Entanglement Witness $2.0$: Compressed/Mirrored Entanglement Witnesses

An entanglement witness is an observable detecting entanglement for a subset of states. We present a framework that makes an entanglement witness twice as powerful due to the general existence of a second (lower) bound, in addition to the (upper) bound of the very definition. This second bound, if non-trivial, is violated by another subset of entangled states. Differently stated, we prove via the structural physical approximation that two witnesses can be compressed into a single one. Consequently, our framework shows that any entanglement witness can be upgraded to a witness $2.0$. The generality and its power are demonstrate by applications to bipartite and multipartite qubit/qudit systems.

Revealing an unknown quantum state of a physical system is of fundamental importance in quantum information theory and any application. Quantum state tomography aims to determine the full knowledge of a given physical system by series of quantum measurements, however, in a lot of times this is only in principle possible or connected to cost-inefficient, demanding and time-consuming experimental procedures even if the systems consists only of for few qubits. Often one is only interested in the question whether the state is entangled or not and what type of entanglement is present, for instance bipartite or genuinely multipartite entanglement. Then so called entanglement witnesses (EW), particular observables, can do the job. Such observables typically require less experimental setups to unambiguously verify the existence of entanglement and even additional information on the type of entanglement may be revealed. This concept relies on he fact that the set of separable states are convex, i.e. EWs correspond to hyperplanes separating some entangled states from the separable set [1]. In high-dimensional and multi-partite systems also the every entanglement structure becomes of interest which has a nested convex structure [2][3][4].
EWs exhibit numerous advantages. They are observables on a single-copy level and are factored into local observables, namely no entanglement resources are needed to realize EWs. These properties makes them suited for experimental investigations. A major drawback is that a prior information is needed to be at hand in advance. This is easily explained when looking at Fig. 1: The entangled state ρ A is detected by the witness W , whereas the states ρ B , ρ C are not. Optimizing the witness W to the witness W − detects also state ρ B to be entangled, but still does not detect the entangled state ρ C . For this state another witness needs to be constructed or state tomography has to be applied. The framework EW 2.0 proves that a cEW W has both lower and upper bounds for separable states, where the bounds correspond to distinct EWs W (±) . The same experimentally obtained information is also capable to detect the entanglement of state ρC in addition to ρA,B.
This paper shows that indeed every EW has a 'twin' without the need to perform additional experimental measurements. In Fig. 1 this is visualized by the EW W (+) detecting ρ C . In other words, one single EW is as much as two EWs or two EWs can be compressed into one. Consequently, the capability of detection entangled states is improved. Since our framework upgrades any EW to two, we call these framework EWs 2.0. Differently stated, EWs have in general upper and lower bounds for which we present examples.
The paper is organized as follows. First, we give a concise introduction to entanglement witnesses, then we introduce the framework EW 2.0. In a further step we show how it relates to the structural physical approximation of EWs. Then we show the power of the framework EW 2.0 by three examples. Example 1 shows how arXiv:1811.09896v1 [quant-ph] 24 Nov 2018 a given entanglement witness applied to bipartite qubits detects via its 'twin' other entangled states which in this case turns out to be exhaustive for all locally maximally mixed states. Example 2 considers physical states with dimensions higher than two and discusses a witness that is stronger than the PPT-criterion, namely being capable to detect bound entanglement. The Example 3 applies the framework EW 2.0 to tripartite systems. Last but not least we show how the framework EW 2.0 can be improved by a structural X-physical approximation followed by a summary and outlook. }, that give descriptions of measurement devices. We can restrict the consideration to normalized EW without loss of generality, i.e., trW = 1, on a Hilbert space An EW W lies at the border of the set of separable states if there exists a separable state σ ∈ S sep such that tr(W σ) = 0 and no finer EW exists, then it is called optimal [6]. It is clear that any EW W can be shifted by subtracting a positive operator such that it may lie at the border ( in Fig. 1 the EW W (−) is finer than W since it detects all entangled states of W and more). An equivalent and operational definition states that W is optimal if and only if there is no other EW detecting all states that can be detected by W . Optimal EWs are the collections of finest EWs. An EW W is called decomposable if it has a form W = A + B Γ for A, B ≥ 0, where B Γ denotes the partial transposition on the operator B. An EW which is not of this form is called non-decomposable and can detect bound entangled states. Bound entanglement has been detected experimentally firstly for two photons entangled in their orbital angular momentum degrees of freedom [5].
Framework of EW 2.0: For a positive operator W ∈ S(H), where S(H is the set of all quantum states, let λ max ( W ) and λ min ( W ) denote the maximal and minimal eigenvalues, respectively. We also introduce U (·) and L(·) as upper and lower bounds that satisfy an optimization over all separable states We call the range, [L( W ), U ( W )], the separability window of W . Obviously, for a particular state σ ∈ S sep we have Therefore we introduce the following definition: We call an observable W a compressed entanglement witness (cEW) if it holds that λ min ( W ) < L( W ) and U ( W ) < λ max ( W ). The upper and the lower bounds are violated by some entangled states, thus both upper and lower bounds can detect entangled states.
A cEW W corresponds to two EWs. The upper bound in Eq. (2) is equivalent to an EW W (+) as follows, Interestingly, EWs W (±) lie on the border of the set of separable states, see Fig. 1. That is, there exist separable states σ (±) ∈ S sep such that Tr(W (±) σ (±) ) = 0. Structural physical approximation: Here we show how our framework EW 2.0 relates to the structural physical approximation (SPA) [7,8] of EWs. For an EW W its SPA can be written as a non-negative observable where minimal p + ∈ (0, 1) is chosen such that W P ≥ 0. Let us call W P a positive SPA (p-SPA) in the sense that an EW W is admixed with a positive fraction 1 − p + > 0.
In a similar vein, we introduce the negative SPA (n-SPA) with a negative fraction of an EW W as follows with maximal p − > 1 such that W Q ≥ 0. Note that there exists an α > 0 such that α I 1 ⊗ I 2 − W is an EW for which its p-SPA finds the n-SPA W Q of the EW W .
Proposition 1. Suppose that two EWs denoted by W (±) that lie on the border of separable states S sep . Then they can be compressed to a single observable W ≥ 0 via the p-SPA to W (+) and the n-SPA to W (−) and the upper and lower bounds to the cEW W are obtained by L( W ) = p + /D and U ( W ) = p − /D, namely Entangled states are detected by violating either of the bounds.
The proof is straightforward. The proposition also shows the relation of two EWs W (±) compressed to a single observable, i.e.
Obviously, the two EWs W (±) decompose the unity and therefore one is generated by the other one by subtracting from the unity. We call two EWs W (±) satisfying the relation in Eq. (6) SPA-mirrored EW.
Generalization to standard EWs: The framework EW 2.0 is so far presented with non-negative operators W ≥ 0, that can be interpreted as quantum states. In the following, let us show the framework EW 2.0 with standard EWs. Proposition 2. Suppose that an observable W is the cEW of two EWs W (±) . Then the EWs W (±) have upper bounds satisfied by all separable states where the upper bounds are given as follows Entangled states are detected by violating either of the bounds.
Example 3 (Three-qubit states) In Ref. [14] a general formalism was introduced to detect different types of entanglement for any number of particles and dimensions. Let us here consider three qubits. For instance, the function (with ρ ij being the elements of the density matrix ρ) is greater zero for all fully separable but also for all biseparable states. Thus a violation of 0 ≤ Q GHZ (ρ) proves that the state ρ is genuinely multipartite entangled. Moreover, the function is maximized for the Greenberger-Horne-Zeilinger state, e.g. |GHZ = 1 √ 2 (|000 + |111 ). The factor 2 is chosen to set Q GHZ (|GHZ GHZ|) = ±1. Now let us apply our procedure to obtain a new lower bound. Different to what we considered so far is that this inequality correspond to a non-linear witness, but of course this witness can be linearised by using (i) √ ρ ii ρ jj ≤ ρii+ρjj 2 and (ii) assuming ρ 18 to be purely real or imaginary. This gives Q lin GHZ = 1−ρ 11 −ρ 88 +2Re{ρ 18 } and obviously still the same optima ±1 for |GHZ . But considering the optimization over all fully separable state σ sep results in Consequently, the upper bound looses its predictive power. Another physically distinct genuine multipartite entangled state is the W -state or Dicke-state, e.g. in the computational basis given by |Dicke = 1 √ 3 {|001 + |010 + |100 } and the function [14] Q Dicke (ρ) = 2( ρ 22 + ρ 33 + ρ 55 2 + √ ρ 11 ρ 44 + √ ρ 11 ρ 66 which is maximal (normalized to 1) for Dicke states |Dicke (but not for |GHZ ). Optimizing over all separable states leads to Indeed the upper bound is again a non-trivial one since max Q Dicke (ρ ent ) = 1.5 (actually the ρ ent equals |GHZ ). Of course, one can also consider the optimization over all fully separable and bi-separable states, then, however, both criteria Q GHZ/Dicke provide no longer non-trivial upper bounds, they are optimal in this sense. X-physical approximation (XPA): Finally, we develop the structure of the framework EW 2.0 by generalizing SPA. The generalization replaces I 1 ⊗ I 2 /D in a positive and negative SPAs in Eqs. (3) and (4) with a fullrank, non-negative and unit-trace operator X ∈ S sep (H), such that given an EW W one defines p-XPA and n-XPA of W as follows with minimal p X < 1 such that P X ≥ 0, and with maximal q X > 1 such that Q X ≥ 0. Note that for a full-rank, non-negative and unit-trace operator X, there always exist such p X and q X [15]. We call P X (Q X ) a p-XPA (n-XPA) to W . Hence, if W is a cEW and X > 0, then we can define a pairs of EWs {W holds which reduces to Eq. (6) for X = I 1 ⊗ I 2 /D. This construction shows that for any separable state X one can produce new pairs of EWs {W Summary and Outlook: To summarize, we have presented the framework of EW 2.0 that compresses entanglement detection of two EWs into a single one or vice versa constructs out of a single EW two EWs. This works because EWs generally also have (non-trivial) upper bounds satisfied for all separable states. Our findings show that the recorded experimental knowledge can be utilized to detect a larger set of entangled states than the one from the very definition of a witness via our procedure. The Examples presented illustrate how the framework EW 2.0 works in detail for different physical systems. Last but not least, we showed how the framework can be generalised by the X-physical approximation, again enlarging the detection capacity of entanglement without the need of additional resources.
Our results pave a new avenue in the theory of entanglement detection leaving open some questions: It is interesting to find how all properties of standard EWs such as optimality, (non-)decomposability, extremality, etc., are related to EW 2.0. Furthermore it would be interesting to characterize a set of EWs that can be generated by a single EW (via X), as well as their relations to the properties of the original EWs. Naturally, the complexity of multipartite systems offers many distinct possibilities to further investigate its convex-nested structure which are key properties for novel quantum algorithms.