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Entanglement detection is, in general, a very hard problem, however, given some information about the state under investigation one can design entanglement witnesses to detect its en-tanglement. In previous section, we mentioned the slipperiness of witnessing bounds due to experimental noise and losses as a major source of difficulty for practical entanglement detec-tion and introduced UEW technique as a soludetec-tion to this challenge. At a fundamental level, however, there are two main assumptions on which any witnessing procedure is built. First, one should “believe” in quantum mechanics, in the sense that the statistics obtained in mea-surements are truly given by the Born rule. This is particularly important if one is willing to make the conclusion that entanglement is some kind of correlations beyond what is allowed by LCSOCC. Of course, this is what we assume to be true throughout the present dissertation.

The second assumption is that one should make sure, using proper investigations and char-acterizations, that any measuring device involved in the process is doing exactly what it is meant to do as described by the theory. For example, suppose that a measuring device is presumed to measure the component of quantum states proportional to the POVM element Πˆ₀ = |ψ₀ihψ₀|. If instead the device makes an erroneous projection onto ˆΠ₁ = |ψ₁ihψ₁| where

|ψ₁i = |ψ₀i + |φi for some arbitrary small error parameter , then the statistic obtained from this device may cause invalid conclusions about the entanglement content of states. This is because the witnessing inequality to be tested is obtained based on the assumption that the statistics are due to projections onto ˆΠ₀ rather than ˆΠ₁. We refer the interested readers to van Enk et al [42] for more details and examples of situations in which such errors lead to incorrect conclusions regarding the entanglement of the state. One can think of this effect as the measuring device deceiving the experimenter.

In order to understand mathematically why such a cheating is possible in principle, we note that every quantum measurement ultimately consist of the following elements: a map f_m: I → M where I = {i}^m_i=1is an index set and M = { ˆΠ_i}^m_i=1is a POVM; a map f_pr : I → P where P = {p_i}^m_i=1 is a probability distribution; finally, a measurement is nothing but the map g_m : M → P such that g_m= f_pr◦ f_m⁻¹. Hence, what we perceive in a measurement procedure is only the map g_m⁻¹ = f_m◦ f_pr⁻¹: we associate the occurrence statistics to each index i ∈ I through the map f_pr⁻¹ and then, by corresponding the index to a POVM element, we relate that statistic to the corresponding POVM element. Hence, it is always possible for an agent to intervene and manipulate every classical element of this process via some classical maps hcl : I → I or ecl : P → P, to either change the indices or the probabilities⁵, to deceive the experimenter.

Importantly, for this to be possible, the intervening agent must know the correspondence f_m

5Note that the former of these maps is ultimately equivalent to the latter one. That is, if f_pr: I → P, then for every map h_cl: I → I there exists a map e_cl: P → P such that f_pr◦ hcl= e_cl◦ fpr.

so that she can appropriately manipulate fpr. This situation makes more sense when it is embedded into the following scenario. Suppose that Alice and Bob are two agents who claim to share an entangled state ˆ%_AB∈ S/ _sep. Charlie, a third agent, plays the role of a referee whose aim is to verify this claim. The obvious strategy is the following.

Protocol 2.2.4.

(i) Charlie obtains some information about the state from Alice and Bob, then he chooses a test operator.

(ii) He decomposes the test operator into local POVM elements as ˆL =P

ijl_ijΠˆ^A_i ⊗ ˆΠ^B_j. (iii) He asks Alice and Bob to carry out the measurements M^A= { ˆΠ^A_i }ⁿ_i=1and M^B = { ˆΠ^B_j}^m_j=1

on their shared state and communicate the outcomes to the referee to obtain the correla-tions p_ij =D ˆΠ^A_i ⊗ ˆΠ^B_jE

∈ P^AB, i.e., he provides f_m: I^A× I^B→ M^A× M^B and asks the agents to build up the map f_pr : I^A× I^B → P^AB.

(iv) By knowing f_mand receiving f_pr from agents, Charlie combines the outcomes to obtain the expectation values D ˆΠ^A_i ⊗ ˆΠ^B_jE

According to our previous clarification, however, Alice and Bob are “untrusted” agents who can potentially abuse their knowledge of the maps f_m to manipulate the probabilities p_ij to deceive the referee. It is worth noticing that this is a very common situation in quantum communication or computation protocols that use entanglement as a resource. In view of the refereed scenario outlined above, the obvious question then would be whether one could modify the witnessing procedure so that it becomes immune to cheating strategies adopted by Alice and Bob. Interestingly, following Buscemi’s work on nonlocal quantum games [43], Branciard et al [44] showed that such a modification of the protocol is indeed possible.

The idea comes about by having a closer look at the refereeing Protocol 2.2.4. In step (iii) of this procedure where Charlie reveals to Alice and Bob the map f_m: I^A× I^B → M^A× M^B, he also discloses what outcomes he expect to see from an entangled state. This clearly gives the opportunity to Alice and Bob to cheat by simply faking the complementary map f_pr: I^A× I^B → P^AB. There is an easy solution to this problem within the context of quantum mechanics: Charlie should simply ask quantum questions [43, 44].

Protocol 2.2.5.

(i) Charlie obtains some information about the state from Alice and Bob and chooses a suit-able test operator.

(ii) He decomposes the test operator into local quantum states {ˆτ_i^A0T} and {ˆω^B_i^0T} for Alice and Bob, respectively, as ˆL=P

iβ_iτˆ_i^A0T⊗ˆω_i^B0T, with T denoting the transposition operation and β_i∈R⁶.

6In this protocol, the indices X₀ and X refer to the question input and shared state Hilbert spaces, respec-tively, for each party A and B, while ˜X = X₀X represents the joint Hilbert space for each party.

(iii) He suggests Alice and Bob to carry out the two-outcome projective measurements M^A˜ = { ˆΠ^X₀^˜, ˆΠ^X₁^˜} for X = A, B, where ˆΠ^X₁^˜ = |Φ⁺iX˜hΦ⁺| with |Φ⁺iX˜ = ^√1

d

P

i|ii_X0|ii_X on their shared state⁷ and the input questions⁸.

(iv) Charlie chooses the pair of questions (ˆτ_i^A0T, ˆω_i^B0T) according to some probability distribu-tion {pi} and sends them to Alice and Bob.

(v) The parties make joint measurements on their quantum state and the incoming quantum question. Then they communicate the outcomes of their measurement in each run to the referee.

(vi) Charlie then combines the outcomes and evaluates the “average reward”⁹

¯

℘( ˆ%_AB| ˆL) = X

i

β_iµ(1, 1|i), (2.36)

in which µ(1, 1|i) is the probability (ratio) of both Alice and Bob answering the value 1 to the ith question given by

µ(1, 1|i) = Tr( ˆΠ^A₁^˜ ⊗ ˆΠ^B₁^˜)(ˆτ_i^A0 ⊗ ˆ%_AB⊗ ˆω_i^B0). (2.37)

(vii) An average ¯℘( ˆ%_AB| ˆL) > g_s implies the entanglement of ˆ%_AB.

The entanglement certification condition in (vii) relies on the fact that, due to ˆL being an entanglement test, for every separable state ˆσ_AB ∈ S_sep the quantity ¯℘(ˆσ_AB| ˆL) in Eq. (2.36) remains below the optimal separable bound g_s [44]. Here, for evaluation of ¯℘( ˆ%_AB| ˆL) Charlie does not need to have any information about the process that has led to the production of the classical answers, namely what is happening in Eq. (2.37). By rules of quantum mechanics, it has been guaranteed that as long as the shared state ˆ%_AB is not entangled it is impossible to produce correlated outcomes µ(1, 1|i) that deceives the referee, even if Alice and Bob commu-nicate classically during the procedure [45, 46]. In this way, Protocol 2.2.5 provides a route towards measurement-device-independent (MDI) entanglement witnessing that does not rely on the detailed quantum description of the measurements.

The most important point is that Charlie does not reveal the encoding f_m : I^A× I^B → M^A× M^B, instead, he uses f_q : I^A × I^B → S_q^A0 × S_q^B0, where S_q^X0 is the local quantum state space, and then reveals the map f_qm : S_q^A0 × S_q^B0 → M^A× M^B to Alice and Bob. Now, the nonorthogonality of the local questions plays a crucial rule in the protocol. Suppose that the quantum questions were locally orthonormal and thus unambiguously distinguishable, i.e., {ˆτ_i^A0T = |ii_A0hi| ∈ S_q^A0} and {ˆω_i^B0T = |ii_B0hi| ∈ S_q^B0}. Then Alice and Bob could easily measure

7Without loss of generality we have assumed that the shared entangled state lives in a finite dimensional Hilbert space which is d-dimensional for both parties; see Theorem2.2.2.

8Note that, the parties can perform any two-outcome local measurements, however, it turns out that the one suggested by Charlie is one of the optimal measurements possible [44]. Moreover, for Alice and Bob to be able to choose their measurement strategy, Charlie has to inform them of the test operator ˆL.

9We follow the terminology used in Ref. [45] for consistency.

the input questions and figure out the encoding of the classical index i. This would in turn imply that f_qm = f_m, and thus the situation would be exactly the same as Protocol 2.2.4 and vulnerable against cheating. The good news, however, is that by Lemma2.2.7a decompositions of a test operator into locally orthonormal expansion basis is not allowed, retaining the minimal security requirements in such protocols. On top of that, the decomposition must be chosen such that the quantum questions forbid unambiguous discrimination with any nonzero probability, otherwise, there will exists a finite chance for agents to cheat [45, 46].