# The rule of conditional probability is valid in quantum theory [Comment on Gelman & Yao's "Holes in Bayesian Statistics"]

The rule of conditional probability is valid in quantum theory P.G.L.

Porta

Mana

Kavli

Institute,

Trondheim

Bernard O. Koopman discussed the falsity of such claims already in The

Introduction in his work is very clear: Ever since the advent of modern quantum mechanics in the late the idea has been prevalent that the classical laws of probability cease, in some sense, to be valid in the new theory. More or less explicit statements to this effect have been made in large number and by many of the most eminent workers in the new physics [. . .]. Some authors have even gone farther and stated that the formal structure of logic must be altered to conform to the terms of reference of quantum physics [. . .]. Such a thesis is surprising, to say the least, to anyone holding more or less conventional views regarding the positions of logic, probability, and experimental science: many of us have been apt – perhaps too naively – to assume that experiments can lead to conclusions only when worked up by means of logic and probability, whose laws seem to be on a different level from those of physical science. The primary object of this presentation is to show that the thesis in question is entirely without validity and is the product of a confused view of the laws of probability. It must be remarked that such claims have hitherto never been supported by any rigorous proof – with explicitly stated definitions and assumptions, well-defined and unambiguous notation, and clear logical and mathematical steps (and Gelman & Yao are no exception). The typical fallacy in the kind of examples presented rests in the neglect of the experimental setup, leading either to an incorrect calculation of conditional probabilities, or to the incorrect claim that the probability calculus yields an equality, where it actually does not. The same incorrect claims can be obtained with completely non-quantum systems , such as drawing from an urn, if the setup is neglected (Kirkpatrick Porta

Mana § IV).

Let us start with such a non-quantum counter-example. A non-quantum counter-example Consider an urn with one B lue and one R ed ball. Two possible drawing setups are given: D a With replacement for blue, without replacement for red.

That is, if blue is drawn, it is put back before the next draw (and the urn is shaken); if red is drawn, it is thrown away before the next draw. of the Pitman-Koopman theorem for sufficient statistics,

Koopman P orta M ana Conditional probability is valid in quantum theory D b With replacement for red, without replacement for blue.

These two setups are obviously mutually exclusive. We can easily find the unconditional probability for blue at the second draw in the setup D a : P ( B | D a ) (cid:3) . (1) Note that this probability can be intuitively found by simple enumeration, à la Boole, considering “possible worlds” if you like. Out of four possible worlds, half of which has blue at the first draw, and the other half has red, we can count that three worlds have blue at the second draw. The conditional probabilities for blue at the second draw, given the first draw, are also easily found: P ( B | B ∧ D a ) (cid:3) P ( B | R ∧ D a ) (cid:3) . (2) We find that P ( B | D a ) (cid:3) P ( B | B ∧ D a ) P ( B | D a ) + P ( B | R ∧ D a ) P ( R | D a ) , (3) which is just the rule of conditional probability. It is in fact just the systematization and generalization of the intuitive “possible worlds” reasoning done above. Next consider the setup D b . We easily find P ( B | D b ) (cid:3) , (4) P ( B | B ∧ D b ) (cid:3) P ( B | R ∧ D b ) (cid:3) , (5) P ( B | D b ) (cid:3) P ( B | B ∧ D b ) P ( B | D b ) + P ( B | R ∧ D b ) P ( R | D b ) . (6) Now compare the unconditional probability for blue at the second draw in the setup D a , with the conditional probabilities for blue at the second draw given the first draw in the setup D b : P ( B | D a ) (cid:44) P ( B | B ∧ D b ) P ( B | D b ) + P ( B | R ∧ D b ) P ( R | D b ) . (7) This inequality is not surprising – we are comparing different setups. It is not an instance of the conditional-probability rule. In fact the probability calculus has nothing to say, a priori, about the relation between the left side and right side, which are conditional on different statements or, if you like, pertain to two different sample spaces. P orta M ana Conditional probability is valid in quantum theory You can call the inequality above “interference” if you want; for further and more involved examples with urns and decks of cards see Kirkpatrick (2003a,b) and

Porta

Mana (2004a § IV).

Now consider another pair of drawing setups: setup D c , with replace- ment for both colours; and setup D d , without replacement for either colour. You can easily find that P ( B | D c ) (cid:3) P ( B | B ∧ D c ) P ( B | D c ) + P ( B | R ∧ D c ) P ( R | D c ) , (8) P ( B | D d ) (cid:3) P ( B | B ∧ D d ) P ( B | D d ) + P ( B | R ∧ D d ) P ( R | D d ) , (9) P ( B | D c ) (cid:3) P ( B | B ∧ D d ) P ( B | D d ) + P ( B | R ∧ D d ) P ( R | D d ) . (10) The first two equalities above are expressions of the conditional- probability rule. The third is not , however. It is simply a peculiar equality contingent on the two specific setups. The probability calculus therefore correctly handles situations leading to inequalities such as (7), and to equalities such as (10). The explicit presence of ‘ D . . . ’, which represents given information, is necessary discussions involving different setups, such as the above. If I ask you “what’s the probability of blue at the second draw?”, you will ask me “in which drawing setup?”. The probability is conditional on the information about the drawing scheme. The inequality (7) is what Gelman & Yao (2020 p. complain about, but in the context of a pair of quantum setups. I do not see how one can complain about it, or claim inconsistencies. It is obviously correct even from an intuitive analysis of the two setups. And the probability calculus correctly leads to it, too. The probability calculus correctly leads also to the equality (10) . As already said, given two mutually exclusive setups, the probability calculus a priori neither commits to an equality nor to an inequality. I will now show that the simple example above is in fact conceptually quite close to the quantum experiment mentioned by Gelman & Yao.

The closeness is especially clear from the experimental and mathematical developments of quantum theory of the past years (at the very least), as the literature cited below shows. P orta M ana Conditional probability is valid in quantum theory The quantum double-slit experiments

The basic argument of Gelman & Yao is that, in a given setup of the quantum double-slit experiment, we have a specific probability distribution for the appearance of an emulsion or excitation on some point of the screen. We can call this a “screen detection”, but please keep in mind that in so doing we are adding an extra interpretation that modern quantum theory does not actually commit to (see discussion and references below). In a different experimental setup we have conditional probabilities for screen detection conditional on slit detection. Now, the probability of the first setup is not equal to the combination of the conditional probabilities of the second setup. But this is exactly what happened in our urn example above, eq. (7) . In the present quantum case we do not have a violation of the conditional probability rule either – if anything it is a confirmation. To see the analogy more clearly, let me present some additional facts from quantum theory. The experimental setup without detectors at the slits and the setup with slit detectors are actually limit cases of a continuum of experimental setups (Wootters & Zurek for a recent review and further references see Banaszek et al. In the general case, such a setup has slit detectors of varying efficiency, denoted by a parameter q ∈ [ , ] that can be chosen in the setup. The possible degrees of efficiency are of course mutually exclusive, so these setups are mutually exclusive. The slit detector has a given efficiency in the following sense: Let us call y the detection position on the screen, and X is the statement that detection occurs at slit (you can translate to random- variable jargon if you prefer). When we prepare the electromagnetic field in a quantum state S , and use ideal detectors with perfect efficiency, the probability of detection at slit is, say p S , and − p S for slit If we use the setup with detectors having efficiency q – denote it by D q – then the probability of detection at slit is p ( X | D q , S ) (cid:3) ( − q ) + q p S , (11) and ( + q ) − q p S for slit The setup with perfect detectors is the limit case q (cid:3) . In the case of zero efficiency, q (cid:3) , there is no relation between the light states and the firing of the slit detectors; that is, we are always fully uncertain as P orta M ana Conditional probability is valid in quantum theory to which detector would fire, no matter how the light state is prepared. These kinds of setup – and many other interesting ones – are quite easy to prepare with the statistically analogous quantum Mach-Zehnder-like interferometers (see the textbooks in footnote below; Leonhardt § Yuen & Shapiro In each setup D q (and given the light state S ) we also have the conditional probability distribution p ( y | D q , S ) for detection at y on the screen, and the conditional probability distributions p ( y | X , D q , S ) for detection at y on the screen, given detection X at the slits. We have p ( y | D q , S ) (cid:3) p ( y | X , D q , S ) p ( X | D q , S ) + p ( y | X , D q , S ) p ( X | D q , S ) . (12) This is an instance of the conditional-probability rule, which is of course valid. This equality also holds for long-run frequencies (see point ( iii ) below). Note that such conditional and unconditional frequencies are experimentally observed. I would like you to convince yourself, though, that the equality above (not the specific values of the frequencies) is not really an experimental fact, since it rests on the very way we measure conditional frequencies. The conditional and unconditional distributions above will of course be different depending on the setup D q and the light state S . But in each instance the rule of conditional probability holds. For example, if q ′ (cid:44) q ′′ , p ( y | D q ′ , S ) (cid:3) p ( y | X , D q ′ , S ) p ( X | D q ′ , S ) + p ( y | X , D q ′ , S ) p ( X | D q ′ , S ) , (13) p ( y | D q ′′ , S ) (cid:3) p ( y | X , D q ′′ , S ) p ( X | D q ′′ , S ) + p ( y | X , D q ′′ , S ) p ( X | D q ′′ , S ) , (14) p ( y | D q ′ , S ) (cid:44) p ( y | X , D q ′′ , S ) p ( X | D q ′′ , S ) + p ( y | X , D q ′′ , S ) p ( X | D q ′′ , S ) . (15) The last inequality, analogous to eq. (7) , comes from experimental observations (see the brief discussion below about the relation with de Finetti’s theorem), and was not in fact not ruled out a priori by the probability calculus. P orta M ana Conditional probability is valid in quantum theory Now let me discuss a couple of very interesting experimental facts about this collection of setups: First, both the conditional y | X and unconditional probability distributions for the screen detection y generally have an oscillatory profile, typical of interference (Wootters & Zurek

Banaszek et al. see also Chiao et al. for other experimental variations). The oscillatory character is maximal for the zero-efficiency setup q (cid:3) and decreases as q increases. For the perfect-detector setup q (cid:3) there is no interference. But we can have quite a lot of interference even when the detection efficiency is quite high, so that for some light states we are almost certain about slit detection; see references above. (The profile depends on the specific light state, of course, which we are assuming fixed.) Second, the unconditional (frequency) distribution observed in the setup D with zero-efficiency slit detectors is experimentally equal to the distribution for screen detection observed in the setup without slit detectors (note that in the latter setup we cannot speak of conditional or unconditional probability, since slit detection does not exist). Third, one conditional distribution observed in the setup with one slit closed is experimentally equal to one in the setup D with perfect slit detectors. (Here we must be careful, because there is no slit detection in the second setup; rather, we speak of appearance or non-appearance at the screen, and in the latter case no conditional distribution is defined.) The equalities in the last two cases should a priori not be expected, because the setups are physically different. Of course one can look for physical, “hidden variables” explanations of such equalities. Experi- mental quantum optics simply acknowledges the fact that two setups are equivalent for such detection purposes, and incorporates this informa- tion into its mathematical formalism, by means of appropriately defined ‘ povm s ’, discussed below. Note the statistical analogy between the cases above and the cases with the setups of the urn examples previously discussed. In each setup, the rule of conditional probability holds (and in the quantum case we can have distributions, conditional and unconditional, with oscillatory profiles). Across different setups, probability theory says that such a rule cannot be applied; and indeed we find inequalities across some setups and equalities across others, both in the quantum and non-quantum case, eqs (7) , (10) . Even more striking statistical analogies appear in P orta M ana Conditional probability is valid in quantum theory the already cited non-quantum counter-examples (Kirkpatrick Porta

Mana § IV). It is also possible to consider situations in which we are uncertain about which measurement setup applies. For example we may not know whether there were slit detectors, or the value q of the detector efficiency. In such situations we introduce probabilities p ( D . . . ) for the possible setups and the conditional-probability rule applies, yielding for example p ( y | S ) (cid:3) (cid:80) q p ( y | D q , S ) p ( D q ) (16) (here our knowledge of the state was assumed to be irrelevant to our inference about the setup). Then, given the measurement outcome, we can make inferences about the setup (Barnett et al. Ziman et al. D’Ariano et al. see also Rigo et al. – for example whether a slit detector was present or not – again using the conditional- probability rule in the guise of Bayes’s theorem.

This kind of inference is especially important in quantum key distribution (Nielsen & Chuang where we try to infer whether a third party was eavesdropping, that is, performing a covert measurement. Again no violations of the probability rules in the quantum realm: quite the opposite, those rules allow us to make important inferences. Further remarks and curiosities about quantum interference experiments I would like to mention a couple more experimental facts – which are, besides, statistically very interesting – to correct some statements by Gelman & Yao in relation to the double-slit experiment. ( i ) It does matter whether many photons are sent at once, or one at a time (cf. Gelman & Yao § point as well as their wavelength, temporal spread, and so on (strictly speaking, the spatio-temporal dependence of the field mode). These details are part of the specification of the light state S mentioned above, and lead to different probabilities distributions of screen detection. For example, in some setups and for some states we can have a detection probability density p ( y ) for the first photon, and a different density for the second photon p ( y | y ) , conditional on the detection of the first – both being different from the cumulative density of detections. Interference phenomena can also be observed P orta M ana Conditional probability is valid in quantum theory in time, not only in space. See e.g. the phenomena of higher-order coherence, bunching, anti-bunching, and many other interesting ones . Quoting

Glauber (1965

Lect. I p. The new light detectors enable us to ask more subtle questions than just ones about average intensities; we can for example, ask questions about the counting of pairs of quanta, and can make measurements of the probability that the quanta are present at an arbitrary pair of space points, at an arbitrary pair of limes. The rules of the probability calculus also apply in all such situations. We can infer, for example, the position of the first photon detection given the second from the conditional probability rule p ( y | y ) ∝ p ( y | y ) p ( y ) . ( ii ) The details about the light source and the setup are not “latent variables”: they specify the quantum state of light and the meas- urement performed on it. They are like the initial and boundary conditions necessary for the specification of the behaviour of any physical system. ( iii ) In view of point ( i ) above, it is important not to conflate the probab- ility distributions for single-photon detections, those for cumulative photon detection, and the frequency distributions of a long-run of such detections (Gelman & Yao § seem to conflate the two). Such distinction is always important from a Bayesian point of view. Quoting

Glauber (1965

Lect. I p. again: There is therefore no possibility in general of replacing the ensemble averages by time averages. [. . .] we shall find that individual meas- urements yield results wholly unlike their ensemble averages. The distinction between particular measurements and their averages may thus be quite essential. I may add that the idea and parlance of “photons passing through slits” are used today only out of tradition; maybe a little poetically. The technical parlance, as routinely used in quantum-optics labs for example (Leonhardt Gerry & Knight has a different underlying picture. The ‘ system ’ in a quantum-optics experiment is not photons, but the modes of the field-configuration operator (note that this is for example Mandel & Wolf

Morgan & Mandel

Paul

Jacobson et al. and textbooks such as Loudon

Mandel & Wolf

Scully & Zubairy

Gerry & Knight

Walls & Milburn P orta M ana Conditional probability is valid in quantum theory not yet Quantum

ElectroDynamics). “Photon numbers” denote the discrete outcomes of a specific energy-measurement operator; “photon states” denote specific states of the field operators. As another example, “entanglement” is strictly speaking not among photons, but among modes of the field operator (van Enk

Several quantum physicists indeed oppose the idea and parlance of “photons”, owing to the confusion they lead to. Lamb wrote in the author does not like the use of the word “photon”, which dates from In his view, there is no such thing as a photon. Only a comedy of errors and historical accidents led to its popularity among physicists and optical scientists. Wald (1994) warns: standard treatments of quantum field theory in flat spacetime rely heavily on Poincaré symmetry (usually entering the analysis implicitly via plane- wave expansions) and interpret the theory primarily in terms of a notion of “particles”. Neither

Poincaré (or other) symmetry nor a useful notion of “particles” exists in a general, curved spacetime, so a number of the familiar tools and concepts of field theory must be “unlearned” in order to have a clear grasp of quantum field theory in curved spacetime. [p. ix] [. . .] the notion of “particles” plays no fundamental role either in the formulation or interpretation of the theory. [p. See also

Davies’s

Particles do not exist (1984). A summary of the modern formalism of quantum theory It may be use- ful to give a summary of how probability enters the modern formalism of quantum theory. See textbooks such as Holevo (2011),

Busch et al. (1995), Peres (2002 especially ch. de Muynck (2004 especially ch. and the excellent text by Bengtsson & Życzkowski (2017). A quantum system is defined by its sets of possible states and possible measurements. A state ρ is represented by an Hermitean, positive- definite, unit-trace matrix ρ (which satisfies additional mathematical properties: Jakóbczyk & Siennicki

Kimura

Kimura & Kossakowski

Bengtsson & Życzkowski called ‘ density matrix ’. States traditionally represented by kets | ψ ⟩ are just special cases of density matrices. A measurement setup M is represented by a set of Hermitean, positive-definite matrices { M r } (of the same order of the Lamb shift,

Lamb & Retherford P orta M ana Conditional probability is valid in quantum theory as the density matrices) adding up to the identity matrix. They are called ‘ positive-operator-valued measures ’, usually abbreviated povm s. Traditional von

Neumann projection operators {| ϕ r ⟩⟨ ϕ r |} are just special cases of povm s. Each matrix M r is associated with an outcome r of the measurement. These outcomes are mutually exclusive. An outcome can actually represent a combination of simpler outcomes, r ≡ ( x , y , z , . . . ) , such as the intensities or firings at two or more detectors. The probability of observing outcome r ≡ ( x , y , . . . ) given the meas- urement setup M and the state S is encoded in the trace-product of the respective matrices: p ( x , y , . . . | M ∧ S ) ≡ tr ( M x , y , . . . ρ ) , (17) These probabilities for all r form a probability distribution. The tradi- tional

Born-rule expression ‘ |⟨ ϕ r | ψ ⟩| ’ is just a special case of the above formula. The probabilities in the formula come from repeated measure- ment observations in the same experimental conditions: we can invoke de Finetti’s (1937; theorem here – the partial-exchangeability variant – and some quantum physicists indeed do (Caves et al. van Enk & Fuchs

Fuchs et al. The trace-product above is just a scalar product in a particular space. How a set of probability or frequency distributions can be encoded in scalar products is explained in a down-to-earth way in Porta

Mana (2003;

Once the probability distribution above is given we can use the full-fledged probability calculus for our inferences. We can for example sum (or integrate) over detector outcomes y , . . . , obtaining the marginal probability for detector outcome x ; or calculate the probability of outcome y conditional on x ; or make inferences about the measurement setup or the state. Again, there are no violations of the probability rules. The formalism (17) is neat in this respect because it allows us to represent such situations through new povm s and density matrices. You can easily check, for example, that the marginal probability for x from eq. (17) can be encoded in the povm { M ′ x } ≡ { (cid:80) y , . . . M x , y , . . . } . A situation of uncertainty between setups M ′ and M ′′ , as in eq. (16) , can be encoded in the povm (cid:8) p ( M ′ ) M ′ r + p ( M ′′ ) M ′′ r (cid:9) . And so on, and similarly for states and their density operators. For systems with infinite degrees of freedom such as electromagnetic fields or electrons (Fermionic fields), the matrices above are replaced by operators defined in particular algebras. A povm element can actually be P orta M ana Conditional probability is valid in quantum theory a space-time-indexed operator. The computational details can become quite complicated, but the same basic ideas apply.

This formalism obviously also includes the specification of post- measurement states (if the system still exists afterwards), transformations, evolutions. I shall not discuss these; see the textbooks cited above. Conclusions I hope that the above discussion and bibliography clearly show that: • the rules of probability theory, including the conditional-probability rule, are fully valid in quantum theory and essential in its modern applications; • some peculiar equalities or inequalities across different experi- mental conditions do not contradict the conditional-probability rule, and they appear just as well in quantum as in non-quantum situations, such as drawing from an urn. Quantum theory already has its physically conceptual difficulties and computational difficulties, as should be clear from the portrait sketched in the present comment. It is pointless – and pedagogically confusing and detrimental, for students of quantum optics for instance – to make it seem even more difficult with false claims of non-validity of probability theory or with distorted pictures of its experimental content. Bibliography (“de X ” is listed under D, “van X ” under V, and so on, regardless of national conventions.) Banaszek,

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