How good is a positive map at detecting quantum entanglement?

This is a Perspective on "The Non-m-Positive Dimension of a Positive Linear Map" by Nathaniel Johnston, Benjamin Lovitz, and Daniel Puzzuoli, published in Quantum 3, 172 (2019).

By Ion Nechita (CNRS, LPT Toulouse).

Since the dawn of quantum mechanics, entanglement has been a central notion of the theory [18], a very important body of work being dedicated to understanding, classifying, measuring, and characterizing this very important property of quantum states. Once it has been recognized that it is computationally hard to decide whether a given quantum state is entangled or separable (i.e. classically correlated), researchers turned their attention to finding entanglement criteria, that is computationally efficient tests which guarantee the presence of quantum entanglement, without being also necessary for it.

In order to define entangled quantum states, we start with the opposite notion, that of separable states: a quantum state $\rho_{AB}$, shared between two parties Alice and Bob is called separable if it can be written as a convex mixture of product states between our two protagonists:
\begin{equation}\label{eq:separable-decomposition}\tag{1}
\rho_{AB} = \sum_{i=1}^r p_i \sigma_A^{(i)} \otimes \sigma_B^{(i)},
\end{equation}
where $\sigma_A^{(1)}, \ldots, \sigma_A^{(r)}$ are states on Alice’s system, the $\sigma_B^{(i)}$ are states on Bob’s system, and the $p_i$ are convex weights. In other words, separable states are the quantum states that Alice and Bob can prepare locally, using only shared randomness. States which are not separable are called entangled. The prototypical entangled state is the maximally entangled state of two qubits:
\begin{equation}\label{eq:maximally-entangled-state}\tag{2}
| \psi_{me} \rangle = \frac{1}{\sqrt 2} (|00\rangle + |11\rangle) \in \mathbb C^2 \otimes \mathbb C^2.
\end{equation}
For unit rank quantum states $\rho = | \psi \rangle \langle \psi |$, called pure, deciding whether the state is separable or entangled is easy: one needs to decide whether the reshaped matrix $\Psi$ has rank one, a task which is computationally cheap, using standard algorithms for computing singular- or eigenvalues. Recall that to a quantum state $|\psi\rangle = \sum_{ij} x_{ij} |ij \rangle$, one associates a $d \times d$ matrix $\Psi = \sum_{ij} x_{ij} |i \rangle \langle j|$. The tensor $\psi$ is called the vectorization of $\Psi$, while $\Psi$ is said to be a reshaping of $\psi$ into a matrix. For the example of the maximally entangled state above, one computes $\Psi_{me} = I_2 / \sqrt 2$, which is a matrix of rank 2, certifying the entanglement of the pure state $|\psi_{me}\rangle$.

In the case of mixed quantum states (positive semidefinite matrices of unit trace), the situation is much more complicated. The decision problem associated to the (weak) membership problem for the set of separable states has been proven by Gurvits [8] to be NP-hard, in some precise way of encoding the size of the input and the required precision. The result of Gurvits means that there is no universal, computationally cheap, and exact criterion for entanglement and separability; one needs to allow for some errors if one desires some fast way of testing entanglement.

Computationally efficient entanglement tests have existed for a long time, the most important being the Peres-Horodecki positive partial transpose (PPT) test [17,10]: if the partial transposition of a quantum state is not positive semidefinite, then the state is entangled:
$$[\operatorname{id} \otimes \operatorname{transp}](\rho_{AB}) \ngeq 0 \implies \rho_{AB} \text{ is entangled.}$$
The operation $[\operatorname{id} \otimes \operatorname{transp}](X)$ is cleverly denoted by $X^\Gamma$, the $\Gamma$ superscript being meant to represent half of the transpose. In the case of the rank one (pure state) case of the maximally entangled case from \eqref{eq:maximally-entangled-state}, the partial transposition criterion gives
$$\frac 1 2 \begin{bmatrix}
1 & 0 & 0 & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
1 & 0 & 0 & 1
\end{bmatrix}^\Gamma =
\frac 1 2 \begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}, \text{ with spectrum $(-1, 1, 1, 1)/2$}.$$
The presence of the negative eigenvalue $-1$ in the spectrum above proves that the maximally entangled state is, indeed, entangled.

In their seminal work [10], the Horodeckis proved a very intriguing result, giving an equivalent characterization of separability: a quantum state $\rho_{AB} \in \mathcal M_d(\mathbb C) \otimes \mathcal M_d(\mathbb C)$ is separable if and only if for every positive map $\Phi : \mathcal M_d(\mathbb C) \to \mathcal M_d(\mathbb C)$, the matrix $[\operatorname{id} \otimes \Phi](\rho_{AB})$ is positive semidefinite. Here, we use the notion of positivity in the $C^*$ algebra sense: a linear map $\Phi$ between to matrix algebras is called positive if it sends positive semidefinite matrices to positive semidefinite matrices, see [16]. Hence, to show that a quantum state $\rho_{AB}$ is entangled, it suffices to find a single positive map $\Phi$ who’s partial action on the quantum state renders it non-positive semidefinite; such a map is called an entanglement witness (to be more precise, in the literature the Choi-Jamio{\l}kowski matrix [12,5] of $\Phi$ is called an entanglement witness [11, Sections VI B 2,3]). The partial transposition criterion corresponds to the very important choice $\Phi = \operatorname{transp}$.

The problem with the Horodeckis’ characterization of separability is that one needs to check the positive semidefinite condition for all positive maps. One can get around this in small dimensions: for $2 \otimes 2$ and $2 \otimes 3$ systems, it is enough to check the positivity of the partial transposition; this fact is a non-trivial result in operator algebra due to Woronowicz [20], see also [2, Section 2.4.5] for a very pleasant treatment of the $2 \otimes 2$ case. In dimension larger than $6 = 2\times 3$, the transposition map does not suffice: there exist PPT entangled states. Moreover, one does not get away with using “more” positive maps, see [19,1,7].

In [13], the authors investigate the duality relation between positive maps and entanglement from a different perspective. Instead of trying to find a subclass of positive maps necessary and sufficient for characterizing entanglement, they look at the set of density matrices which are certified as entangled by a given positive map. The larger this set is, the better is the given fixed map at detecting entanglement.

Precisely, the authors introduce the non-$m$-positive
dimension
of a positive map $\Phi:\mathcal M_d(\mathbb C) \to \mathcal M_d(\mathbb C)$, which measures how large a subspace of $\mathbb C^d \otimes \mathbb C^d$ can be if every quantum state supported on the subspace is non-positive semidefinite under the partial action of
$\Phi$. Equivalently, this is the maximal number of negative eigenvalues that the adjoint map $\operatorname{id}_m \otimes \Phi^*$ can produce from a positive semi-definite input, where the identity map acts on $m \times m$ matrices. This number, denoted $\nu_m(\Phi)$ has been previously computed by one of the authors in the case of the transposition map [14]:
$$\nu_m(\operatorname{transp}_d) = (m-1)(d-1).$$

The authors proceed then to study many properties of the quantities $\nu_m$, and, importantly, to define a regularized version thereof,
$$\nu(\Phi):=\lim_{m \to \infty} \frac{\nu_m(\Phi)}{m}.$$
It is important to show that the limit above exists, this being a non-trivial result. The authors proceed then to give important lower bounds for the quantities $\nu_m$ and $\nu$, which are then applied to some important classes of positive maps in quantum information theory, such as the reduction map [4], the Choi map [6], or the Breuer-Hall map [3,9].

Overall, the results in [13] are an important contribution to the theory of positive linear maps, which is, mainly due to the close relation to entanglement theory, a very active field of research. Recently, many important mathematical contributions have been made from research groups with a quantum theory background ([15] or [21] just to mention two recent ones), evidence of the strong interactions between the two fields.

► BibTeX data

► References

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