By analyzing specific attacks on a protocol, one finds limitations on the secret key rates that can be obtained. The attacks studied here on the COW protocol also allow one to get a feeling on how the protocol should perform. In particular, we expect to find secret key rates that scale linearly with the transmission coefficient t, just like for other protocols robust against photon-number-splitting attacks; however,
1To be consistent with the results presented in [D], we now show the key rate per pulse, contrary to the previous subsection where it was defined per pair of pulses. There is simply a factor of 2 between the two definitions.
2Note however, that there also exists attacks to which COWm2 in particular is more vulnerable.
See [51].
0 20 40 60 80 100 10−5
10−4 10−3 10−2
Distance (km)
Secret key rate
BSA on COW, V = 1 2PA on COW @ V = 0.98 2PA on COWm1 @ V = 0.98 1PA on COWm2 @ V = 0.98
Figure 2.3: Secret key raterper pulse, for the BSA and for 2-pulse (2PA) and 1-pulse attacks (1PA) on the three protocols COW, COWm1 and COWm2, for V = 0.98 andQ= 0, as a function of the distance (valid for large distances only). Parameters:
η= 0.1; 0.25 dB km−1 of losses;f ≃0 (for COW);tB≃1.
the above USD attacks show that a careful analysis of the detection statistics is necessary to get this key rate.
Proving the security of the COW protocol against the most general attacks re-mains work in progress. One should certainly get inspiration from the techniques that have been developed to prove the security of more standard protocols (for a review, see [17]), and adapt these tools to the specificities of distributed-phase-reference schemes. Also, it might prove useful to consider modified and more sym-metric versions of the COW protocol, as was done in [D], to which it might be possible to apply a De Finetti kind of argument [27].
Chapter 3
Testing Leggett’s model for non-local correlations
Quantum theory predicts that nature can produce non-local correlations, in the sense that they violate Bell inequalities [3]. This fascinating feature has been widely confirmed, up to a few standard loopholes, in many beautiful experiments [54] with entangled photons or ions (see for example [5, 55, 56, 57, 58]). However, while the quantum formalism provides precise rules to calculate these correlations, it doesn’t give any explanation on how they occur, or where they come from.
To get a hint on what is essential to quantum correlations, it is interesting to look for alternative non-local models that either reproduce or can be shown to be incompatible with the quantum predictions. In this prospect, Leggett proposed in 2003 such an alternative model [11], and precisely proved its incompatibility with quantum theory: he derived inequalities that are necessarily satisfied by his model, but that can be violated quantum mechanically.
Here we present a simple approach to this model, along with new inequalities for testing it. We also go beyond Leggett’s model, and show that one cannot ascribe even partially defined individual properties to the components of a maximally entangled pair.
3.1 Quantum correlations from “simpler” ones
We shall use here the formal description of a correlation, which can be written as a conditional probability distribution P(α, β|~a,~b), where α, β are the outcomes observed by two partners, Alice and Bob, when they perform measurements labeled by~a and~b, respectively. On the abstract level,~a and~b are merely inputs freely and independently chosen by Alice and Bob. On a more physical level, Alice and Bob hold two subsystems described by an entangled quantum state; in the simple case of qubits, the inputs are naturally characterized by vectors on the Bloch sphere, hence the notation~a,~b.
3.1.1 Decomposing non-local correlations
How should one understand nonlocal correlations, in particular those correspond-ing to entangled quantum states? A natural approach consists in decomposcorrespond-ing P(α, β|~a,~b) into a statistical mixture of hopefully simpler correlations:
P(α, β|~a,~b) = Z
dλρ(λ) Pλ(α, β|~a,~b) . (3.1) We shall be concerned here with the case of binary outcomes α, β = ±1, though generalizations are possible. In the binary case, the correlations can conveniently be written as
Pλ(α, β|~a,~b) = 1 4
³1 +α MλA(~a,~b) +β MλB(~a,~b) +αβ Cλ(~a,~b)´
. (3.2)
This expression allows one to clearly distinguish the marginals MλA(~a,~b) =X
α,β
α Pλ(α, β|~a,~b) , MλB(~a,~b) = X
α,β
β Pλ(α, β|~a,~b), (3.3) and the correlation coefficient
Cλ(~a,~b) =X
α,β
αβ Pλ(α, β|~a,~b). (3.4) We shall only be interested in correlations Pλ which, like quantum correlations, do neither allow signaling from Alice to Bob, nor from Bob to Alice, i.e. correlations fulfilling the so-calledno-signaling condition. This condition has a simple form here:
Alice’s marginal is independent of Bob’s input,MλA =MλA(~a) and symmetrically for Bob’s marginal, MλB =MλB(~b). Accordingly, in non-signaling correlations, only the correlation coefficient in (3.2),Cλ(~a,~b), can be nonlocal, while the marginals depend only on local variables.
3.1.2 Bell’s locality
Bell’s locality assumption that Pλ(α, β|~a,~b) factorizes into PλA(α|~a)· PλB(β|~b) can then be simply expressed as
Cλ(~a,~b) =MλA(~a)·MλB(~b) . (3.5) However, it is well known that quantum correlations violate Bell inequalities [3], and therefore they cannot be of that form.
3.1.3 Leggett’s model
In 2003 Leggett proposed another model of the form (3.1) to help us understand quantum correlations [11]. The basic assumption is that locally everything happens as if each single quantum system would always be in a pure state; non-locality can
only come from the correlation between the measurement results, expressed by the term Cλ(~a,~b). In our formalism this translates directly to the assumption that the supplementary variables λ describe the hypothetical pure states of Alice and Bob’s qubits, denoted by normalized vectors~u, ~v on the Bloch sphere:
λ =|~ui ⊗ |~vi , (3.6)
and the local marginals have the usual form as predicted by quantum physics:
MλA(~a) =h~u|~a ~σ|~ui=~u·~a , (3.7) MλB(~b) =h~v|~b ~σ|~vi=~v·~b . (3.8) If the qubits are encoded in the polarization of photons, as in Leggett’s initial idea, then the assumption is that each photon should be perfectly polarized (in the directions ~u and~v), and the local marginals should fulfil Malus’ law.
Let us recall that, in orthodox quantum theory, the singlet state is such that the properties of the pair are sharply defined (the state is pure), but the properties of the individual qubits are not. In this perspective, Leggett’s model is an attempt of keeping the correlations while reintroducing sharp properties at the individual level as well.