We now analyse what are the physical reasons for the different performances of the protocol in the settings that we considered. We first develop an analysis in terms of Landauer’s principle and then we discuss the presence of contextuality for quantum strategies that achieve a probability of success higher than the Bell bound.
4.3.1 Connection to Landauer’s principle
We have seen that under the assumption that only reversible gates are employed, the CHSH* game acts as a witness that distinguishes quantum and classical systems, and systems of different dimension. How reasonable is it to restrict the operations to reversible transformations? As described in subsection4.1.3, it was first argued by Landauer that irreversible operations are not fundamental and that every irreversible classical operation on logical bits must be accompanied by a rise in the entropy of the non-information bearing degrees of the system or its environment [70]. This holds because in order to build an irreversible gate out of fundamentally reversible operations, we need to discard or erase information.
We have seen that erasure is a powerful tool that allows to win the CHSH* game with certainty. Reversible classical and quantum settings lead to distinct lower values for the game. This can be seen as a reflection of the non-classical nature of quantum information storage and measurement.
Moreover, following Landauer’s approach, in the optimal strategy presented for the irre- versible setting, winning the game with certainty requires the erasure of one bit for only one of the four input combinations of a and b. On average, the heat generated by the protocol is therefore 14kT log22. With similar considerations, we can imagine an optimal quantum strategy in the unitary setting as implementing a partial erasure. We can quantify the heat generated by this partial erasure as, on average, 140.41kT log22 corresponding to the probability of success
1 4+ 1 4+ 1 4+ 1 4( √
2−1) ≈ 0.85. In other words, to increase the winning probability for the game in the classical reversible setting to unity, 14kT log22 information would need to be erased, whereas to do so for the unitary setting only 1
40.59kT log22 must be erased.
about the irreversible function a· b. The quantum resource in this protocol is the qubit’s ability to simulate two classical bits (one of which is going to be erased). This is made even more explicit in Figure 4.9, which compares the state spaces of a pair of bits, a single qubit and a single bit. In particular, in the optimal quantum strategy the single-qubit state space (that mimics the two-bit state space) encodes the four possible input combinations as four quantum states. The measurement then extracts one bit of information. Since the four states are not all pairwise orthogonal, the system is not storing two independent bits prior to the measurement and can therefore perform better than the reversible classical and Clifford settings.
4.3.2 Connection to Contextuality
We have already argued that the CHSH* game never shows non-locality and contextuality in its standard definitions due to Kochen-Specker [33] and Spekkens [34] defined in subsection
3.1.2. We here report a notion of transformation non-contextuality, recently introduced by Mansfield and Kashefi in [71], where the contexts are sequences of transformations. They called it sequential transformation non-contextuality (STNC) and it refers to the fact that the same transformation in different sequences of transformations must have the same ontological representation. More precisely, if we consider a finite sequence C = (Ui)ti=1 of unitaries Ui, the
ontological representation of the unitary ΓUi, is the same in any other sequence of unitaries C 0,
ΓUi(C)= ΓUi(C0). (4.5)
We are here also assuming that the sequential composition is reflected at the ontological level, i.e. ΓUt...U1 = ΓUt◦· · ·◦ΓU1. Despite being different from the other notions of non-contextuality, STNC still encodes the same counterfactual spirit of them, where pre-existing properties asso- ciated to each experimental procedure (here unitary transformations) must not depend on the contexts (here sequences) they belong to.
This notion of contextuality is useful as a resource for computational advantages in some particular computational models, called l2−TBQC.5 l2-TBQC is a computational model con- sisting of a classical control computer that can only perform mod2-linear computation, and
5
it can interact with a resource (possibly quantum) to enhance its computational power. The CHSH* protocol is an example of l2-TBQC protocol. The natural ontological model to asso- ciate to l2−TBQC has ontic space Zn
2 for some n ∈ N and transformations that are mod2−
linear. Since unitaries must be represented by invertible functions at the ontological level, this implies that their action must correspond to addition of vectors inZn
2, e.g.
ΓU(λ) = λ + u
for some u∈ Zn
2. We call this ontological model as l2−ontological model.
In [71] Mansfield and Kashefi proved that in a l2-TBQC protocol sequential transformation contextuality (with the assumption of l2−ontology) is necessary to enable quantum advantage over classical resources for the task of probabilistically computing any non-linear function. Of course, if we drop the assumption of l2−ontology the result does not hold, as already an ontological model for classical physics – which is intrinsically sequential transformation non-contextual – can reproduce a protocol that performs non-linear functions (which concerns problems belonging to the complexity class P ). The point here is to impose natural restrictions on generic ontological models that reflect the artificial nature of the computation encoded by the protocol (in this case the restriction to mod 2 linear computation). The result is that either the natural assumption of STNC or the here natural assumption of l2−ontology is incompatible with quantum mechanics.
This work is relevant to the CHSH* game, since the result above applies to the CHSH* game too, which therefore shows sequential transformation contextuality. The result above can also be stated in our case as the following no-go theorem: a sequential transformation non- contextual l2−ontological model cannot in general reproduce the performance of the CHSH* game in the unitary setting.
Lastly, we report an interesting connection between contextuality and the entropic costs associated to the (partial) erasures considered in the previous subsection.6 Let us define with Landauer’s Erasure (LE) the fraction of erasure (in terms of the unit – kT log22 – of bit erased) associated to a given strategy that performs better than the optimal reversible classical
6
This part refers to some preliminary results developed mainly with Shane Mansfield and still not contained in any published work.
strategy. More formally, let us define, in accordance with [120], the average distance between two boolean functions f, g :Zn
2 → Z2 as d(f, g) = 21n|{i ∈ 2n|f(i) 6= g(i)}|, i.e. the fraction of the number of inputs for which the two functions differ. We also define the non-linearity ν(f ) of a function f : Zn
2 → Z2 as the distance between the function f and the closest Z2-linear
function g :Zn 2 → Z2, ν(f ) = min g {d(f, g)|g : Z n 2 → Z2 is Z2-linear}. (4.6)
The Landauer’s erasure, LE ∈ [0, 1], associated to the probability of success of a strategy for the task of computing a non-linear function f, is defined as
LE = psuc− p
rev suc
ν(f ) , (4.7)
where prev
suc denotes the probability of success of the optimal reversible classical strategy. For
example, in the CHSH* game, when considering the optimal quantum strategy, the Landauer’s erasure is LE = cos2(π8)−0.75
0.25 = 0.41, which corresponds to the quantity already discussed in
the previous subsection. Notice that this value represents an upper bound on the Landauer’s erasures for the quantum strategies (unitary setting). Moreover, the definition (4.7) can be rearranged in the relation
pfail ≥ (1 − LE)ν(f), (4.8)
considering that prev
suc= 1−ν(f) and that the probability of failure is pfail = 1−psuc. The reason
for doing this is that the relation (4.8) above is exactly the same relation for the contextual fraction (CF) – a way of quantifying contextuality in the sheaf-theoretic approach [119] – in [120, Theorem 3], pfail ≥ (1 − CF )ν(f). We leave the study of the relation between these
two quantities for future reasearches. However, we suggest that the analogy of these two relations can lead to possible applications. In realistic scenarios involving reversible quantum computation and actual irreversible processes, we can consider a relation involving both the contributions due to the contextualilty and to the erasures: pfail ≥ (1 − CF − LE)ν(f). This
consideration suggests a way of measuring the amount of contextuality (CF) in the computation, as, after n rounds of the experiment, the probability of failure is known and the average entropic
cost (encoded by LE) can be measured. Moreover, it seems that the notions of erasure and contextuality are interchangeable in these scenarios by rebalancing the amount of computation which is quantum and the one which is purely irreversible.