# Cornelia Spee

PostdocRoom: B-106

Phone: +49 271 740 3797

## Preprints

See also arxiv

Cornelia Spee, Katharina Schwaiger, Géza Giedke and Barbara Kraus
*
Mode-entanglement of Gaussian fermionic states
*

arXiv:1712.07560

We investigate the entanglement of n-mode n-partite Gaussian fermionic states (GFS). First, we identify a reasonable definition of separability for GFS and derive a standard form for mixed states, to which any state can be mapped via Gaussian local unitaries (GLU). As the standard form is unique two GFS are equivalent under GLU if and only if their standard forms coincide. Then, we investigate the important class of local operations assisted by classical communication (LOCC). These are central in entanglement theory as they allow to partially order the entanglement contained in states. We show, however, that there are no non-trivial Gaussian LOCC (GLOCC). That is, any GLOCC transformation can be accomplished via GLUs. To still obtain insights into the various entanglement properties of n-mode n-partite GFS we investigate the richer class of Gaussian stochastic LOCC. We characterize Gaussian SLOCC classes of pure states and derive them explicitly for few-mode states. Furthermore, we consider certain fermionic LOCC and show how to identify the maximally entangled set (MES) of pure n-mode n-partite GFS, i.e., the minimal set of states having the property that any other state can be obtained from one state inside this set via fermionic LOCC. We generalize these findings also to the pure m-mode n-partite (for m>n) case.

Jannik Hoffmann, Cornelia Spee, Otfried Gühne and Costantino Budroni
*
Structure of temporal correlations of a qubit
*

arXiv:1712.01234

In quantum mechanics, spatial correlations arising from measurements at separated particles are well studied. This is not the case, however, for the temporal correlations arising from a single quantum system subjected to a sequence of generalized measurements. We first characterize the polytope of temporal quantum correlations coming from the most general measurements. We then show that if the dimension of the quantum system is bounded, only a subset of the most general correlations can be realized and identify the correlations in the simplest scenario that can not be reached by two-dimensional systems. This leads to a temporal inequality for a dimension test, and we discuss a possible implementation using nitrogen-vacancy centers in diamond.

## Publications

J. I. de Vicente, C. Spee, D. Sauerwein and B. Kraus
*
Entanglement manipulation of multipartite pure states with finite rounds
of classical communication
*

Phys. Rev. A *95*,
012323
(2017),
arXiv:1607.05145

We studied pure state transformations using local operations assisted by finitely many rounds of classical communication ($LOCC_{\mathbb{N}}$) in C. Spee, J.I. de Vicente, D. Sauerwein, B. Kraus, arXiv:1606.04418 (2016). Here, we first of all present the details of some of the proofs and generalize the construction of examples of state transformations via $LOCC_{\mathbb{N}}$ which require a probabilistic step. However, we also present explicit examples of SLOCC classes where any separable transformation can be realized by a protocol in which each step is deterministic (all-det-$LOCC_{\mathbb{N}}$). Such transformations can be considered as natural generalizations of bipartite transformations. Furthermore, we provide examples of pure state transformations which are possible via separable transformations, but not via $LOCC_{\mathbb{N}}$. We also analyze an interesting genuinely multipartite effect which we call locking or unlocking the power of other parties. This means that one party can prevent or enable the implementation of LOCC transformations by other parties. Moreover, we investigate the maximally entangled set restricted to $LOCC_{\mathbb{N}}$ and show how easily computable bounds on some entanglement measures can be derived by restricting to $LOCC_{\mathbb{N}}$.

C. Spee, J. I. de Vicente, D. Sauerwein and B. Kraus
*
Entangled pure state transformations via local operations assisted by
finitely many rounds of classical communication
*

Phys. Rev. Lett. *118*,
040503
(2017),
arXiv:1606.04418

We consider generic pure $n$-qubit states and a general class of pure states of arbitrary dimensions and arbitrarily many subsystems. We characterize those states which can be reached from some other state via Local Operations assisted by finitely many rounds of Classical Communication ($LOCC_{\mathbb{N}}$). For $n$ qubits with $n>3$ we show that this set of states is of measure zero, which implies that the maximally entangled set is generically of full measure if restricted to the practical scenario of $LOCC_{\mathbb{N}}$. Moreover, we identify a class of states for which any $LOCC_{\mathbb{N}}$ protocol can be realized via a concatenation of deterministic steps. We show, however, that in general there exist state transformations which require a probabilistic step within the protocol, which highlights the difference between bipartite and multipartite LOCC.

M. Hebenstreit, C. Spee and B. Kraus
*
The MES of tripartite qutrit states and pure state separable
transformations which are not possible via LOCC
*

Phys. Rev. A *93*,
012339
(2016),
arXiv:1511.03158

Entanglement is the resource to overcome the restriction of operations to Local Operations assisted by Classical Communication (LOCC). The Maximally Entangled Set (MES) of states is the minimal set of n-partite pure states with the property that any truly n-partite entangled pure state can be obtained deterministically via LOCC from some state in this set. Hence, this set contains the most useful states for applications. In this work we characterize the MES for generic three qutrit states. Moreover, we analyze which generic three qutrit states are reachable (and convertible) under LOCC transformations. To this end we study reachability via separable operations (SEP), a class of operations that is strictly larger than LOCC. Interestingly, we identify a family of pure states that can be obtained deterministically via SEP but not via LOCC. To our knowledge these are the first examples of transformations among pure states that can be implemented via SEP but not via LOCC.

C. Spee, J. I. de Vicente and B. Kraus
*
The maximally entangled set of 4-qubit states
*

J. Math. Phys. *57*,
052201
(2016),
arXiv:1510.09164

Entanglement is a resource to overcome the natural restriction of operations used for state manipulation to Local Operations assisted by Classical Communication (LOCC). Hence, a bipartite maximally entangled state is a state which can be transformed deterministically into any other state via LOCC. In the multipartite setting no such state exists. There, rather a whole set, the Maximally Entangled Set of states (MES), which we recently introduced, is required. This set has on the one hand the property that any state outside of this set can be obtained via LOCC from one of the states within the set and on the other hand, no state in the set can be obtained from any other state via LOCC. Recently, we studied LOCC transformations among pure multipartite states and derived the MES for 3- and generic 4-qubit states. Here, we consider the non-generic 4-qubit states and analyze their properties regarding local transformations. We prove that most SLOCC classes show a similar behavior as the generic states, however we also identify three classes with very distinct properties. The first consists of the GHZ and W class, where any state can be transformed into some other state non-trivially. In particular, there exists no isolation. On the other hand, there also exist classes where all states are isolated. Last but not least we identify an additional class of states, whose transformation properties differ drastically from all the other classes. Our investigations do not only identify the most relevant classes of states for LOCC entanglement manipulation, but also reveal new insight into the similarities and differences between separable and LOCC transformations and enable the investigation of LOCC transformations among arbitrary four qubit states.

C. Spee, J. I. de Vicente and B. Kraus
*
Remote entanglement preparation
*

Phys. Rev. A *88*,
010305(R)
(2013),
arXiv:1302.2471

We introduce a new multipartite communication scheme, with the aim to enable the senders to remotely and obliviously provide the receivers with an arbitrary amount of multipartite entanglement. The scheme is similar to Remote State Preparation (RSP). However, we show that even though the receivers are restricted to local unitary operations, the required resources for remote entanglement preparation are less than for RSP. In order to do so we introduce a novel canonical form of arbitrary multipartite pure states describing an arbitrary number of qubits. Moreover, we show that if the receivers are enabled to perform arbitrary local operations and classical communication, the required resources can drastically be reduced. We employ this protocol to derive robust entanglement purification protocols for arbitrary pure states and show that it can also be used for sending classical information.

Julio I. de Vicente, Cornelia Spee and Barbara Kraus
*
The maximally entangled set of multipartite quantum states
*

Phys. Rev. Lett. *111*,
110502
(2013),
arXiv:1305.7398

Entanglement is a resource in quantum information theory when state manipulation is restricted to Local Operations assisted by Classical Communication (LOCC). It is therefore of paramount importance to decide which LOCC transformations are possible and, particularly, which states are maximally useful under this restriction. While the bipartite maximally entangled state is well known (it is the only state that cannot be obtained from any other and, at the same time, it can be transformed to any other by LOCC), no such state exists in the multipartite case. In order to cope with this fact, we introduce here the notion of the Maximally Entangled Set (MES) of n-partite states. This is the set of states which are maximally useful under LOCC manipulation, i.e. any state outside of this set can be obtained via LOCC from one of the states within the set and no state in the set can be obtained from any other state via LOCC. We determine the MES for states of three and four qubits and provide a simple characterization for them. In both cases, infinitely many states are required. However, while the MES is of measure zero for 3-qubit states, almost all 4-qubit states are in the MES. This is because, in contrast to the 3-qubit case, deterministic LOCC transformations are almost never possible among fully entangled four-partite states. We determine the measure-zero subset of the MES of LOCC convertible states. This is the only relevant class of states for entanglement manipulation.

## Book contributions

C. Spee, J. I. de Vicente and B. Kraus*Few-body entanglement manipulation*

In: (Quantum [Un]speakables II, Springer (2016))

url