Introduction

The predictions of quantum theory are incompatible with any physical model that satisfies a natural principle of locality, as shown by Bell1,2. Specifically, the correlations obtained by performing local measurements on an entangled quantum state violate an inequality, Bell’s inequality, which is satisfied by all local correlations. Understanding the link between entanglement and Bell nonlocality is a longstanding and challenging problem2,3. While the observation of Bell inequality violation implies the presence of entanglement, it is still not known whether all entangled states can lead to Bell inequality violation.

While nonlocality turns out to be a generic feature of all entangled pure states4,5, the situation is however much more complex for mixed states. There exist mixed entangled states which are local, as they admit a local hidden variable model6, even for the most general type of non-sequential measurements7. Nevertheless, it turns out that certain local entangled states can violate a Bell inequality when a more general scenario is considered. If pre-processing by local operations and classical communication (LOCC) is performed before the local measurements, the ‘hidden nonlocality’ of some local entangled states can be revealed8,9,10. Alternatively, when several copies of the state can be jointly measured in each run of the Bell test, nonlocality can be super-activated11,12. Finally, the nonlocality of certain local entangled states can be revealed by placing several copies of the state in a quantum network13,14.

In the most general case, an arbitrary number of copies of the state can be pre-processed by LOCC operations before performing the Bell test. Hence the problem becomes intimately related to entanglement distillation15. A bipartite entangled state is said to be distillable if, from an arbitrary number of copies, it is possible to extract pure entanglement by LOCC16. It thus follows that any entangled state that is distillable can lead to Bell inequality violation.

There exist however entangled states which are not distillable, so-called ‘bound entangled’ states17, shown to be relevant for instance in quantum many-body systems18,19,20. Hence the phenomenon of entanglement displays a form of irreversibility. On the one hand, entanglement is required to produce a bound entangled state, that is, the state cannot be produced via LOCC. On the other hand, no pure entanglement can ever be extracted from a bound entangled state by LOCC. This leads naturally to the question of whether bound entangled states can also violate a Bell inequality. In 1999, Peres21 first discussed this problem and conjectured that bound entanglement can never lead to Bell inequality violation. Originally, the conjecture was formulated using the notion of partial transposition22, one of the most useful tools for characterizing entanglement23, directly related to symmetry under time reversal and to distillability of entanglement17. Specifically, Peres suggested that entangled quantum states with positive partial transpose (PPT)22, and hence no distillable entanglement, can never give rise to nonlocality. An alternative formulation of the conjecture is that any entangled state that leads to Bell inequality violation must be non-positive under partial transposition (NPT). Peres’ intuition was that distillability of entanglement is equivalent to nonlocality, that is, the violation of a Bell inequality by measurements on a quantum state necessarily implies that some pure entanglement can be distilled out of this state.

In recent years, an intense research effort has been devoted to this problem. Several works provided evidence in favour of the Peres conjecture, showing that the violation of important classes of Bell inequalities implies distillability24,25 and non-positivity under partial transposition26,27. On the other hand, weaker versions of the conjecture were disproven, in the multipartite case28,29,30, and more recently considering the notion of quantum steering31,32,33. However, Peres’ original conjecture remained open, and has become known as one of the main conjectures in quantum information theory. Solving this problem is thus an important challenge as it would lead to a deeper understanding of how different manifestations of the phenomenon of entanglement relate to each other (see Fig. 1).

Figure 1: Relation between different fundamental manifestations of quantum entanglement.
figure 1

Bell nonlocality, non-positivity under partial transposition, and entanglement distillability represent three facets of the phenomenon of entanglement. Understanding the connection between these concepts is a longstanding problem. It is well known that entanglement distillability implies both nonlocality25 and non-positive partial transpose17. Peres21 conjectured that nonlocality implies non-positivity under partial transposition and entanglement distillability; hence represented by the dashed arrows. The main result of the present work is to show that this conjecture is false, as indicated by the red crosses. To complete the diagram, it remains to be seen whether non-positive partial transpose implies distillability, one of the most important open questions in entanglement theory45,46. If this conjecture turns out to be false, it would remain to be seen whether non-positive partial transpose implies Bell nonlocality.

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Here, we disprove the original Peres conjecture. Specifically, we present a bipartite entangled state which is PPT, hence bound entangled, but which can nevertheless violate a Bell inequality. This shows that Bell nonlocality is fundamentally different from both entanglement distillability and non-positivity under partial transposition. Finally, we show that bound entanglement can be useful in nonlocality-based quantum information protocols, in particular for device-independent randomness certification34,35.

Results

Bound entangled state

We start by constructing the bound entangled state, and will later show that it violates a simple Bell inequality. We consider a state of two qutrits, that is, of local Hilbert space dimension d=3. Note that there are no PPT entangled states for qubit–qubit and qubit–qutrit systems23. Specifically, we consider an entangled state of the form

The eigenvalues are , and the eigenvectors are given by

where . The state ρ is part of a family of states recently discussed in ref. 31. Importantly the above choice of eigenvalues and eigenvectors ensures that the state ρ is invariant under the partial transposition map22, that is, PT(ρ)=(⊗TB)(ρ)=ρ, where TB denotes the transposition operation on the second subsystem. This ensures that the state ρ is PPT, that is, PT(ρ)0, and therefore undistillable17.

Bell inequality violation

Nevertheless, the state ρ is entangled, hence bound entangled, as it can lead to Bell inequality violation. To prove this, we consider a Bell test with two distant observers, Alice and Bob. Alice chooses between three measurement settings, xε{0, 1, 2}, and Bob among two settings, yε{0, 1}. Alice’s settings yield a binary outcome, aε{0, 1}. Bob’s first setting (y=0) has a ternary outcome, bε{0, 1, 2}, and his second setting (y=1) is binary, bε{0, 1}. The experiment is thus characterized by the joint probability distribution p(ab|xy). These statistics can be reproduced by a local model if they admit a decomposition of the form:

where λ represents the shared local variable distributed according to the density μ (λ). For the Bell test considered here, all statistics of the above form satisfy the Bell inequality36:

where pA(a|x) and pB(b|y) denote Alice’s and Bob’s marginal distributions. Hence a violation of the above inequality, that is, I>0, implies the presence of nonlocality.

In particular, this can be achieved by performing judiciously chosen local measurements on the bound entangled state ρ. The local measurement operators, acting on , are denoted Ma|x for Alice and Mb|y for Bob. The measurement operators of Alice are rank-1 real-valued projectors M0|x=|Ax›‹Ax|, with

where p=1/5. We have that M1|x=−M0|x, where denotes the identity operator in . Bob’s first measurement is given by (for b=0, 1) by

and M2|0=−M0|0−M1|0. For Bob’s second setting, we take M0|1=|2›‹2| and M1|1=−M0|1. The resulting statistics is given by the probability distribution

These statistics do not admit a decomposition of the form (3), as they lead to a violation of the Bell inequality (4), given analytically by

This proves that a bipartite bound entangled state can give rise to nonlocality, thus disproving the Peres conjecture (see Fig. 2).

Figure 2: Building a counter example to the Peres conjecture.
figure 2

To disprove the conjecture, we construct a quantum state ρ with the following properties: (1) ρ is positive under partial transposition (PPT) and (2) ρ is Bell nonlocal. Property (1) follows here from the fact that ρ is invariant under partial transposition, and implies that ρ cannot be distilled. Property (2) follows from the fact that the statistics resulting from local measurements on ρ violate a simple Bell inequality.

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To derive this result, we followed a numerical optimization method based on semi-definite programming (SDP)37, briefly outlined in the Methods section. The construction described above is however analytical, and was reconstructed from the output of the optimization procedure. In fact, slightly higher Bell violations, up to IPPT =2.6526 × 10−4, could be found numerically for two-qutrit PPT states. Moreover, using the SPD techniques of ref. 38, an upper bound on the largest possible violation obtainable from PPT states is here found to be , hence leaving the possibility open for a slightly higher violation using PPT states of arbitrary Hilbert space dimension.

Finally, it is worth pointing out that our result implies that the set of PPT states violating Bell inequality (4) is of non-zero measure. Although the Bell violation we observe is small, it is nevertheless finite. Hence it follows that any state obtained by adding a sufficiently small (but finite) amount of an arbitrary separable state to the bound entangled state ρ, will also violate the Bell inequality. As the set of separable states is of non-zero measure, the result follows.

Randomness certification

The fact that a bound entangled state can violate a Bell inequality suggests potential applications in quantum information processing, in particular in nonlocality-based tasks. Here we consider randomness expansion based on Bell nonlocality34,35, in which true quantum randomness can be certified without relying on a detailed knowledge about the functioning of the devices used in the protocol. The security of the protocol is therefore called ‘device-independent’. Following the techniques of refs 39, 40, we obtained lower bounds on the amount of randomness that can be certified from the nonlocal statistics. The randomness is captured by the probability of guessing the outcome of Bob’s (or Alice’s) measurement pg(b|y). Note that this guessing probability is computed by a maximization over all possible realizations that are compatible with the observed data p(ab|xy), see refs 39, 40 for details. To quantify the randomness it is useful to consider the min-entropy, Hmin=−log2pg(b|y), which represents the number of random bits that can be extracted per run of the Bell test (using adequate post-processing) from Bob’s measurement setting y. The results, summarized in Table 1, show that randomness can indeed be certified using a bipartite bound entangled state. Note that in practice, implementing such a protocol would be extremely challenging, as the Bell violation is very small, and hence very sensitive to noise.

Table 1 Device-independent randomness certification using bound entangled states.
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Finally, it would be interesting to see if bound entanglement is useful for other quantum information protocols based on nonlocality. First, given its usefulness in quantum key distribution (QKD)41, it would be interesting to see if bound entanglement could also be used to establish a secret key in the context of device-independent QKD42. Second, our bound entangled state could be useful in communication complexity, a task which is strongly connected to quantum nonlocality. Using the techniques of ref. 43 (see also ref. 44), it should be possible to construct a communication complexity problem for which bound entanglement helps reducing the amount of communication compared to classical resources.

Discussion

To summarize, we have shown that bipartite bound entangled states can lead to Bell inequality violation, thus disproving the longstanding conjecture of Peres. This represents significant progress in our understanding of the relation between entanglement and Bell nonlocality, demonstrating in particular that nonlocality does not imply non-positivity under partial transposition or entanglement distillability (see Fig. 1). The main open question now is whether all bound entangled states can give rise to Bell inequality violation, which would imply that entanglement and nonlocality are basically equivalent. From a more applied perspective, we also showed that bound entanglement can be useful in nonlocality-based quantum information tasks, in particular device-independent randomness certification.

Methods

Numerical method

Consider a Bell inequality of the form

with real coefficients cab|xy and local bound L. To find efficiently violations of such an inequality for PPT states of local Hilbert space dimension d, we use the semi-definite programming procedure described in Box 1.

Additional information

How to cite this article: Vértesi, T. & Brunner, N. Disproving the Peres conjecture by showing Bell nonlocality from bound entanglement. Nat. Commun. 5:5297 doi: 10.1038/ncomms6297 (2014).