Interpretation and connection with the no-go theorem

state is incoherentDdoes nothing to the exponential and we recover the GF associated to the TPM prescription.

One should also note that this formalism can be generalized for many collisions by concate- nating multiple graphs, in the same fashion of Section7.6.

Finally, consideringw⌘ q+ uwe have P(w)=

Z

d u dq (w u q)P(q, u), (8.43)

P(q, u)=X

(q q[ ]) ( u u[n⁰,n])PQBN[ ], (8.44)

where u[n⁰,n]= n⁰ n. Combining the above P(w)=X

(w w[ ])PQBN[ ], (8.45)

wherew[ ]= q[ ]+ u[n⁰,n].

this reasoning, consider

PQBN[ ]= X

↵₀

p↵₀pµ|h↵0|ni|²|hµ⁰n⁰|U|µ↵₀i|²; (8.46)

this prescription depends on the system’s state since we have to know the basis in which the system is diagonal to measure it, that is, we violate (I) of the no-go theorem. Yet, this particular violation does not make the QBN distribution negative. Let me rewrite the above in terms of traces

PQBN[ ]=X

↵₀

Tr ⇧_n⇧_↵0 Tr⇣

U^†⇧_µ0 ⌦⇧_n0U⇧µ⌦⇧_↵0⇢E ⌦⇢S

⌘. (8.47)

Rather than a single trace, we have the product of two, which can be interpreted as two sepa- rate protocols (or experiments), in each experiment the protocol applied to|↵₀iis distinct, i.e., in practice one needs two copies of|↵0i. Matching outcomes↵0 are then interpolated through Bayesian inference. Apparently, there seems to be no way to write the above asP[ ]=Tr(K ⇢).

But, should it not be possible, would this make Eq. (8.47) not plausible? Consider a hypothetical situation in which we want to measure a quantum system with an usual mercury thermometer [experiment (1)]. At some point in history there were experiments establishing a correspon- dence table between the height of the mercury column and the Celsius scale [experiment (2)].

This correspondence table is given in the context of experiment (1), which is concerned with the unitary interaction between the mercury and the system of interest. By the time of experiment (1), nobody cares about providing a POVM which describes experiments (1) and (2) altogether.

Nevertheless, one uses a subjective belief — the prior validation of experiment (2) — encoded in the thermometer marks. The crucial point is that it does not necessarily make sense to provide a POVM to an inference-based method and this does not hurt our everyday scientific practice.

By the contrary, Science is filled with Bayesian reasoning and it does not need to be embedded in quantum theory. To better qualify Bayesian reasoning, I refer to the following quote.

“The abandonment of superstitious beliefs about the existence of Phlogiston, the Cosmic Ether, Absolute Space and Time, . . . , or Fairies and Witches, was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs.(...)

Probabilistic reasoning—always to be understood as subjective—merely stems from our

being uncertain about something. It makes no di↵erence whether the uncertainty relates to an unforseeable future, or to an unnoticed past, or to a past doubtfully reported or forgotten; it may even relate to something more or less knowable (by means of a computation, a logical deduction, etc.) but for which we are not willing or able to make the e↵ort; and so on.” (de Finetti [147])

Although de Finetti takes a rather liberal stance regarding the meaning of probabilities, it would still be discomforting to conclude that no quantum experiment could sample the QBNs discussed thus far. Fortunately, this is not the case [146]. To do that, we need to realize that the traces in Eq. (8.47) can be thought as being performed over di↵erent copies of the system’s Hilbert space,HS !HS₁ ⌦HS₂. Doing so, we can write a single trace

P_QBN[ ]=X

↵0

p↵₀Trn⇣

[U^†⌦1_S

2]⇧µ⁰⌦⇧_n⁰⌦1_S

2[U⌦1_S

2]⇧µ⌦1_S

1 ⌦⇧_n⌘

⇢_E ⌦|↵₀↵₀ih↵₀↵₀|o .

(8.48) Then, define the broadcast state⇢bro ⌘ P

↵0 p↵0|↵0↵0ih↵0↵0| 2L HS1 ⌦HS2 and the operator K ⌘ [U^†⌦1_S

2]⇧_µ0⌦⇧_n0⌦1_S

2[U⌦1_S

2]⇧_µ⌦1_S

1 ⌦⇧_n2L HE ⌦HS₁ ⌦HS₂ and we have

PQBN[ ]= Tr(K ⇢_E ⌦⇢_bro). (8.49)

In the original no-go paper [42] the authors have suggested a state-dependent protocol to measure coherent work based on using multiple copies of the quantum system. In [146] the connection between QBNs and the broadcast state suggests that entanglement between copies allows going beyond the original proposal. The shortcoming of this method is that you cannot sample the QBN distribution from the original system of interest. Note that we do not achieve the form suggested in (ii) of the no-go theorem (pg. 114), P_QBN[ ] = Tr(K ⇢_E ⌦⇢); instead, we resorted to the preparation of a particular entangled state to book-keep copies of a projected state, so that an extra measurement could be performed on the copy while the original state keeps evolving without su↵ering the back-action from this extra measurement. Another way to achieve this procedure is to perform post-selection; instead of considering a special initial preparation⇢bro, we consider a “measurement”⇧_↵0⌦⇧_↵0 [146]. Indeed, we consider⇧_↵0⌦⇧_↵⁰₀ and discard whatever data with↵⁰₀ , ↵₀. The QBN we have considered here is subtle because it combines actual measurements ( -TPM) with inferred possibilities (which stem either from prior characterization/di↵erent experiments or broadcast states/post-selection).

As a take-home message, we have discussed that the initial reasoning behindP_QBN[ ], ar- guably, does not need to be embedded in a single quantum experiment to make sense. However, it is not inconsistent with such program since one can provide a POVM to extract quantum Bayesian networks. I also emphasize that the no-go result is not contradicted by the QBN for- malism, since it depends on the system’s initial state and⇢_S = ↵⇢⁰_S +(1 ↵)⇢⁰⁰_S doesnotimply thatPQBN(⇢S)=↵PQBN(⇢⁰_S)+(1 ↵)PQBN(⇢⁰⁰_S), again violating one of the formulations of (I) in pg.114.