Witnessing random unitary and projective quantum channels: Complementarity between separable and maximally entangled states

Modern applications in quantum computation and quantum communication require the precise characterization of quantum states and quantum channels. In practice, this means that one has to determine the quantum capacity of a physical system in terms of measurable quantities. Witnesses, if properly constructed, succeed in performing this task. We derive a method that is capable to compute witnesses for identifying deterministic evolutions and measurement-induced collapse processes. At the same time, applying the Choi-Jamiołkowski isomorphism, it uncovers the entanglement characteristics of bipartite quantum states. Remarkably, a statistical mixture of unitary evolutions is mapped onto mixtures of maximally entangled states, and classical separable states originate from genuine quantum-state reduction maps. Based on our treatment, we are able to witness these opposing attributes at once and, furthermore, obtain an insight into their different geometric structures. The complementarity is further underpinned by formulating a complementary Schmidt decomposition of a state in terms of maximally entangled states and discrete Fourier-transformed Schmidt coefficients.

Figure 1

The mapping of the Choi-Jamiołkowski isomorphism $\mathcal{J}$ is depicted via the arrows. On the one hand, the deterministic evolution together with a classical statistical description (RU maps) is mapped onto the set of bipartite states which are formed by pure entangled states with equally weighted Schmidt coefficients, i.e., ME states. On the other hand, the maps which correspond to the genuine quantum description of the measurement process (RP maps) have the image of the set of classically correlated (separable) states.

Figure 2

The (gray) ball depicts the volume of all mixed quantum states which is bounded by states of the form (14) for $d=3$. The octahedron (blue) represents the set of separable states and the cube (red) describes ME states.

Figure 3

The maximal projection of a pure bipartite state $|\psi \rangle$ with the Schmidt coefficients $\vec{\sigma }$ onto separable (left plot) and ME (right plot) is shown. The (gray) sphere indicates the normalization of the state, $\parallel \vec{\sigma }{\parallel }_2^2=1$. Whenever the overlap $\langle \psi \left|\widehat{\varrho }\right|\psi \rangle$ of a bipartite state is outside of one of those surfaces, we have an inseparable (left) or non-ME (right) state $\widehat{\varrho }$.

Figure 4

The interpretations of norms of the vector of Schmidt coefficients $\vec{\sigma }$ are shown. For normalized states, we have $\parallel \vec{\sigma }{\parallel }_2=1$. If, in addition, the top row is fulfilled, we have a separable (S) or ME state. Choosing a pure state to be a witness, we get the upper bounds for separable or ME states in the middle row. The relation between the Schmidt coefficients and complementary Schmidt coefficients $\vec{\tau }$, with $\parallel \vec{\tau }{\parallel }_2=1$, under the GDFT [Eq. (57)] is given in the bottom row.