The wavefunction is real but nonphysical: A view from counterfactual quantum cryptography
Abstract
Counterfactual quantum cryptography (CQC) is used here as a tool to assess the status of the quantum state: Is it real/ontic (an objective state of Nature) or epistemic (a state of the observer’s knowledge)? In contrast to recent approaches to wave function ontology, that are based on realist models of quantum theory, here we recast the question as a problem of communication between a sender (Bob), who uses interactionfree measurements, and a receiver (Alice), who observes an interference pattern in a MachZehnder setup. An advantage of our approach is that it allows us to define the concept of “physical”, apart from “real”. In instances of counterfactual quantum communication, reality is ascribed to the interactionfreely measured wave function () because Alice deterministically infers Bob’s measurement. On the other hand, does not correspond to the physical transmission of a particle because it produced no detection on Bob’s apparatus. We therefore conclude that the wave function in this case (and by extension, generally) is real, but not physical. Characteristically for classical phenomena, the reality and physicality of objects are equivalent, whereas for quantum phenomena, the former is strictly weaker. As a concrete application of this idea, the nonphysical reality of the wavefunction is shown to be the basic nonclassical phenomenon that underlies the security of CQC.
I Introduction
Quantum superposition, as exemplified by the familiar doubleslit experiment, lies at the heart of quantum weirdness. The experiment contains, according to Feynman, the “only mystery” in quantum mechanics Feynman et al. (1963). The interference observed behind the doubleslit indicates that the particle, when moving from the source, travels in some sense along both slits. Classical waves of water or sound show such interfering behavior, but they are collective phenomena, and can be attributed to distinct localized masses or molecules travelling through the two slits. In the quantum case, the interference can be observed even when the source is so attentuated that there is practically only one particle between the doubleslit plane and the screen. Dirac summarized the situation by saying “Each photon then interferes only with itself” Dirac (1930).
Here we revisit this phenomenon, prompted in part by recent interest in studying the nature of the quantum state. The basic, oldest and arguably most controversial question here is: Is the quantum state real, i.e., Does it objectively exist and represent a state of Nature (cf. eg., Kastner (2012)); or, Is it epistemic, i.e., Does it represent the observer’s state of knowledge? In the framework of ontological models of quantum mechanics Harrigan and Spekkens (2010), which is based on the hidden variable theories of Bell Bell (1987) and KochenSpecker Kochen and Specker (1967), a model is ontic if every pair of pure quantum states corresponds to nonoverlapping ontic supports, and is epistemic otherwise. Recently, a number of arguments have been put forth within this framework in favor of the ontic view Pusey et al. (2012); Colbeck and Renner (2012); Patra et al. (2013); Colbeck and Renner () under certain assumptions Lewis et al. (2012); Aaronson et al. (2013); Ghirardi and Romano (2013a, b, ); Leifer (2014).
Here we will assess the status of the quantum state from a very different perspective, one that differs from the above type of approach to wave function ontology in three basic ways: (1) our result is not based on the framework of realist or any other models of quantum theory, but instead one that is based on operational considerations about communication involving interactionfree measurements Elitzur and Vaidman (1993); (2) The definition of “reality” is not explicitly tied to assumptions like independence, nosignaling, etc., but is direct and intuitive; (3) Our approach allows us to define not only a concept of “reality”, but also another, that of “physicality”. We think that this is crucial, because our work shows that the distinction between physicality and reality lies at the heart of what makes the quantum state nonclassical.
Ii Interactionfree measurements and counterfactual communication
Interactionfree measurements (IFMs) in quantum theory allow one to obtain knowledge of the presence or absence of an object without interrogating it directly. For example, an absorber placed in one of arms of a MachZehnder interferometer is detected by its disturbance of the destructive interference that would have resulted in its absence Elitzur and Vaidman (1993). IFM can be used as the basis for what are called counterfactual computation Jozsa (1998); Hosten et al. (2006) and counterfactual crytpography Noh (2009); Shenoy et al. (2013), where the absorber is present or absent depending on the outcome of a computer or choice of a communicator. Here ‘counterfactual’ essentially means that Bob’s blockade in one of the arms of a Michelson interferometer causes a detection of the particle away from the blockade. Singleparticle nonlocality lies at the heart of this phenomenon, since without this, the blockade would localize a particle near it.
Recent advances include counterfactual communication based on the quantum Zeno effect Salih et al. (2013) (cf. also Vaidman (2014); Salih et al. (2014)) and the extension of the counterfactual bipartite protocols to the tripartite case Shenoy H. et al. (2014); Salih (). The N09 protocol of Noh (2009) has been experimentally implemented Brida et al. (2012), and its security proofs discussed by various authors Yin et al. (2010); Zhang et al. (2012a, b).
For the purpose of this work, it suffices for any one outcome to be counterfactual, and we use the recently proposed semicounterfactual quantum cryptographic scheme Shenoy et al. (2013), where only one of the outcomes is counterfactual. This experimental setup for counterfactual communication is based on a Michelsontype interferometer. One of the arms is taken to be an external one, connecting Alice and Bob, and the other arm is an internal arm within Alice’s station (Figure 1).
A singlephoton from a source in Alice’s station hits her beamsplitter BS and is split along the two arms and . Alice and Bob each possesses a switch which is randomly in one of two modes: absorb () and reflect with a Faraday mirror (). The state of the particle after the beamsplitter is:
(1) 
where () is the creation operator for the light mode in the internal (external) arm, and is the vacuum state of those two modes. The operators for Alice’s detector modes, in terms of those for the arm modes, is given by:
(2) 
where correspond to creation operators for modes corresponding to detector (Figure 1). The three possiblities of Alice’s and Bob’s choice are:

Alice and Bob both apply : This results in an interference with a bright fringe at detector and a dark fringe at :
(3) 
Exactly one of them applies and the other : This produces a click in detector or each equal probability:
(4) where () indicates Alice (Bob) applying , while () indicates Alice (Bob) applying .

Both apply : neither detector nor clicks:
(5)
Ideally, a bit is communicated when Alice observes , for in this case she knows for certain that Bob applied an operation anticorrelated with hers. The efficiency of the protocol is Shenoy et al. (2013).
The counterfactual situation arises in events when Bob applies and Alice applies . In this case, if the detector clicks, then it carries 1bit of information of Bob’s measurement choice (that it was ), even though the photon did not physically travel and interact with his detector, as evidenced by his lack of detection. It is in this sense that this 1bit information is counterfactual, and the physical interpretation of this communication forms the objective of this work.
The other possibility for a event, that Bob applied and Alice , is not counterfactual with respect to Bob (hence the characterization of the protocol of Ref. Shenoy et al. (2013) as semicounterfactual.)
Iii Physical interpretation
We distinguish two concepts to be used to describe the nature of the quantum state: the term “physical”, as used in counterfactual communication to qualify the travel of a particle, and the term “real”, as has been used recently in quantum foundations. We will define these two concepts based on simple, operational criteria inspired by the above experiment.
iii.1 Reality
Consider the communication scenario described in Figure 2. This is an unwrapped version of the path taken by a particle from Alice’s source, via Bob, back to Alice in Figure 1. Alice knows that a certain entity is emitted from the source at position along the indicated path towards position . Bob’s action at the intermediate position is to block or to not block its passage during its transit. Suppose is a real thing, like a classical ball. Then if Bob applies blocking action , Alice does not receive the ball at , while if he applies the forwarding action , she does. Thus Bob’s action can be deterministically inferred by Alice through local observation.
On the other hand, suppose is epistemic and not a real thing. Then it represents a probability distribution
(6) 
as perceived by Alice, with being the probability that the particle travels from towards Alice through , and , the probability that it does not travel from the source along this path towards Bob.
If Alice finds the particle at , then Bob clearly did not apply . But if she does not recover the particle, it does not necessarily imply that Bob applied , since there is a nonvanishing probability that the particle did not travel from to . In other words, an epistemic necessarily precludes Alice’s deterministic retrodiction of Bob’s blocking action.
If we denote by “no” Alice’s nondetection of the particle at , then , and by Bayesian argument:
(7) 
where is the probability that Bob applies . Eq. (7) shows that only if . Otherwise, , implying that Alice cannot deterministically infer that Bob applied action . More generally, “yes” indicates a specific type of outcome observed by Alice, conditioned on her any local operation. For example, if she turns on a knob, and a beep appears on a monitor, this could correspond to “yes”, while “no” would correspond to the nonappearance of such a beep.
This motivates the following definition of reality: We say that is real and not epistemic, if and only if there is some detection event at such that Alice can deterministically infer by local observation in her station that Bob applied the blocking action during the ’s transit:
(8) 
where Real() is the proposition that asserts the reality of . Note that this condition for reality is quite operational: it is based on no interpretational framework for but is stated in terms of Bob’s actions and Alice’s observations in a laboratory.
We now apply the above definition to a classical object. If is (say) a physical ball transmitted along the indicated path in Figure (2) with in Eq. (6). Then is real according to the above criterion, since every nondetection event at allows Alice to infer Bob’s absorbing action. Further, if is epistemic (corresponding to in Eq. (6)), then it fails to be real according to our criterion (8).
Now consider the setup in Figure 1. To map the scheme of Figure 2, we let both coordinates and to be with Alice, and with Bob. We denote as “yes” the events conditioned on Alice applying the operation locally. Thus the observation of conditioned on her applying corresponds to outcome “no” in Eq. (7). In such cases, Bob’s blocking action is deterministically inferred by Alice, since
(9)  
by Bayes’ theorem, and since by Eq. (3), while . According to the above criterion, then the wave function that leaves the source and travels to Bob, must be real.
iii.2 Physicality
It is in counterfactual quantum communication that we encounter the idea of “transfer of information without any physical travel of particles” or “nonphysical transmission of information”. In the experiment of Figure 1, we saw that it arises naturally from the way communication occurs from Bob to Alice when he measures the photon via interactionfree measurement, and she observes .
By contrast, in all foundational approaches to the status of the wavefunction, it is only the reality of the wavefunction, rather than its physicality, that is called into question. We want to stress that this is the significant contribution of quantum cryptography, in particular the counterfactual variety, to this discussion.
To be precise, the particle’s transmission is called nonphysical in counterfactual cryptography when Bob’s communication generates no detection on his apparatus. Generalizing this, the essence of physicality here, then, is that a physical object should be detected by an ideal detector.
This motivates us to identify physicality with physical detectability: We say that is physical if and only if for every transmission event , if Bob applies the absorption action during ’s transit, then it necessarily leads to his detection (provided he has an ideal detector), else (i.e., he applies ), it leads to a detection by Alice. In other words, a physical is always detected by an intercepting ideal detector:
(10) 
where “Y” indicates Bob’s detector registering a detection. Here Physical() is the proposition which asserts that is physical. This condition, like Eq. (8) for reality, is operational: it is not based on any interpretational framework for , but is stated in terms of Alice’s and Bob’s actions and observations.
Note that if is physical, then from the second condition in (10), we find that is also real. Thus physicality entails reality. On the other hand, the converse is not true. only entails that there are some events such that if Bob applies , Alice finds “yes”. If this does not hold true in all cases, then it fails to be physical, opening up the possibility of nonphysical but real .
In Figure 2, consider to be a classical real object. Then and Bob’s action always leads to a detection; while his action leads to Alice’s observation of “yes”, thereby satisfying the physicality condition (10). A classical epistemic object fails (10) because in the probabilistic event when no particle travels down to Bob, his action does not generate a detection for him. Thus, classical objects are either real and physical or epistemic and nonphysical. In other words, reality and physicality are equivalent in this formalism.
iii.3 Nonphysical reality
The defintions Eq. (8) and (10) allow the existence of objects that are realphysical, realnonphysical and epistemicnonphysical in the present framework. As noted above, classical conflates reality with physicality. Thus the only classical possibilities are realphysical and epistemicunphysical. Quantum mechanics allows the additional existence of realnonphysical .
Combining the results from the Subsections (III.1) and (III.2), we conclude that the quantum wave function that propagates from Alice to Bob in the onward leg is real according to criterion (8) but nonphysical according to criterion (10). On the other hand, as noted above:
(11) 
so that the class of real objects is strictly weaker than that of physical objects. Thus an epistemic (i.e., unreal) object is necessarily nonphysical, according to our criterion.
Since the equivalence of reality and physicality holds within the domain of classical objects, objects that are real but nonphysical are nonclassical. These ideas are depicted in Figure 3, where the set of real (physical) objects is called Real (Physical).
Note that classical waves (say water waves) show an interference behavior similar to the singlephoton system. Yet water waves are both real and physical: here can be taken to represent the relative mass of water moving down each arm, and “yes” can represent Alice observing a depletion of water mass at her end, while represents Bob’s blocking action. It immediately follows that classical water waves are physical, and consequently real by criteria (8) and (10).
Iv Security via nonphysical reality of the wave function
We now claim that nonphysical reality underlies the security of the semicounterfactual cryptography protocol of Section II in the precise sense that the security check for the protocol consists in verifying that the reality condition (8) holds, while the physicality (10) fails. Suppose an eavesdropper Eve attacks the exposed arm () in Figure 1, with the aim of testing whether the particle travelled along that path. To the extent she can detect the particle’s location, she commits the particle to one or the other arm, and thereby enforces classical behavior. The degree of failure of Alice to verify nonphysicalreal behavior in the data, alerts Alice and Bob to Eve’s presence.
Let us consider this more quantitatively, a detailed study of which is presented in Shenoy et al. (2013). Eve attacks the particle in arm at a point along the solid line in the onward leg, by first interacting it with a probe of hers prepared in the initial state using interaction :
(12) 
where and represent nonorthogonal states, with , where is a security parameter in the range . Eve’s intention is to subsequently measure the probe, and use its outcome in conjunction with Alice’s announcement, to optimize her guess of the shared secret bit.
If Alice announces , then Eve measures her probe according to the following positive operatorvalued measure (POVM) Nielsen and Chuang (2000), which can be shown to be optimal:
(13) 
where outcome indicates a conclusive outcome, while outcome is inconclusive. From (13), it follows that on an ensemble of and , the probability that Eve obtains a conclusive answer is . Clearly, this is also Eve’s information on the (unprocessed) secret key.
Eve’s presence is therefore revealed through the departure from perfect interference, as observed by Alice, which from Eqs. (12) and (2) is
(14) 
implying a visibility of . If condition (8) is perfectly satisfied, then, conditioned on Alice applying , if Bob applies , then a “yes” outcome, namely detection, must arise. To the degree that , the condition (8) fails, and the second condition in (10) also fails. To summarize, security arises because Eve’s attack physicalizes, and thus classicalizes the particle, which is detected by Alice and Bob who are monitoring nonphysicalreal behavior.
To complete the discussion, the error produced in the key is that Alice’s and Bob’s inputs are not anticorrelated when a click happens Shenoy et al. (2013):
(15)  
since where , where , and . The mutual information between Alice and Bob is , where is the Shannon binary entropy. The condition for positive key rate is that Csizár and Körner (1978), or
(16) 
which implies that the error rate must be less than about 21% for security, and thence, for proof of nonphysicality of the wave function.
With more powerful models of Eve’s attack, this error threshold would be lowered. However, the basic idea– that Eve’s attack physicalizes, and thus classicalizes the particle– should intuitively be the same.
V Conclusions and Discussions
We have used counterfactual quantum cryptography (CQC) as a tool to study the ontology of the quantum wave function. This stands in contrast to approaches based on ontological theories, where a concept of “reality” for is defined, and under certain arguable assumptions, arguments in favor of the quantum state being real are presented. Furthermore, they do not appear to throw sufficient light on what exactly is weird about quantum mechanics.
In the present approach, we gave operational definitions of “reality” and of “physicality” of . In particular, the reality of is inferred from the fact of genuine communication from Bob to Alice during counterfactual communication, while its nonphysicality is inferred from the fact that Bob makes no physical detection of the particle during that communication. Thereby we are able to draw conclusions that go beyond earlier approaches: (a) that the quantum state is real; (b) that it is not physical; (c) that this nonphysical reality is the mark of nonclassical phenomena, since reality and physicality coincide for classical phenomena.
Of course, the status of the wave function (as being real or epistemic) does not depend on Bob’s choice of or . Nor does it depend on whether Bob is located at the end of arm or . What may conclude is that the each of the superposed states in Eq. (1), and , is by itself realnonphysical, and thus, so too the particle state state in Eq. (1) is also realnonphysical. We may therefore conclude that the quantum state is quite generally realnonphysical. In retrospect, we may reflect in this new light on the wisdom of Feynman’s observation with regard to the doubleslit experiment, mentioned in the opening paragraph. Our approach suggests that in the production of fringes in the doubleslit experiment, there is indeed some “real stuff” travelling down both slits, but it is not physical. This explication thus puts (or so we hope!) a name on the mystery alluded to by Feynman.
In counterfactual communication, Bob’s choice has a nonlocal influence on Alice’s local observation. This manifestation of singleparticle nonlocality is consistent with relativity, since Alice must wait for the time required by Bob’s particle to return, before deciding whether he applied or , and thus respects signal locality.
From the perspective presented here, the essential mark of nonclassical behavior, and the ‘mystery’ of the quantum, is the nonvanishing gap between the real and the physical. An analogous situation has been reported also in the study of quantum correlations, where nonclassicality is identified with the gap between the communication cost of nonlocal correlations P and the signal accessible within P Aravinda and Srikanth (2014). Now is just the signal required to simulate the correlations using resources from a deterministic hidden variable (DHV) theory. Thus the ‘mystery’ highlighted by this latter gap is that the signal that arises at the ontological level cannot be used for signaling at the physical level (cf. Spekkens (2005); Kak (2014)).
Our work showed that the nonphysical reality of the wave function is not an abstruse philosophical notion, but has the concrete application of being responsible for security in CQC. Finally, we venture that it is the lack of distinction in the literature between the real and the physical aspect that is responsible for the historical difficulty in interpretting the physical significance of the quantum state. In the discussion pertaining to the doubleslit experiment, at first one has the intuitive feeling that there is something real traveling down both slits. One then subconsciously maps this real thing to something physical. But clearly the possibility of the quantum wave as a physical entity is one that we would consciously reject. Thus, psychologically speaking, a person thinking about quantum foundations is caught in the perpetual dilemma of deciding whether or not the quantum state is real. It is our belief that our work resolves this dilemma.
Acknowledgements.
RS acknowledges support from the DST project SR/S2/LOP02/2012.References
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