{{\rm{\Phi }}}^{+}\right\rangle $ , and ${U}_{s,\,t}=$ ${\sum }_{j=0}^{d-1}{\zeta }_{d}^{{sj}}| j\rangle \langle j\oplus t| ,s,t=0,1,\cdots ,d-1$ , are Weyl operators, ${\zeta }_{d}={{\rm{e}}}^{\tfrac{2\pi \sqrt{-1}}{d}}$ and $j\oplus t$ denotes $(j+t)\mathrm{mod}d$ . Denoting $A\geqslant B$ if A − B is positive for operators A and B, we have $\begin{eqnarray}{\rho }_{\mathrm{Bell}\ }\geqslant c\left|{{\rm{\Phi }}}_{c}\right\rangle \left\langle {{\rm{\Phi }}}_{c}\right|,\end{eqnarray}$
where $c=\max \left\{{p}_{s,\,t}:s,\,t=0,\,1,\,\cdots ,\,d-1\right\},\tfrac{1}{{d}^{2}}\leqslant c\leqslant 1$ , and $\left|{{\rm{\Phi }}}_{c}\right\rangle $ is the corresponding maximally entangled state. Consequently, we obtain $\begin{eqnarray}\begin{array}{rcl}J\left({\rho }_{\mathrm{Bell}}\right) & = & \displaystyle \sum _{\alpha =1}^{n}\mathrm{Tr}\left({{\mathscr{P}}}_{\alpha }\otimes \overline{{{\mathscr{P}}}_{\alpha }}{\rho }_{\mathrm{Bell}}\right)\\ & \geqslant & \displaystyle \sum _{\alpha =1}^{n}\mathrm{Tr}\left[\left.{{\mathscr{P}}}_{\alpha }\otimes \overline{{{\mathscr{P}}}_{\alpha }}\left(c\left|{{\rm{\Phi }}}_{c}\right\rangle \left\langle {{\rm{\Phi }}}_{c}\right|\right)\right]\right.\\ & = & \displaystyle \frac{{d}^{2}-1}{n-1}\displaystyle \frac{{dc}}{n}.\end{array}\end{eqnarray}$
If ${\rho }_{\mathrm{Bell}}$ is an entangled state via theorem $\begin{eqnarray}c\gt \displaystyle \frac{{d}^{2}-2d+n}{({d}^{2}-1)d},\end{eqnarray}$
then $J\left({\rho }_{\mathrm{Bell}}\right)\gt \tfrac{{d}^{2}-2d+n}{{n}^{2}-n}$ . In turn, taking $n={d}^{2}$ , that is, $c\gt \tfrac{2}{d+1}$ . Similarly, $J\left({\rho }_{\mathrm{Bell}}\right)\gt \tfrac{2}{d(d+1)}$ and ${\rho }_{\mathrm{Bell}\ }$ must also be entangled.Now, compare theorem 4 and the ETF criterion, and one has $\mathrm{Tr}\left({{\mathscr{P}}}_{k}\right)=\mathrm{Tr}\left({{\mathscr{Q}}}_{k}\right)=\tfrac{d}{n}$ , $R\left({\rho }_{\mathrm{Bell}\ }\right)$ is written as
$\begin{eqnarray}\begin{array}{rcl}R\left({\rho }_{\mathrm{Bell}\ }\right) & = & \displaystyle \sum _{\alpha =1}^{n}| \displaystyle \frac{1}{d}\displaystyle \sum _{s,\,t=0}^{d-1}{c}_{{st}}\\ & & \times \,\displaystyle \sum _{i,\,j=0}^{d-1}\mathrm{Tr}\left({{\mathscr{E}}}_{\alpha }{U}_{{st}}| i\rangle \langle j| {U}_{{st}}^{\dagger }\right)\mathrm{Tr}\left({{\mathscr{Q}}}_{\alpha }| i\rangle \langle j| \right)-\displaystyle \frac{1}{{n}^{2}}| ,\end{array}\end{eqnarray}$
and the right-hand side of the relation in equation (47) is $\tfrac{{d}^{2}-2d+n}{{n}^{2}-n}-\tfrac{1}{n}$ by calculation.Furthermore, the following situations should be considered and analyzed, such that
(i) ${c}_{{ab}}\geqslant \tfrac{d}{n}$ . Taking ${{\mathscr{Q}}}_{\alpha }=\overline{{U}_{{ab}}^{\dagger }{{\mathscr{E}}}_{\alpha }{U}_{{ab}}}$ , it is easy to derive4 , ρBell is entangled, if and only if,
$\begin{eqnarray}R\left({\rho }_{\mathrm{Bell}\ }\right)\geqslant \displaystyle \frac{{d}^{2}-1}{n-1}\displaystyle \frac{d}{n}{c}_{{ab}}-\displaystyle \frac{1}{n}.\end{eqnarray}$
According to theorem $\begin{eqnarray}\displaystyle \frac{{d}^{2}-1}{n-1}\displaystyle \frac{d}{n}{c}_{{ab}}-\displaystyle \frac{1}{n}\gt \displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n}-\displaystyle \frac{1}{n},\end{eqnarray}$
i.e. ${c}_{{ab}}\gt \tfrac{{d}^{2}-2d+n}{({d}^{2}-1)d}$ .(ii) ${c}_{{fg}}\lt \tfrac{d}{n}$ . Similarly, let ${{\mathscr{Q}}}_{\alpha }=\overline{{U}_{{fg}}^{\dagger }{{\mathscr{E}}}_{\alpha }{U}_{{fg}}}$ , and one can get
$\begin{eqnarray}R\left({\rho }_{\mathrm{Bell}\ }\right)\lt \displaystyle \frac{1}{n}-\displaystyle \frac{{d}^{2}-1}{n-1}\displaystyle \frac{d}{n}{c}_{{fg}}.\end{eqnarray}$
By theorem 4 , ρ Bell is entangled if4 , it is not difficult to find that the probe efficiency of the ETF depends on the parameter n.
$\begin{eqnarray}\displaystyle \frac{1}{n}-\displaystyle \frac{{d}^{2}-1}{n-1}\displaystyle \frac{d}{n}{c}_{{fg}}\gt \displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n}-\displaystyle \frac{1}{n},\end{eqnarray}$
i.e. ${c}_{{fg}}\lt \tfrac{-{d}^{2}+2d+n-2}{d({d}^{2}-1)}$ . Based on (i) and (ii), one can directly summarize that if $\begin{eqnarray}{c}_{{ab}}\gt \max \left\{\displaystyle \frac{d}{n},\displaystyle \frac{{d}^{2}-2d+n}{({d}^{2}-1)d}\right\}=\displaystyle \frac{d}{n},\end{eqnarray}$
or $\begin{eqnarray}{c}_{{fg}}\lt \min \left\{\displaystyle \frac{d}{n},\displaystyle \frac{-{d}^{2}+2d+n-2}{d({d}^{2}-1)}\right\}=\displaystyle \frac{-{d}^{2}+2d+n-2}{d({d}^{2}-1)}\end{eqnarray}$
holds, then ρBell is entangled. However, ETF criteria only detect the entanglement for ${c}_{{ab}}\gt \tfrac{{d}^{2}-2d+n}{{n}^{2}-n}$ via similar methods. Based on theorem The d-dimensional Werner states, that is [41]
$\begin{eqnarray}{\rho }_{W}=\displaystyle \frac{1}{{d}^{3}-d}\left[(d-g){I}_{{d}^{2}}+({dg}-1)\eta \right],\end{eqnarray}$
where g satisfies $-1\leqslant g\leqslant 1$ , and the operator η is given by $\begin{eqnarray}\eta =\displaystyle \sum _{i,\,j=0}^{d-1}| {ij}\rangle \langle {ji}| .\end{eqnarray}$
Besides, the Werner states ${\rho }_{W}$ are entangled if $-1\leqslant g\lt 0$ .Similar to example 1, taking ${{\mathscr{E}}}_{\alpha }={{\mathscr{Q}}}_{\alpha }$ , one has
$\begin{eqnarray}\begin{array}{rcl}F({\rho }_{W}) & = & \displaystyle \sum _{\alpha =1}^{n}\mathrm{Tr}\left({{\mathscr{E}}}_{\alpha }\otimes {{\mathscr{Q}}}_{\alpha }{\rho }_{w}\right)\\ & = & \displaystyle \frac{d-g}{{d}^{3}-d}\displaystyle \sum _{\alpha =1}^{n}\mathrm{Tr}\left({{\mathscr{E}}}_{\alpha }^{2}{I}_{{d}^{2}}\right)+\displaystyle \frac{{dg}-1}{{d}^{3}-d}\displaystyle \sum _{\alpha =1}^{n}\mathrm{Tr}\left({{\mathscr{Q}}}_{\alpha }^{2}\eta \right)\\ & = & \displaystyle \frac{d(d-g)}{n(n-1)}+\displaystyle \frac{d({dg}-1)}{n(n-1)}.\end{array}\end{eqnarray}$
On the one hand, by using the ETF criterion, one has
$\begin{eqnarray}F({\rho }_{W})\gt \displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n},\end{eqnarray}$
if and only if, $g\gt \tfrac{n-d}{{d}^{2}-d}$ . Therefore, the entanglement of the Werner state cannot be detected by ETF criteria.With this in mind, we now apply theorem 4 , The left- and right-hand sides of relation (30) are, respectively,4 is more efficient than the ETF criterion for the d-dimensional Werner states.
$\begin{eqnarray*}\begin{array}{rcl}F\left({\rho }_{W}\right) & = & \Space{0ex}{2.5ex}{0ex}| \displaystyle \frac{d(d-g)}{n(n-1)}+\displaystyle \frac{d({dg}-1)}{n(n-1)}-\displaystyle \frac{1}{n}\Space{0ex}{2.5ex}{0ex}| \\ & & \,\mathrm{and}\ \displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n}-\displaystyle \frac{1}{n}.\end{array}\end{eqnarray*}$
Since $\begin{eqnarray}\begin{array}{rcl}F\left({\rho }_{W}\right) & = & \Space{0ex}{2.5ex}{0ex}| \displaystyle \frac{d(d-g)}{n(n-1)}+\displaystyle \frac{d({dg}-1)}{n(n-1)}-\displaystyle \frac{1}{n}\Space{0ex}{2.5ex}{0ex}| \\ & & \gt \displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n}-\displaystyle \frac{1}{n},\end{array}\end{eqnarray}$
if and only if, $g\gt \tfrac{n-d}{{d}^{2}-d}$ or g < $\tfrac{{d}^{2}-2d-n+2}{d-{d}^{2}}-1$ . Thereby, the detection ability of theorem It is worth noting that $g=\tfrac{2}{d}-1$ , if and only if, n = d2, i.e. GSIC-POVMs, and theorem 4 can detect the entanglement of $\left({\rho }_{W}\right)$ for $-1\leqslant g\lt \tfrac{2}{d}-1$ ; this result is consistent with [40]. Besides, if taking n = d, it is not difficult to find that entanglement of Werner states can be detected only when the parameter g = − 1. Therefore, the utilization of GSIC-POVM can be considered the best solution.
6. Entanglement detection via ETF in many-body quantum systems
Naturally, the above entanglement criteria can also be generalized to many-body quantum states. With ETFs, the following criteria are readily available.
For any density matrix ρ in ${\left({{\mathscr{C}}}^{d}\right)}^{\otimes L}$ , suppose that $\{{{\mathscr{P}}}_{i}^{j}\}{}_{i=1}^{n}$ , $j=1,2,\cdots ,L$ , are L same sets of ETF in ${{\mathscr{C}}}^{d}$ , respectively. Here, ρ is fully separable, if and only if,
$\begin{eqnarray}{{\mathscr{S}}}_{6}(\rho )\leqslant \displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n},\end{eqnarray}$
where $\begin{eqnarray}{{\mathscr{S}}}_{6}(\rho )=\displaystyle \sum _{i=1}^{n}\mathrm{Tr}({\otimes }_{j=1}^{L}{{\mathscr{P}}}_{i}^{j}\rho ).\end{eqnarray}$
Similarly, consider the following separable state,
$\begin{eqnarray}\rho ={\otimes }_{j=1}^{L}\left|{\varphi }_{j}\right\rangle \left\langle {\varphi }_{j}\right|.\end{eqnarray}$
Thus, via equation ( $\begin{eqnarray}{{\mathscr{S}}}_{6}(\rho )=\displaystyle \sum _{i=1}^{n}\displaystyle \prod _{j=1}^{L}\mathrm{Tr}\left({{\mathscr{P}}}_{i}^{j}\left|{\varphi }_{j}\right\rangle \left\langle {\varphi }_{j}\right|\right).\end{eqnarray}$
Note that $\begin{eqnarray}0\leqslant \mathrm{Tr}\left({{\mathscr{P}}}_{i}^{j}\left|{\varphi }_{j}\right\rangle \left\langle {\varphi }_{j}\right|\right)\leqslant \displaystyle \frac{d}{n}.\end{eqnarray}$
Moreover, for L non-negative real numbers, it satisfies the inequality [39], $\begin{eqnarray}{x}_{1}{x}_{2}\cdots {x}_{L}\leqslant {\left(\displaystyle \frac{{x}_{1}^{2}+{x}_{2}^{2}+\cdots +{x}_{L}^{2}}{L}\right)}^{\tfrac{L}{2}},\end{eqnarray}$
thus one has $\begin{eqnarray}\begin{array}{rcl}{{\mathscr{S}}}_{6}(\rho ) & \leqslant & \displaystyle \sum _{i=1}^{n}{\left\{\displaystyle \frac{1}{L}\displaystyle \sum _{j=1}^{L}{\left[\mathrm{Tr}\left({{\mathscr{P}}}_{i}^{j}\left|{\varphi }_{j}\right\rangle \left\langle {\varphi }_{j}\right|\right)\right]}^{2}\right\}}^{\tfrac{L}{2}}\\ & \leqslant & \displaystyle \sum _{i=1}^{n}\displaystyle \frac{1}{L}\displaystyle \sum _{j=1}^{L}{\left[\mathrm{Tr}\left({{\mathscr{P}}}_{i}^{j}\left|{\varphi }_{j}\right\rangle \left\langle {\varphi }_{j}\right|\right)\right]}^{2}\\ & = & \displaystyle \frac{1}{L}\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^{L}{\left[\mathrm{Tr}\left({{\mathscr{P}}}_{i}^{j}\left|{\varphi }_{j}\right\rangle \left\langle {\varphi }_{j}\right|\right)\right]}^{2}.\end{array}\end{eqnarray}$
Based on equation (10), one has $\begin{eqnarray}{{\mathscr{S}}}_{6}(\rho )\leqslant \displaystyle \frac{{d}^{2}-2d+n}{{n}^{2}-n}.\end{eqnarray}$
Taking any $n={d}^{2}$ , it is easy to obtain ${{\mathscr{S}}}_{6}(\rho )\leqslant \tfrac{1}{6}$ . Obviously, the detection range is the same for both two-body and many-body quantum systems. On the other hand, if the dimensions are different in many-body quantum systems, then we have the following result.Let ${{\mathscr{P}}}_{i}$ be L different sets of ETF in ${{\mathscr{C}}}^{{d}_{i}}$ . Assume that ρ is a density matrix in ${\left({{\mathscr{C}}}^{{d}_{i}}\right)}^{\otimes L}$ . If ρ is fully separable, then
$\begin{eqnarray}{{\mathscr{S}}}_{7}(\rho )\leqslant \displaystyle \frac{1}{L}\displaystyle \sum _{i=1}^{L}\displaystyle \frac{{d}_{i}^{2}-2{d}_{i}+{n}_{i}}{{n}_{i}^{2}-{n}_{i}},\end{eqnarray}$
where $\begin{eqnarray}{{\mathscr{S}}}_{7}(\rho )=\mathop{\max }\limits_{\left\{{{\mathscr{P}}}_{i}^{j}\right\}\subseteq {{\mathscr{P}}}_{i}}\displaystyle \sum _{i=1}^{n}\mathrm{Tr}({\otimes }_{j=1}^{L}{{\mathscr{P}}}_{i}^{j}\rho ),\end{eqnarray}$
and $n=\min \{{d}_{1},{d}_{2},\cdots ,{d}_{L}\}$ .Similar to the proof of theorem 6 , there is no redundancy here. Besides, the Cauchy–Schwarz inequality can also be applied to optimize the separability of many-body high-dimensional quantum systems.
7. Conclusion
In this paper, based on ETFs and some inequality, we presented several new entanglement detection criteria in different two-body quantum systems. Furthermore, a series of more efficient entanglement criteria are proposed based on ρ − ρA ⨂ ρB. In addition, the lower bound relation between the proposed entanglement criteria is proved, respectively. Finally, some common quantum states, including Bell diagonal states and Werner states, are used to prove that the proposed criteria are somewhat powerful and comprehensive.
As is well known, it is an interesting problem to construct new and more efficient divisibility criteria using different measurements. For instance, the concept of mutually unbiased frames has been introduced as a general framework for studying unbiasedness [67]. We also hope that the proposed entanglement detection framework can provide insightful hints for solving the existence problems of MUBs and SIC-POVMs from the perspective of entanglement. In addition, with advances in physics experimental techniques, complex measurements or designs with interesting incompatible features may become available resources in enabling quantum information processing.
Conflict of interest statement
The authors declare that there are no conflicts of interest.
Acknowledgments
This work is supported by the Natural Science Foundation of Sichuan Province (Grant No. 25QNJJ4066).