Abstract
Characterization of entanglement in composite quantum systems is a difficult task. Already in the bipartite case, it was proven to be NP-hard [Gur03] (Gurvits, Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing, STOC ’03, 2003). However, as we have already seen in Sect. 2.1.2, there exist several criteria which give sufficient conditions to certify that a state is entangled, of which the most celebrated one is the Positive under Partial Transposition (PPT) criterion. Nevertheless, the characterization of states which are both PPT and entangled remains elusive.
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Notes
- 1.
- 2.
Since, for any observable M, one has \(\mathrm {Tr}[M\rho _{\overline{S}}]=\mathrm {Tr}[(\mathbbm {1}_S\otimes M)\rho _{\mathbf {A}}]\), by taking \(M=| \phi \rangle \!\langle \phi |\), with \(| \phi \rangle \in \ker \rho _{\overline{S}}\), the following identity holds: \(0=\langle \phi |\rho _{\overline{S}}| \phi \rangle =\mathrm {Tr}[| \phi \rangle \!\langle \phi |\rho _{\overline{S}}]=\mathrm {Tr}[(\mathbbm {1}_S\otimes | \phi \rangle \!\langle \phi |)\rho _{\mathbf {A}}]=\sum _{i}\mathrm {Tr}[| i \rangle \!\langle i |\otimes | \phi \rangle \!\langle \phi |\rho _{\mathbf {A}}]\). Hence, \(\sum _i \langle i |\langle \phi |\rho _{\mathbf {A}}| i \rangle | \phi \rangle =0\) and, because \(\rho _{\mathbf {A}}\succeq 0\), every summand has to be zero. Hence, \(| i \rangle | \phi \rangle \in \ker \rho _{\mathbf {A}}\) for every \(| i \rangle \). By linearity, \(| \psi \rangle | \phi \rangle \in \ker \rho _{\mathbf {A}}\) for any \(| \psi \rangle \in \mathcal{H}_\mathcal{S}\).
- 3.
A fact that is easily seen when expressing the vectors in the Dicke states basis defined in Eq. (2.27), where the identity \(P_{\mathbf {A}}(| D_{|S|}^k \rangle | D_{|\overline{S}|}^{l} \rangle )=| D_n^{k+l} \rangle \) holds.
- 4.
A generic property in mathematics is one that holds for typical examples. From the point of view of measure theory, a generic property is one that holds almost everywhere, or with probability 1. Topologically, or from the point of view of algebraic geometry, a generic property holds on a dense open set. Equivalently, it does not hold in a nowhere dense set (a set whose closure has empty interior).
- 5.
This assumption is crucial; otherwise we could not discard that all the m roots that are close to infinity for small r go to the unspecified roots for large r and that the n roots that are close to zero for large r go to the unspecified roots for small r, thus never crossing the unit circle.
- 6.
For finite-dimensional spaces, since \(\mathcal{H}^*\otimes \mathcal{K}\cong {\hom }(\mathcal{H}, \mathcal{K})\), where \(\mathcal{H}^*\) is the dual space of \(\mathcal{H}\); that implies we can arrange the coefficients of \(| \Phi \rangle \) in a \((k+1)\times (k+1)\) matrix.
- 7.
It suffices to take n different values of \(\alpha \), denoted \(\alpha _i\), because the determinant of a \(n\times n\) Vandermonde matrix is \(\prod _{1\le i < j \le n}(\alpha _i-\alpha _j)\).
- 8.
It is sufficient to check the sign of the minimal eigenvalue of a O(n) number of matrices that have size \(O(n^2)\), where n is the number of qubits, as we shall see in Remark 3.26.
- 9.
The correction \(\mathrm {Tr} h\) is added just to impose \(\mathrm Tr \rho (x)=1\) for all x.
- 10.
Here we use the identity \(A\otimes B \mathrm {vec(X)} = \mathrm {vec}(A X B^T)\), where the vectorization operator acts as \(\mathrm {vec}| a \rangle \langle b | = | a \rangle | b \rangle \) and it is extended by linearity. The partial transposition acts just as a permutation of the components of h.
- 11.
The case of four qubits was studied in detail in [Aug+10].
- 12.
Because \(\mathcal{D}^{\mathrm {PPT}}\) arises as the intersection of \(\mathcal{D}^{PPT, S}\) for all the considered bipartitions \(S|\overline{S}\) of \(\mathbf {A}\), \(x_*\) corresponds to the smallest x such that the one of the eigenvalues of \(\rho _{\mathbf {A}}^{T_{S}}\) changes its sign for the first bipartition \(S|\overline{S}\) that this sign change happens.
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Tura i Brugués, J. (2017). PPT Entangled Symmetric States. In: Characterizing Entanglement and Quantum Correlations Constrained by Symmetry. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-49571-2_3
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