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Characterizing Entanglement via Uncertainty Relations

We derive a family of necessary separability criteria for finite-dimensional systems based on inequalities for variances of observables. We show that every pure bipartite entangled state violates some of these inequalities. Furthermore, a family of bound entangled states and true multipartite entangled states can be detected. The inequalities also allow to distinguish between different classes of true tripartite entanglement for qubits. We formulate an equivalent criterion in terms of covariance matrices. This allows us to apply criteria known from the regime of continuous variables to finite-dimensional systems.

We derive a family of necessary separability criteria for finite-dimensional systems based on inequalities for variances of observables. We show that every pure bipartite entangled state violates some of these inequalities. Furthermore, a family of bound entangled states and true multipartite entangled states can be detected. The inequalities also allow to distinguish between different classes of true tripartite entanglement for qubits. We formulate an equivalent criterion in terms of covariance matrices. This allows us to apply criteria known from the regime of continuous variables to finite-dimensional systems. The detection of entanglement is one of the fundamental problems in quantum information theory. From a theoretical point of view one can try to answer the question whether a given entirely known state is entangled or not, but despite a lot of progress in the last years [1], no general solution of this problem is known. In experiments, one aims at detecting entanglement without knowing the state completely. Bell inequalities [2] and entanglement witnesses [3] are the main tools to tackle this task. Both rely on inequalities for mean values of observables, and violation of them implies entanglement. Entanglement criteria based on inequalities for variances of observables have also been studied, mainly designed for continuous variable systems [4]. Recently, it has been shown that variances of observables of a special type can detect entanglement also in finite dimensional bipartite systems [5]. Even an example of bound entanglement, allowing detection in this way, has been found [6].
In this paper, we present a generalization of this approach by considering arbitrary observables. This leads to new results in various directions. First, we show how the entanglement of every pure bipartite entangled state can be detected with our method. Second, we present inequalities for the detection of a class of bound entangled states. Third, we show how multipartite entanglement can be detected. Our approach even admits to distinguish different classes of true multipartite entanglement. Finally, we show that our criteria are equivalent to a criterion in terms of covariance matrices. This formulation allows us to translate separability criteria known from continuous variables [7,8] to finite-dimensional systems.
We start with the following observation. Let ̺ be a density matrix, and let M be an observable. We denote the expectation value of M by M ̺ := T r(̺M ) and the variance (or uncertainty) of M by We suppress the dependence on ̺ in our notation, when there is no risk of confusion. If ̺ is a pure state the variance is zero iff ̺ is an eigenstate of M. Now we have: Lemma 1. Let M i be some observables and ̺ = k p k ̺ k be a convex combination (i.e. p k ≥ 0, k p k = 1) of some states ̺ k within some subset S. Then holds. We call a state "violating Lemma 1" iff there are no states ̺ k ∈ S and no p k such that Eq. (2) is fulfilled.
Proof. This fact is known, see e.g. [5,9]. The inequality holds for each M i : ✷ Please note that this Lemma has a clear physical meaning: One cannot decrease the uncertainty of an observable by mixing several states.
In most cases we will be interested in separable states. They can be written as a convex combination of product states. So, unless stated otherwise, we will assume that in Lemma 1 the set S denotes the set of product states. Then violation of this Lemma implies entanglement of the state ̺, which can be detected with uncertainties. In fact, the main idea of this paper is to show that Eq. (2) yields strong sufficient entanglement criteria, if we choose the M i appropriately.
For completeness, let us remind the reader of the socalled "Local Uncertainty Relations" (LURs), introduced by Hofmann and Takeuchi [5]. Let A i be observables on Alice's space of a bipartite system. If they do not share a common eigenstate, there is a number C A > 0 such that i δ 2 (A i ) ̺A ≥ C A holds for all states ̺ A on Alice's space. Hofmann and Takeuchi showed: Proposition 1. Let ̺ be separable and let A i , B i , i = 1, ..., n be operators on Alice's (resp. Bob's) space, fulfilling holds.
The LURs provide strong criteria which can by construction be implemented with local measurements. Nevertheless, they have some disadvantages: it is not clear which operators A i and B i one should choose to detect a given entangled state. Also, LURs can by construction characterize separable states only; they do not apply for other convex sets. Finally, it is not clear how to generalize them to multipartite systems. In fact, no LUR for the detection of true multipartite entanglement in finite dimensional systems is known so far.
A way to overcome these disadvantages is to consider nonlocal observables. For an experimental implementation one can decompose any nonlocal observable into local operators i.e. write it as a sum of projectors onto product vectors [10]. Each of the terms in this sum can then be measured locally. The measurement of a nonlocal observable in this way has recently been implemented [11]. Let us start with the case of two qubits: Proposition 2. Let |ψ 1 = a|00 + b|11 be an entangled two qubit state written in the Schmidt decomposition, with a ≥ b. Then, there exist M i such that for |ψ 1 , We only need to prove the bound (4) for a pure product vector |φ . We have To see this, we expand |ψ i in a product basis |ψ i = kl C i kl |kl and also any product vector: ✷ Note that Eq. (4) allows to detect a mixed state of the form This two-qubit example reveals already a main idea for the construction of the M i : We take one M i as the projector onto the range of ̺, and the others as projectors onto a basis of the kernel. This construction suffices to detect all bipartite pure entangled states, many bound entangled states and multipartite entanglement. Proof. Let U be the space orthogonal to |ψ 1 . It is clear that U contains at least one entangled vector |ψ e . We can choose a basis |ψ i , i = 2, ..., N M of U which consists only of entangled vectors. (To do this, we choose an arbitrary, not necessarily orthogonal, basis. If it contains product vectors, we perturb them by adding ε|ψ e ). Then we take M i = |ψ i ψ i |. The only possible common eigenstates of the M i are the |ψ i . So for product states the sum over all uncertainties is bounded from below, while it is zero for |ψ 1 .
✷ It is interesting that the existence of an entangled basis in the kernel of a state ̺ suffices to derive uncertainty relations also for a class of bound entangled states. Here, the class of bound entangled states which we want to consider are those arising from an unextendible product basis (UPB) [12]. They can be constructed as follows: Let {|φ i = |e i |f i , i = 1, ..., n} be a UPB, i.e. all |φ i are (pairwise orthogonal) product vectors, not spanning the whole space, and there is no product vector orthogonal to all |φ i . Then the state is a bound entangled state. N denotes the normalization. Proof. Let U be the subspace spanned by the UPB. Then U must contain at least one entangled vector. To see this, note that subspaces containing only product vectors are of the form {|v = |a |b } with a fixed |a (or |b ) for all |v . But then there would be product vectors orthogonal to U. Due to the existence of an entangled vector in U, there is, according to the proof of Proposition 3, an entangled basis |ψ i ; i = 1, ..., n of U. We take The common eigenstates of the M i are not product vectors, and we have i δ 2 (M i ) ̺UP B = 0.
✷ Before we demonstrate that our method is also useful for multipartite systems, we recall some facts about three qubit systems [13,14]. A three-qubit state is called fully separable iff it can be written as a convex combination of triseparable pure states, and it is called biseparable (BS) iff it is a convex combination of pure states, which are separable with respect to a bipartite splitting. Otherwise, it is called fully entangled. There are two classes of fully entangled states which are not convertible into each other by stochastical local operations and classical communication. These classes are called the GHZ-class and the W-class, and the W-class forms a convex set inside the GHZ-class [14]. The following inequalities are formulated so, that they can be checked with local measurements: Proposition 5. Let ̺ be a three qubit state. We define a sum of variances as If E < 1/2, the state ̺ is fully tripartite entangled. If E < 3/8, the state ̺ belongs even to the GHZ-class. For the GHZ-state |GHZ = 1/ √ 2(|000 + |111 ) we have E = 0.
The method presented here can be extended to more parties. For four qubits one can write down 16 orthogonal GHZ-states of the type |ψ i = (|x 2 . If we then define M i = |ψ i ψ i | it follows as in the proof of Prop. 5. that for all biseparable states i δ 2 (M i ) ≥ 1/2 holds. The same idea can be used to construct uncertainty relations for an arbitrary number of qubits, but the number of M i increases exponentially. Now we want to put our results in a more general framework. Let ̺ = k p k ̺ A k ⊗ ̺ B k be a separable state and let M i , for i = 1, ..., n be some observables. We can define two functionals Z(̺)[X] and W (̺)[X] with X = (x 1 , ..., x n ) ∈ Ê n by: Z is called the generating functional of the moments and W generates the cumulants [16]. We have Z(̺) = , and due to the concavity of the logarithm it follows that holds. The lhs of this equation has a minimum at X = 0. Thus, the matrix of the second derivatives with respect to X of the lhs is positive semidefinite at X = 0. The matrix of the second derivatives of W is the, so-called, covariance matrix (CM) γ(̺, M i ). Its entries are: where we have denoted ∂ i := ∂ ∂xi . We will mention some properties of the CM later. First, we can state: Lemma 2. Let ̺ be separable, and let M i be observables. Then there exist product states ̺ A k ⊗ ̺ B k and p k such that holds. We call a state "violating Lemma 2" iff there are no product states ̺ A k ⊗ ̺ B k and convex weights p k , such that Eq. (9) is fulfilled. Proposition 6. γ has the following properties: (i) The entries are given by: The diagonal entries are the variances: γ ii = δ 2 (M i ).
Proof. (i) can be directly calculated from the definition with the help of standard formulas for differentiating operators [17] and the power expansion of the logarithm. (ii) can be proven by calculating ✷ Of course, one could exhibit other properties of γ, but these two properties suffice for our purpose: Theorem 1. A state ̺ violates Lemma 1 with S being the set of separable states iff it violates Lemma 2. The observables leading to a violation may differ.
Proof. A state violating Lemma 1 violates also Lemma 2 with the same M i , since the sum of all variances is the trace of γ. To show the other direction, let us assume that γ violates Lemma 2 and look at the set of symmetric matrices of the form T := { k p k γ(̺ A k ⊗ ̺ B k , M i ) + P } where P ≥ 0 is positive. T is convex and closed. We have γ / ∈ T, and due to a corollary of the Hahn-Banach-Theorem [18] there exists a symmetric matrix W and a number C such that T r(W γ) < C while T r(W µ) > C for all µ ∈ T [19]. Since T r(W P ) ≥ 0 for all P ≥ 0, we have W ≥ 0. Now we use the spectral decomposition and write W ij = l λ l a l i a l j with λ l ≥ 0. Let us define N l = √ λ l i a l i M i . Then T r(W γ) = l δ 2 (N l ) < C according to Eq. (11), while for all convex combinations of product states k p k l δ 2 (N l ) ̺ A k ⊗̺ B k > C holds. This is a violation of Lemma 1 for the N l .
✷ So Lemma 2 is equivalent to Lemma 1. But the advantage of Lemma 2 is that it allows us to relate the uncertainty relations to entanglement criteria known from continuous variables, mainly Gaussian states. Let us note some facts about Gaussian states [7,8]. The nonlocal properties of a Gaussian state are completely encoded in a real symmetric CMγ, which is the CM of the canonical conjugate observables position and momentum. A Gaussian stateγ in a bipartite system is separable iff there are CMsγ A ,γ B for Alice and Bob such thatγ ≥γ A ⊕γ B [7,20]. In fact, the separability problem for Gaussian states was solved in Ref. [8], where a mapγ 1 →γ 2 was defined which maps a separable (resp. entangled) CM to another separable (resp. entangled) CM. By iterating this map, one gets a CM for which separability is easy to check. We can now formulate criteria similar to [7,8]: Proposition 7. Let ̺ be separable and let A i , i = 1, ..., n and B i , i = 1, ..., m be observables on Alice's resp.
Bob's space. Define M i = A i ⊗ ½, i = 1, ..., n and M i = ½ ⊗ B i , i = n + 1, ..., n + m. Then there are states ̺ A k and ̺ B k and convex weights p k such that if we define holds. Moreover: We write γ(̺, M i ) in a block structure: where A = γ(̺, A i ) (resp. B = γ(̺, B i )) is an n × n-(resp. m × m)-matrix. Then we have [21]: Proof. Eq. (12) follows from Lemma 2 and the fact that for these special M i for product states: Eqns. (14,15) arise from the first step of the algorithm in [8]. We arrived already after the first step at a block diagonal matrix and thus at a fixed point. We prove it as in [8] with a Lemma proven there: Let γ be matrix with a block structure as in (13). Equivalent are [21]: (a) γ ≥ 0; (b) ker(B) ⊆ ker(C) and A − CB −1 C T ≥ 0; (c) ker(A) ⊆ ker(C T ) and B−C T A −1 C ≥ 0. Applying the equivalence (a)-(b) to Eqs. (12,13) yields with Proposition 6, A − C(B − κ B ) −1 C T ≥ κ A ≥ 0. With the equivalence (b)-(c) we get B − κ B − C T A −1 C ≥ 0, which proves Eq. (15). The proof of (14) is similar. ✷ In view of the first part of the paper we can state that any pure entangled state and any bound entangled UPB state can be detected with the methods of Lemma 2. Identifying more general classes of states which allow a detection leads to the difficult problem of characterizing the possible k p k γ(̺ A k ⊗ ̺ B k , M i ) in Eq. (9), resp. the κ A/B in Eqns. (14,15). We only know some properties of them: Taking A i = B i = σ i , i = x, y, z for two qubits, we know that i δ 2 (A i ) ≥ 2 [5], thus T r(κ A ) ≥ 2. Applying this to the Werner states ̺(p) = p|ψ ψ| + (1 − p)½/4 with |ψ = (|01 − |10 )/ √ 2 one can calculate that they are detected via Eq. (14) for p ≥ 1/ √ 3. In conclusion, we have developed a family of necessary separability criteria based on inequalities for variances. We have shown that they are strong enough to detect the entanglement in various experimental relevant situations. We have formulated an equivalent criterion in terms of covariance matrices, which enabled us to connect continuous variable systems with finite-dimensional systems. The question whether there are entangled states which do not violate any uncertainty relation, is a very interesting one; here, it remains open.