Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems

The notion of steering was first introduced by Schrödinger in 1935 in order to capture the essence of the Einstein–Podolsky–Rosen argument [1]. It describes the ability of one experimenter, Alice, to remotely affect the state of another experimenter, Bob, through local actions on her system supported by classical communication. Steering is based on a quantum correlation strictly between entanglement and non-locality, meaning that not every entangled state can be used for steering and not every steerable state violates a Bell inequality [2].

Recently, it has been shown that steering plays a fundamental role in various quantum protocols and in entanglement theory. In the former, steering characterizes systems useful for one-sided device-independent quantum key distribution [3], subchannel discrimination [4] and randomness generation [5]. Concerning entanglement theory, steering has been used to find counterexamples to Peres conjecture, which was an open problem for more than fifteen years [6,7,8]. Steering is also known to be closely related to incompatibility of quantum measurements. Namely, any set of non-jointly measurable observables is useful for demonstrating steering [9,10], and every incompatibility problem can be mapped into a steering problem in a one-to-many manner [11,12,13].

The extension of steering to multipartite systems has also been proposed. In the multipartite setting the concept of steering has some ambiguity in it. Whether one is interested in the typical spooky action at a distance [14,15,16] or in a more detailed semi-device independent entanglement verification scheme [17,18], one ends up with two different definitions. Here we are interested in the latter scenario as it relates more closely to our approach.

To detect steerability of a given bipartite quantum state might turn into a cumbersome task. The question of steerability (with given measurements on Alice’s side and tomography on Bob’s side) can be formulated as a semidefinite program (SDP) [19,20,21] and as such one could imagine that the task is straightforward and easy to implement. Whereas SDP methods provide a powerful tool for steering detection, they are often restricted to systems with only a few measurements and small dimensions due to computational limitations. One should mention, though, that SDP methods can be used to set bounds for steering even in a scenario with a continuum of measurements [22,23,24]. Another way of detecting steering is through criteria based on correlations [2,25,26,27,28,29]. Whereas these criteria are mostly analytical and straightforward to evaluate, they are also often either not optimal or limited to qubit systems.

In Ref. [30], steering criteria are developed from entropic uncertainty relations (EURs). The criteria are based on generalized entropies, hence, forming an extension of the entropic criteria in Refs. [31,32]. The work falls into the second category of the aforementioned classification of steering criteria, and, as pointed out by the authors, the criteria of Ref. [30] manage to beat other correlation-based methods either in applicability or in detection power. In this work we extend the analysis of Ref. [30] to Rényi entropies and to tripartite steering scenarios. We discuss in detail the question of steerability with local and global measurements in the tripartite setting. Please note that recently similar efforts have been pursued in the context of Rényi entropies [33].

This work is organized as follows. First, we introduce the concept of steering for bipartite and tripartite systems in Section 2. Second, we present some useful entropies for the characterization of steering, followed by bounds of EURs in Section 3, where we also propose some bounds for Tsallis entropies, obtained from numerical investigations. We explain the criteria for the detection of steering from EURs in Section 4. In Section 5 we provide a connection to existing entanglement criteria. In Section 6 we investigate the optimal parameters from generalized entropies for the detection of steering, followed by the application of the criteria to some common examples. Finally, in Section 7 we extend the criteria to the tripartite case, and apply it to noisy GHZ and W states. We conclude the paper with some final remarks.

In a bipartite steering scenario, Alice and Bob share a quantum state, Alice performs local actions (measurements) on her part of the state and Bob is left with non-normalised states (or a state assemblage) depending on Alice’s choice of measurement and her reported outcomes. The task for Bob is to verify if his assemblages could be prepared using a separable state or not [29]. In a more formal manner, we can assume that Alice performs a measurement A with outcome i on her part of the system, while Bob performs a measurement B with outcome j on his part. From that, they can obtain the joint probability distribution of the outcomes. If for all possible measurements A and B one can express the joint probabilities in the form then the shared state is called unsteerable. Here, $p\left(i\right|A,\lambda )$ is a general probability distribution, while $p_Q\left(j\right|B,\lambda )={\mathrm{Tr}}_B\left[B\left(j\right){\sigma }_{\lambda }\right]$ is a probability distribution originating from a quantum state ${\sigma }_{\lambda }$. Furthermore, $B\left(j\right)$ denotes a measurement operator, i.e., $B\left(j\right)\ge 0$ and ${\sum }_{j}B\left(j\right)=\mathbb{𝟙}$, and ${\sum }_{\lambda }p\left(\lambda \right)=1$, where $\lambda$ is a label for the hidden quantum state ${\sigma }_{\lambda }$. A model as in Equation (1) is called a local hidden state (LHS) model, and if it exists, Bob can explain all the results through a set of local states $\left\{{\sigma }_{\lambda }\right\}$ which is only altered by the classical information about Alice’s performed measurement and the recorded outcome. Otherwise, the state is called steerable. One should notice that for a state to be unsteerable, one has to prove the existence of an LHS model for all possible measurements on Alice’s side and for a tomographically complete set on Bob’s side, whereas for proving steerability it suffices to find a set of measurements for Alice and Bob for which the probabilities cannot be expressed as Equation (1).

For multipartite systems, LHS models can be extended in different ways. For simplicity, let us consider the case of tripartite systems. In addition to the notation used before, we assume that Charlie performs measurements C with outcomes labelled with k. Then, one possibility is to ask if Alice can steer the state of Bob and Charlie. If for all possible measurements A, B and C the joint probability distribution can be expressed as the system is called unsteerable from Alice to Bob and Charlie. Here, $p_Q\left(j\right|B,\lambda )p_Q\left(k\right|C,\lambda )=\mathrm{Tr}[B\left(j\right)\otimes C\left(k\right)({\sigma }_{\lambda }^B\otimes {\sigma }_{\lambda }^C)]$, where the hidden states of Bob and Charlie are factorizable. We require the factorizability in order to distinguish the tripartite scenario from a bipartite one (i.e., Bob and Charlie being a single system) where unsteerability is defined as with $p_Q(j,k|B,C,\lambda )=\mathrm{Tr}[B\left(j\right)\otimes C\left(k\right){\sigma }_{\lambda }^{BC}]$. Please note that the factorizability requirement includes all hidden state models using separable states through a redefinition of the hidden variable space. From a physical point of view, Equation (2) corresponds to tests of full separability with untrusted Alice; whereas Equation (3) corresponds to tests of biseparability in the $A|BC$ cut with untrusted Alice.

Another possibility is to ask whether the joint probability distribution of measurements performed by Alice, Bob and Charlie, can be expressed as which means that the system is unsteerable from Alice and Bob to Charlie with factorizable post-processing, meaning that we assume the post-processings on one party to be independent of that of the other party. This extra assumption is one possibility to distinguish, between bipartite and tripartite scenarios. Please note that one could also require non-signalling instead of factorizability of the post-processings. In a purely bipartite scenario an unsteerable joint probability distribution would be given by

Similarly to the above scenario, Equation (4) corresponds to tests of full separability with untrusted Alice and Bob; whereas Equation (5) corresponds to tests of biseparability in the $AB|C$ cut with untrusted Alice and Bob.

One should notice that, in the steering scenario from Alice and Bob to Charlie, there is a difference whether Alice and Bob decide to perform global or local measurements. A simple example of this difference can be explored in the framework of super-activation of steering [34]. Here, the authors show that while one copy of a quantum state is unsteerable, many copies of the same state become steerable, in the sense that steerability is “activated”. Namely, consider a state ${\varrho }_{ABC{C}^{\prime }}={\varrho }_{AC}\otimes {\varrho }_{B{C}^{\prime }}$, where ${\varrho }_{B{C}^{\prime }}$ is a copy of ${\varrho }_{AC}$, and ${\varrho }_{AC}$ is unsteerable, but its steerability can be super-activated (where only two copies is already enough [34]). For this state, local measurements give an unsteerable state assemblage, whereas, because of super-activation, it is steerable with global measurements.

In this section we present the detailed derivation of the generalized entropic steering criteria proposed in Ref. [30]. Here we show the results for Shannon and Tsallis entropies, as has been made in Ref. [30], and also extend the criteria for Rényi entropy. Please note that the proof is not at all restricted to these functions as it can be applied to all functions which satisfy the following properties: (i) (pseudo-)additivity for independent distributions; (ii) state independent EUR; and (iii) joint convexity of the relative entropy. In the following we present our proof for specific entropies, and the method becomes clear from its application to each of them.

At this point, it is interesting to connect our approach with the entanglement criteria derived from EURs [53]. In Ref. [53], it has been shown that for separable states the following inequality holds. Here, $A_1$ and $A_2$ ($B_1$ and $B_2$) are observables on Alice’s (Bob’s) laboratory, and Bob’s observables obey an EUR $S_q\left(B_1\right)+S_q\left(B_2\right)\ge {\mathcal{B}}_B^{\left(q\right)}$. Differently from our approach, $S_q(A_m\otimes B_m)$ is the entropy of the probability distribution of the outcomes of the global observable $A_m\otimes B_m$. Please note that for a degenerate $A_m\otimes B_m$ the probability distribution differs from the local ones. For instance, measuring ${\sigma }_z\otimes {\sigma }_z$ gives four possible local probabilities $p_{++},p_{+-},p_{-+},p_{--},$ but for the evaluation of $S(A_m\otimes B_m)$ one combines them as $q_{+}=p_{++}+p_{--}$ and $q_{-}=p_{+-}+p_{-+}$, as these correspond to the global outcomes.

There are some interesting connections between our derivation of steering inequalities and this entanglement criterion. First, the proof in Ref. [53] is based on EURs for Bob’s observables (the same as our criteria), and this is the only quantum restriction in the criterion, so Equation (49) is a steering inequality, meaning that all probability distributions of the form in Equation (1) fulfil it. Second, in Ref. [53] it was observed that the criterion is strongest for values $2\le q\le 3$, which seems to be the case also for our criteria (shown later). Third, for special scenarios (e.g., Bell-diagonal two-qubit states and Pauli measurements), Equation (49) and Equations (40) and (45) give the same results. However, it does not hold for more general scenarios.

The approach of Ref. [53] has been slightly improved in Ref. [56], where the main idea is to recombine the probability distribution in a different way (see below). Also, the criteria in Ref. [56] are more general in the sense that can be applied to any symmetric and concave function.

Similar to the case of Ref. [53], the criteria from Ref. [56] can be also applied to steering. To see this, let us first explain the main ideas in [56]. In this work, they consider concave and symmetrical (i.e., invariant under the permutation of variables) functions $f:{\mathbb{R}}^n⟶\mathbb{R}$. For simplicity, define where $e=\left\{|e_i\rangle \right|i=1,\dots ,n\}$ is an orthonormal basis of the n-dimensional Hilbert space in which the state ${\rho }_0$ acts. Then, one can construct a probability matrix $P=\left(p_{ij}\right)$, where the elements are defined as $p_{ij}=\langle e_i^A|\langle e_j^B|\rho |e_j^B\rangle |e_i^A\rangle$, where $e_A=\left\{|e_i^A\rangle \right|i=1,\cdots ,n_A\}$ and $e_B=\left\{|e_j^B\rangle \right|j=1,\cdots ,n_B\}$ are orthonormal bases of the $n_{A\left(B\right)}$-dimensional Hilbert space ${\mathcal{H}}_{A\left(B\right)}$.

Then, define a permutation matrix $Q=\left(q_{ij}\right)$, where $\{q_{i1},\dots ,q_{i{n}_B}\}$ is a permutation of an $n_B$-element set $S=\{s_1,\dots ,s_{n_B}\}$ for $i=1,\dots ,n_A$. For example, if we consider the case of $n_A=n_B=3$, three examples of possible constructions of Q are

Now, define where $\delta (a,b)$ is the Kronecker function. Here, the argument of this function is the combination of the probabilities given a permutation matrix Q. If we take the third example in Equation (51), we have

Here one can see that the combination of the probabilities will depend on the permutation matrix Q.

Given the above definitions, the authors prove the following bound for product states (${\varrho }_{AB}={\varrho }_A\otimes {\varrho }_B$), holding for any permutation matrix Q and any concave symmetrical function f. The bound is an entanglement criterion for pure states. Here, one can notice that the right-hand side of Equation (54) is independent of the space ${\mathcal{H}}_A$, giving some hint that the criterion actually detects steerability of the state.

Using the notation $e_k^A=\left\{|e_{i_k}^A\rangle \right|i=1,\dots ,n_A\}$ and $e_k^B=\left\{|e_{j_k}^B\rangle \right|j=1,\dots ,n_B\}$ for different bases of ${\mathcal{H}}_{A\left(B\right)}$, the authors prove that for any separable state ${\varrho }_{AB}$ holds for arbitrary symmetrical concave functions $f_k$, permutation matrices $Q_k$ and bases $e_k^{A\left(B\right)}$. Equation (55) is a general entanglement criterion based on symmetrical concave functions $f_k$. In order to find the optimal criteria, an optimization over all possible permutation matrices should be performed. A specific criterion is given for the case where $f_k$ is replaced by the Shannon entropy, and the bound in Equation (55) is related to EURs.

Now we show that the above entanglement criterion is actually a steering criterion, given that Equation (55) can be obtained if one considers an LHS model. Note first that one can include general measurements into the above considerations by defining where the operators ${\left\{M_i\right\}}_i$ form a positive operator valued measure (POVM) and ${\varrho }_0$ is a quantum state. Then, taking an unsteerable state ${\rho }_{AB}$ and labelling by $N_i$ the POVM elements of Alice’s measurements, one has for a fixed hidden variable $\lambda$

Hence,

On the second line we use concavity of the function f, and in the third line we use symmetry. Taking the sum over all hidden variables $\lambda$ gives

Considering more measurements one has which is exactly the same criteria of Equation (55). This means that the entanglement criteria proposed in Ref. [56] are actually steering criteria.