To illustrate this, we consider the celebrated MZI, shown in Fig. 1, since most literature WPDRs have been formulated for this interferometer. In the simplest case, one sends in a single photon towards a 50/50 (that is, symmetric) beam splitter, BS 1 , which results in the state , then a phase φ is applied to the lower arm giving the state . Finally, the two paths are recombined on a second 50/50 beam splitter BS 2 and the output modes are detected by detectors D 0 and D 1 . Fringe visibility is then defined as

where is the probability for the photon to be detected at D 0 ; maximizes this probability over φ, whereas . In this trivial example, one has . However, many more complicated situations, for which the analysis is more interesting, have been considered in the extensive literature; we now illustrate how these situations fall under the umbrella of our framework with some examples.

As a warm-up, we begin with the simplest known WPDR, the predictability–visibility trade-off. Predictability quantifies the prior knowledge, given the experimental setup, about which path the photon will take inside the interferometer. More precisely, where p guess (Z) is the probability of correctly guessing Z. Non-trivial predictability is typically obtained by choosing BS 1 to be asymmetric. In such situations, the following bound holds4,5:

This particularly simple example is a special case of Robertson’s UR involving s.d.30,31,40,41. However, ref. 41 argues that equation (8) is inequivalent to a family of EURs where the same (Rényi) entropy is used for both uncertainty terms, hence one gets the impression that entropic uncertainty is different from wave–particle duality. On the other hand, ref. 41 did not consider the EUR involving the min- and max-entropies. For some probability distribution P={p j }, the unconditional min- and max-entropies are given by H min (P)=−log max j p j and . We find that equation (8) is equivalent to

which is an EUR proved in the seminal paper by Maassen and Uffink23, and corresponds to E 1 and E 2 in equation (5) being trivial. The entropies in equation (9) are evaluated for the state at any time while the photon is inside the interferometer. It is straightforward to see that and in Methods we prove that

Plugging these relations into equation (9) gives equation (8).

Let us move on to a more general and more interesting scenario where, in addition to prior which-path knowledge, one may obtain further knowledge during the experiment due to the interaction of the photon with some environment F, which may act as a which-way detector. Most generally, the interaction is given by a completely positive trace preserving map , with the input system being Q at time t 1 and output systems being Q and F at time t 2 , see Fig. 1. The final state is , where the superscripts (1) and (2) indicate the states at times t 1 and t 2 . We do not require to have any special form to derive our WPDR, so our treatment is general.

The path distinguishability is defined by , where p guess (Z|F) is the probability for correctly guessing the photon’s path Z at time t 2 given that the experimenter performs the optimally helpful measurement on F. We find that equation (1) is equivalent to

where the entropy terms are evaluated for the state , which corresponds to E 1 =F and E 2 being trivial in equation (5). First, it is obvious from the operational meaning of the conditional min-entropy in the first of equations (6) that we have , and second we use our result equation (10) to rewrite equation (11) as equation (1). As emphasized in ref. 2, we note that equation (1) and its entropic form equation (11) do not require BS 1 to be symmetric. Hence, accounts for both the prior Z knowledge associated with the asymmetry of BS 1 as well as the Z information gained from F.

The above analysis shows that equations (1) and (8) correspond to applying the preparation UR at time t 2 (just before the photon reaches BS 2 ). Preparation uncertainty restricts one’s ability to predict the outcomes of future measurements of complementary observables. Thus, to experimentally measure or more generally , the experimenter removes BS 2 and sees how well he/she can guess which detector clicks, see Fig. 4a. Of course, to then measure , the experimenter reinserts BS 2 to close the interferometer. We emphasize that this procedure falls into the general framework of preparation uncertainty.

Figure 4: Path prediction versus path retrodiction, in the MZI. (a) In the predictive scenario, the second beam splitter is removed and Alice tries to guess which detector will click. (b) In the retrodictive scenario, a blocker is randomly inserted into one of the interferometer arms and Alice tries to guess which arm was blocked (given the knowledge of which detector clicked). Full size image

On the other hand, URs can be applied in a conceptually different way. Instead of two complementary output measurements and a fixed input state, consider a fixed output measurement and two complementary sets of input states. Namely, consider the input ensembles from equations (2) and (3), now labelled as Z i ={|0›,|1›} and W i ={|w ± ›}, where i stands for ‘input’, to indicate the physical scenario of a sender inputting states into a channel. Imagine this as a retrodictive guessing game, where Bob controls the input and Alice has control over both F and the detectors. Bob chooses one of the ensembles and flips a coin to determine which state from the ensemble he will send, and Alice’s goal is to guess Bob’s coin flip outcome. Assuming BS 1 is 50/50, the two Z i states are generated by Bob blocking the opposite arm of the interferometer, as in Fig. 4b, while the W i states are generated by applying a phase (either φ 0 or φ 0 +π) to the lower arm.

It may not be common knowledge that this scenario leads to a different class of WPDRs, therefore we illustrate the difference in Fig. 4. For clarity, we refer to introduced above as output distinguishability, whereas in the present scenario we use the symbol and call this quantity input distinguishability, defined by

where is Alice’s probability to correctly guess Bob’s Z i state given that she has access to F and she knows that detector D 0 clicked at the output. Likewise, we define the notion of input visibility via:

which quantifies how well Alice can determine W i given that she knows D 0 clicked.

Now the uncertainty principle says there is a trade-off: if Alice can guess the Z i states well, then she cannot guess the W i states well and vice versa. In other words, Alice’s measurement apparatus, the apparatus to the right of the dashed line labelled t 1 in Fig. 1, cannot jointly measure Bob’s Z and W observables. EURs involving von Neumann entropy have previously been applied to the joint measurement scenario27,42, we do the same for the min- and max-entropies to obtain (see Methods for details)

which can now be applied to a variety of situations.

As an interesting application of equation (14), we consider the scenario proposed in ref. 33 and implemented in refs 34, 35, 36, where the photon’s polarization P acts as a control system to determine whether or not BS 2 appears in the photon’s path and hence whether the interferometer is open or closed, see Fig. 5. Since P can be prepared in an arbitrary input state , such as a superposition, this effectively means that BS 2 is a ‘quantum beam splitter’, that is, it can be in a quantum superposition of being absent or present. The interaction coupling P to Q is modelled as a controlled unitary as in Fig. 5. In this case, the two visibilities are equivalent (see Methods)

Figure 5: Quantum beam splitter in the MZI. In this scenario, the second beam splitter is in a superposition of ‘absent’ and ‘present’, as determined by the polarization state at time t 2 . The quantum beam splitter (QBS) can be modelled as a controlled unitary, U PQ =|H›‹H| P ⊗ Q +|V›‹V| P ⊗U(R), where U(R) is the unitary on Q associated with an asymmetric beam splitter with reflection probability R. Polarization-resolving detectors (PBS=polarizing beam splitter) on the output modes help to reveal the ‘quantumness’ of the QBS. Full size image

where we assume the dynamics are path preserving, that is, and , where is the reduced channel on Q, which implies that , that is, off-diagonal elements get scaled by a complex number κ with |κ|≤1. In equation (15), is evaluated for any pure state input from the XY plane of the Bloch sphere (for example, |+›). Now we apply equation (14) to this scenario and use equation (15) to obtain:

which extends a recent result in ref. 13 to the case where F is non-trivial. This general treatment includes the special case where , corresponding to a closed interferometer with an asymmetric BS 2 . Ref. 3737 experimentally tested this special case. However, ref. 37 did not remark that their experiment actually tested a relation different from equation (1), namely, they tested a special case of equation (16).

Similarly, ref. 34 tested equation (16) rather than equation (1), but they allowed to be in a superposition. At first sight, this seems to test the WPDR in the case of a quantum beam splitter (QBS), but it turns out that neither the visibility nor the distinguishability depends on the phase coherence in and hence the data could be simulated by a classical mixture of BS 2 being absent or present. Nevertheless, our framework provides a WPDR that captures the coherence in , by conditioning on the polarization P at the interferometer output (see Methods). For example, defining the polarization-enhanced distinguishability, , which corresponds to choosing E 1 =FP, we obtain the novel WPDR:

which captures the beam splitter’s coherence (see Supplementary Note 4 for elaboration) and could be tested with the set-up in ref. 34.

The above examples use the environment solely to enhance the particle behaviour. To give a corresponding example for wave behaviour, that is, where system E 2 in equation (5) is non-trivial, the main result in ref. 11 is a WPDR for the case when the environment F is measured (after it has interacted with the quanton) and the resulting information is used to enhance the fringe visibility. This scenario is called quantum erasure, since the goal is to erase the which-path information stored in the environment to recover full visibility. This falls under our framework by taking E 2 to be the classical output of the measurement on the environment. For elaboration, see Supplementary Note 3, where we also cast the main results of ref. 10 (Supplementary Note 2) and ref. 12 (Supplementary Note 3) within our framework.

In summary, we have unified the wave–particle duality principle and the entropic uncertainty principle, showing that WPDRs are EURs in disguise. We leave it for future work to extend this connection to multiple interference pathways6. The framework presented here can be applied universally to binary interferometers. Our framework makes it clear how to formulate novel WPDRs by simply applying known EURs to novel interferometer models, and these new WPDRs will likely inspire new interferometry experiments. We note that all of our relations also hold if one replaces both min- and max-entropies with the well-known von Neumann entropy. Alternatively, one can use smooth entropies29,39, and the resulting smooth WPDRs may find application in the security analysis of interferometric quantum key distribution43, which often exploits the Franson set-up (Fig. 2).