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Multipartite entanglement detection via correlation minor norm

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Entanglement is a uniquely quantum resource giving rise to many quantum technologies. It is therefore important to detect and characterize entangled states, but this is known to be a challenging task, especially for multipartite mixed states. The correlation minor norm (CMN) was recently suggested as a bi-partite entanglement detector employing bounds on the quantum correlation matrix. In this paper, we explore generalizations of the CMN to multipartite systems based on matricizations of the correlation tensor. It is shown that the CMN is able to detect and differentiate classes of multipartite entangled states. We further analyze the correlations within the reduced density matrices and show their significance for entanglement detection. Finally, we employ matricizations of the correlation tensor for introducing a measure of global quantum discord.

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E.C. was supported by the Israeli Innovation Authority under Projects 70002, 73795 and the Eureka Program, by Elta Systems Ltd., by the Pazy Foundation, by the Israeli Ministry of Science and Technology, and by the Quantum Science and Technology Program of the Israeli Council of Higher Education. This research was supported by the Fetzer-Franklin Fund of the John E. Fetzer Memorial Trust and by Grant No. FQXi-RFP-CPW-2006 from the Foundational Questions Institute and Fetzer Franklin Fund, a donor-advised fund of Silicon Valley Community Foundation. We thank Etai Flint for the illustrations in Figs. 2, 5 and 6. We also acknowledge the useful tutorial in which helped us generating Fig. 4.

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Appendix A: Proofs of some theorems

1.1 A. Proving the bounds on fully separable states

Under the SFNF assumption, the singular values of \(\mathcal {C}_{flat}\) and \(\mathcal {W}_{flat}\) (of the dVH criterion [10]) will be the same, up to an extra singular value:

$$\begin{aligned} \sigma (\mathcal {C}_{flat}) = \left[ \sigma _0 = \prod _i \left( \frac{1}{\sqrt{d_i}} \right) , \sigma _1, \ldots , \sigma _{d^2-1} \right] \;, \;\sigma (\mathcal {W}_{flat}) = \left[ \sigma _1, \ldots , \sigma _{d^2-1} \right] .\nonumber \\ \end{aligned}$$

This is due to the fact that the singular values do not change under substitution of rows/columns and removal of rows/columns of zeros. Furthermore, the extra singular value is due to the main tensor vertex: \({\langle A_0\otimes B_0 \otimes \cdots \otimes N_0\rangle }\), which is not zero under SFNF.

Thus, we may bound the CMN for different cases:

Proof of Theorem 3

We wish to claim that \(\sigma _0 = \prod _i \left( \frac{1}{\sqrt{d_i}} \right) \) is among the h largest singular values, we may use Eq. (15) to claim that:

$$\begin{aligned} \sigma _{h} \le \frac{1}{h}\prod _{j=1}^N\sqrt{\frac{d_j-1}{d_j}}, \end{aligned}$$

because the singular values are in descending order, so in the “worst case” they are equal. Thus, for our bounds, we claim that:

$$\begin{aligned} \sigma _0 = \prod _{i=1}^N \left( \frac{1}{\sqrt{d_i}} \right) \ge \frac{1}{h}\prod _{i=1}^N\sqrt{\frac{d_i-1}{d_i}} \Rightarrow h \ge \prod _{i=1}^N\sqrt{d_i-1}. \end{aligned}$$

Thus, a multipartite state in SFNF, under the any bi-partition A|B, holds that:

$$\begin{aligned} \begin{aligned}&\mathcal {M}_{h,p=\infty } = \prod _{k=0}^{h-1} \sigma _k \left( \mathcal {C}_{flat} \right) = \prod _i \left( \frac{1}{\sqrt{d_i}} \right) \prod _{k=1}^{h-1} \sigma _k \left( \mathcal {W}_{flat} \right) \\&\qquad \qquad \;\; \le \prod _i \left( \frac{1}{\sqrt{d_i}} \right) \left( h-1 \right) ^{-\left( h-1 \right) } \left[ \sum _{k=1}^{h-1} \sigma _k \left( \mathcal {W}_{flat} \right) \right] ^{h-1} \\&\qquad \qquad \;\; \le \prod _i \left( \frac{1}{\sqrt{d_i}} \right) \left( h-1 \right) ^{-\left( h-1 \right) }\left[ \prod _i\left( \frac{d_i-1}{d_i}\right) \right] ^{\frac{h-1}{2}}. \end{aligned} \end{aligned}$$

wherein \(C_h\) is the compound matrix, and we have used the inequality of arithmetic and geometric means in the fourth transition and Eq. (15) the last transition. Using our definitions for \(\alpha \) and \(\beta \), we are done. \(\square \)

Proof of Theorem 4

For a multipartite state in SFNF, under the any bi-partition A|B, we obtain:

$$\begin{aligned}&\mathcal {M}_{h,p=1} = S_h \left( \alpha , \sigma _1, \ldots , \sigma _{d^2-1} \right) \nonumber \\&\qquad \qquad \;\, = \, \alpha S_{h-1} \left( \sigma _1, \ldots , \sigma _{d^2-1} \right) + S_h \left( \sigma _1, \ldots , \sigma _{d^2-1} \right) \end{aligned}$$

Let us denote \( s :=\sum _{k=1}^{d^2-1} \sigma _k \). Clearly \( s \le \beta \). Moreover, the vectors \( \vec {\sigma } :=\left( \sigma _1, \ldots , \sigma _{d^2-1} \right) \) and \( \vec {e} :=\frac{s}{d^2-1} \left( 1, \ldots , 1 \right) \) both sum up to s; thus, \( \vec {\sigma } \succeq \vec {e} \) (\(\succeq \) denotes majorization). Since the symmetric polynomials \( S_h \) are Schur concave, we obtain:

$$\begin{aligned} S_h \left( \sigma _1, \ldots , \sigma _{d^2-1} \right) \le S_h \left( \frac{s}{d^2-1}, \ldots , \frac{s}{d^2-1} \right) . \end{aligned}$$

Next, we use the fact that \( S_h \) is monotonically increasing in each of its variables, alongside the inequality \(s \le \beta \), to obtain:

$$\begin{aligned} S_h \left( \sigma _1, \ldots , \sigma _{d^2-1} \right) \le S_h \left( \frac{\beta }{d^2-1}, \ldots , \frac{\beta }{d^2-1} \right) . \end{aligned}$$

Substitution in Eq. (27) yields:

$$\begin{aligned} \mathcal {M}_{h,p=1}&\le \alpha S_{h-1} \left( \frac{\beta }{d^2-1}, \ldots , \frac{\beta }{d^2-1} \right) + S_h \left( \frac{\beta }{d^2-1}, \ldots , \frac{\beta }{d^2-1} \right) \nonumber \\&= S_h \left( \alpha , \frac{\beta }{d^2-1}, \ldots , \frac{\beta }{d^2-1} \right) \end{aligned}$$

where \(\beta \) is always repeated \(d^2-1\) times. \(\square \)

1.2 B. Proving of the CMN as a global quantum discord measure

Proof of Theorem 6

Consider a mutipartite state partitioned into a bi-partite state A and B, with the respective matricization of the correlation tensor \(\mathcal {C}\). We may further consider a local measurement on one of the parties the construct A, for example, if A consists of two parties we may measure \(\mathbbm {1} \otimes \sigma _X\). The evolution of the correlation matrix under such a measurement is given by: \(\mathcal {C}' =\mathcal {A}_i\mathcal {C}\), where \(\mathcal {A}_i\) is a \(d^2_{A} \times d^2_{A}\) matrix (the construction of \(\mathcal {A}_i\) is exactly the same as in Theorem 1 of [23]). Now, by Theorem 6.7(7) in [26], for all \(k \in \left\{ 1, \ldots , d_A^2 \right\} \) we have

$$\begin{aligned} \sigma _k \left( \mathcal {A}_i\mathcal {C} \right) \le \left\Vert \mathcal {A}_i \right\Vert _1 \sigma _k \left( \mathcal {C} \right) . \end{aligned}$$

If we were to measure on B, the matrix multiplication would be on the right, and the the construction still holds:

$$\begin{aligned} \sigma _k \left( \mathcal {C}\mathcal {B}_i \right) \le \left\Vert \mathcal {B}_i \right\Vert _1 \sigma _k \left( \mathcal {C} \right) . \end{aligned}$$

Due to the fact that \(\Pi \) consists of local projective measurement, which can be interchanged, we may continue measuring the state, where all measurements will comprise \(\Pi \). Each measurement will further decrease the value of the singular values of the (post-measurement) correlation matrix, yielding:

$$\begin{aligned} \sigma _k \left( \mathcal {F}_A\mathcal {C} \mathcal {F}_B \right) \le \left\Vert \mathcal {F}_A \right\Vert _1\left\Vert \mathcal {F}_B \right\Vert _1 \sigma _k \left( \mathcal {C} \right) . \end{aligned}$$

wherein \(\mathcal {F}_A = \prod _i\mathcal {A}_i\), \(\mathcal {F}_B = \prod _i\mathcal {B}_i\) represent the transformation undergone by the correlation matrix under \(\Pi \), and \(\mathcal {A}_i\), \(\mathcal {B}_i\) are commutative, as they represent non-local measurements.

Because \( \mathcal {A}_i \) is a projection, it holds that: \( \left\Vert \mathcal {A}_i \right\Vert _1 = \max _j \sigma _j \left( \mathcal {A} \right) = 1 \), and the same for a measurement on B. Therefore, \( \sigma _k \left( \mathcal {F}_A\mathcal {C} \mathcal {F}_B \right) \le \sigma _k \left( \mathcal {C} \right) \) for all k, and we conclude that \( \mathcal {M}_{h,p} \left( \rho \right) \ge \mathcal {M}_{h,p} \left( \rho ' \right) \) for all hp, using the fact that the CMNs are all monotonically non-decreasing w.r.t. the singular values \( \sigma _k \).

Suppose \(\rho \) has zero discord, which happens if and only if there exists a measurement \(\Pi \) that does not disturb the state, i.e., there exist a matrices \( \mathcal {F}_A\) and \(\mathcal {F}_B \) such that \( \mathcal {F}_A\mathcal {C} \mathcal {F}_B = \mathcal {C} \). Then, for this choice of measurement, we have \( \mathcal {M}_{h,p} \left( \rho \right) - \mathcal {M}_{h,p} \left( \rho ' \right) = 0 \). By the non-decreasing property for the CMN we have proven above, this is indeed the maximum, hence \( \mathcal {D}_{h \le 2, p} \left( \rho \right) = 0 \). \(\square \)

Appendix B: Some states used in this work

First, let us present the SIC-POVM with Bell state: \( \rho _1 = \sum _{i=1}^4 \frac{1}{4}\rho ^{\textit{SIC-POVM}}_i\otimes \left| \psi ^{Bell}_i\right\rangle \left\langle \psi ^{Bell}_i\right| \). Wherein:

$$\begin{aligned} \rho ^{SIC-POVM}_1= & {} 0.5 \begin{bmatrix} \frac{\sqrt{3}+1}{\sqrt{3}} &{} \frac{\sqrt{3}}{3}(1+i)\\ \frac{\sqrt{3}}{3}(1-i) &{} \frac{\sqrt{3}-1}{\sqrt{3}} \end{bmatrix} \nonumber \\ \rho ^{\textit{SIC-POVM}}_2= & {} 0.5 \begin{bmatrix} \frac{\sqrt{3}-1}{\sqrt{3}} &{} \frac{\sqrt{3}}{3}(1-i)\\ \frac{\sqrt{3}}{3}(1+i) &{} \frac{\sqrt{3}+1}{\sqrt{3}} \end{bmatrix} \nonumber \\ \rho ^{\textit{SIC-POVM}}_3= & {} 0.5 \begin{bmatrix} \frac{\sqrt{3}+1}{\sqrt{3}} &{} \frac{\sqrt{3}}{3}(-1-i)\\ \frac{\sqrt{3}}{3}(-1+i) &{} \frac{\sqrt{3}-1}{\sqrt{3}} \end{bmatrix} \nonumber \\ \rho ^{\textit{SIC-POVM}}_4= & {} 0.5 \begin{bmatrix} \frac{\sqrt{3}-1}{\sqrt{3}} &{} \frac{\sqrt{3}}{3}(-1+i)\\ \frac{\sqrt{3}}{3}(-1-i) &{} \frac{\sqrt{3}+1}{\sqrt{3}} \end{bmatrix} \nonumber \\ \psi ^{Bell}_1= & {} \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix} \nonumber \\ \psi ^{Bell}_2= & {} \frac{1}{\sqrt{2}} \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix} \nonumber \\ \psi ^{Bell}_3= & {} \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 0 \\ 0 \\ -1 \end{bmatrix} \nonumber \\ \psi ^{Bell}_4= & {} \frac{1}{\sqrt{2}} \begin{bmatrix} 0 \\ 1 \\ -1 \\ 0 \end{bmatrix} \end{aligned}$$

As said, the SIC-POVM states can be presented as Bloch vectors:

$$\begin{aligned} r_1^{\textit{SIC-POVM}}= & {} \frac{1}{\sqrt{3}} \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix} \nonumber \\ r_2^{\textit{SIC-POVM}}= & {} \frac{1}{\sqrt{3}} \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} \nonumber \\ r_3^{\textit{SIC-POVM}}= & {} \frac{1}{\sqrt{3}} \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix} \nonumber \\ r_4^{SIC-POVM\textit{SIC-POVM}}= & {} \frac{1}{\sqrt{3}} \begin{bmatrix} -1 \\ -1 \\ -1 \end{bmatrix} \end{aligned}$$

Note that in order for our construction to work, the indices of the states cannot be changed.

The state \(\rho _2\) appears in [27] as a .mat file, as it was numerically generated.

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Lenny, R., Te’eni, A., Peled, B.Y. et al. Multipartite entanglement detection via correlation minor norm. Quantum Inf Process 22, 292 (2023).

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