Introduction

The duality between wave and particle properties of a microscopic physical system is a founding principle of quantum mechanics1,2. A canonical illustration is provided by a single particle travelling through a double slit or a Mach–Zehnder interferometer: an attempt to gain information about the path taken by the particle inevitably reduces the visibility of the interference pattern3,4,5,6,7,8,9,10,11. The purpose of this work is to analyse the trade-off between interference visibility and which-way information for a particle equipped with an internal degree of freedom, for example, spin. In such a general case, the interaction with an environment that acquires which-way information can transform in a non-trivial manner the joint path–spin state of the particle. This opens up an interesting question of how to infer the amount of which-way information deposited in the environment from visibility measurements. Another issue is, to what extent manipulating the spin subsystem can control the information about the path taken by the particle.

A common approach to quantify the amount of which-way information is to use distinguishability D, defined as the maximum difference between the probabilities of correct and incorrect identification12 of the path taken by the particle inside the interferometer, based on the state of the environment. We assume here that both paths are equiprobable. In the spinless case, the ability for the particle distributed between two paths to interfere is characterized by the visibility V, which measures the modulation depth of interference fringes after the paths are combined at the interferometer exit. These two quantities are related by the inequality6

When the particle travelling through the interferometer has an internal structure, the amount of available which-way information depends in principle on the preparation of the spin subsystem and the specifics of the interaction with the environment. Here we demonstrate that in this case the strongest bound on the distinguishability D is obtained by replacing V with a quantity named generalized visibility that depends on the initial spin preparation and the effective quantum channel experienced by the particle resulting from the interaction with the environment. Further, we present a systematic method to construct estimates for the generalized visibility based on directly measurable quantities, that is, interference visibilities for particular spin preparations at the input and selections of spin states before combining the paths at the interferometer exit. This provides an efficient strategy to find an upper bound on the available which-way information without performing full quantum process tomography13,14,15. We illustrate the general results with a proof-of-principle single-photon experiment that also demonstrates how which-way information can be erased by introducing mixedness in the spin preparation.

Results

The inequality

The quantum mechanical system considered here is a particle present in one of two distinguishable paths, denoted respectively as |0Q and |1Q. In addition, the particle is equipped with a d-dimensional internal degree of freedom – spin – which we will treat as a subsystem S. The complete state of the particle prepared in the interferometer is characterized by a certain density operator . Both paths are equiprobable, that is, . Inside the interferometer, the particle interacts with an environment E producing a joint state . The interaction is constrained in that it cannot transfer the particle between the paths. In this setting, which-way information contained in the environment corresponds to the ability to discriminate between two normalized density matrices , i=0, 1. In the general case of mixed states, distinguishability is given explicitly by the expression12,16

where ||·|| denotes the trace norm. Let us stress here that this quantity is not conditioned upon any measurement performed eventually on subsystems QS.

Our aim is to infer an upper bound on D from the properties of the particle, that is, subsystems QS, after the interaction with the environment. The state of the particle at this stage can be written as a result of an action of a certain quantum channel17 Λ on the input state , with . Because the particle is not transferred between interferometer arms, for any spin operator , we have

Our central result is that D satisfies the inequality:

where VG, named generalized visibility, reads:

Here is a normalized maximally entangled state of two replicas of the spin subsystem, I is the identity channel, is the identity operator on the spin subsystem and are normalized spin states for the particle in either arm of the interferometer before the interaction with the environment. The complete proof of the inequality is presented in Methods.

Examples

The inequality (4) resembles that for a spinless particle given in equation (1), but the concept of generalized visibility covers a much wider range of physical scenarios. In order to illustrate its breadth, let us consider here three cases. First, when no interaction with the environment occurs, Λ01=I. In this case , that is, no matter what internal states for individual paths are initially chosen, the environment does not gain any which-way information.

Second, let us take , where is a fixed unit-trace hermitian operator. Physically, this occurs when the internal degree of freedom is entirely transferred to the environment and replaced with a state . Then which is the standard expression for quantum fidelity18,19 between density operators and . Consequently, the only way to ensure that no which-way information leaks out is to associate both the interferometer paths with identical internal states.

Finally, suppose that

where T denotes transposition. This channel gives . The resulting expression has non-intuitive properties: for a particle prepared in any pure state the generalized visibility equals 1/d, which is strictly less than one for a non-trivial spin subsystem when d≥2. Generalized visibility reaches one only when the two paths are associated with maximally mixed spin states, , which follows immediately from the inequality between arithmetic and quadratic means. This indicates that which-way information deposited in the environment may be erased by introducing classical noise in the input state of the spin subsystem. To illuminate this point, let us take a single photon travelling through a Mach–Zehnder interferometer and consider its polarization, spanned by two orthogonal states |hS and |υS, as the two-dimensional internal degree of freedom. Let the interaction of the photon with the environment be given by a unitary map

where |enE, n=1,…,4, are four orthogonal states of the environment and ψih=Sh|ψiS, ψiυ =Sυ|ψiS are the probability amplitudes in the rectilinear basis. It is easy to verify that this map indeed induces , although the polarization state in an individual arm is scrambled to a completely mixed state, .

If we prepare |ψ0S=|ψ1S=|hS, the two states of the environment correlated with the paths

are partly distinguishable. Similarly, for the vertical input polarization, |ψ0S = |ψ1S = |υS, the environment states are

However, if the input polarization is prepared in a completely mixed state , one sees immediately that both the paths are associated with the same mixed environment state and which-way information is not available anymore. The inequality (4) demonstrates that this effect is an intrinsic feature of the channel under consideration rather than its specific realization given in equation (7).

Estimating generalized visibility

Generalized visibility VG is given by a rather intricate expression involving the spin preparation and the effective quantum channel experienced by the particle. In principle, full information about the channel can be obtained from quantum process tomography13,14,15, but this approach may be resource consuming, especially for a high dimension of the internal subsystem S. We will now give a recipe how to construct estimates for VG from direct visibility measurements for specific spin preparations and selections of individual spin components before interfering the particle paths.

Consider the following procedure. The particle is prepared in one of states

where the normalized kets and describe the spin state in the upper and lower interferometer path. The index μ labels different preparations. After the interaction with the environment, the spin subsystem is filtered individually in each interferometer arm to extract probability amplitudes corresponding to normalized spin states and . These components are subsequently made indistinguishable by a suitable spin transformation and interfered on a balanced beam splitter with a relative phase shift φ. The probabilities of detecting the particle at the two output ports ± of the beam splitter read

Here,

is the overall probability of detecting the particle in the filtered components. The modulation depth of interference fringes is characterized by the fractional visibility, given explicitly by

For the quantities introduced here, we always have |Vμν| ≤ pμν. Full-depth modulation of fringes is observed for the equality sign.

Let us now take any set of complex coefficients αμν such that a linear combination of rank-one operators can be written as

for a certain operator acting on the duplicated spin subsystem that satisfies . We show in Methods that under these assumptions the generalized visibility is bounded by a linear combination of fractional visibilities with the same coeffcients αμν:

Combining this result with equation (4) yields a family of bounds on distinguishability in the form

Special cases of equation (16) are intuitive. Consider a scenario shown in Fig. 1. We use only a single preparation (henceforth in this example we drop the superfluous label μ), and the sets of filter states in the upper and lower arms form two orthonormal bases for the spin subsystem with ν indexing the basis elements. In equation (14), we can take arbitrary phase factors as coefficients αν=, with the operator satisfying the condition . According to equation (15), we have , which maximized over phases θν estimates the generalized visibility by a sum of absolute values of fractional visibilities:

Figure 1: Projective measurement of fractional visibilities.
figure 1

A particle is prepared in the spin state in the upper arm and in the lower arm. After interaction with the environment described by the channel Λ the phase between interferometer arms is modulated by φ. Next, the spin states are filtered in the orthonormal bases for the upper arm and for the lower arm, and individual components are interfered pairwise to determine fractional visibilities Vν,ν=0, 1,…,d–1.

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If for each filter we measure the maximum possible fractional visibility |Vν|=pν, the right-hand side of equation (17) becomes one, because completeness of the bases and implies that . In other words, when the interaction with the environment introduces different phase shifts between distinguishable components of the spin subsystem but maintains overall coherence between the paths, we can observe full visibility by suitably sorting spin components at the output and interfering them pairwise. This suffices to verify that no which-way information is deposited in the environment. However, as noted earlier, the interplay between the path and the spin subsystems can exhibit more intricate behaviour. We illustrate this point with a single-photon experiment.

Experiment

To expose more complex aspects of the trade-off between which-way information and interference visibility for a particle with an internal structure, we investigated experimentally interference of a single photon in a noisy Mach–Zehnder interferometer shown in Fig. 2, using photon polarization as the internal subsystem S. Single photons with 810 nm central wavelength were generated by type-II spontaneous parametric down-conversion in a 30-mm long periodically poled potassium titanyl phosphate crystal pumped with 14 mW of 405 nm wavelength light from continuous wave diode laser, and heralded by detection of orthogonally polarized conjugate photons from the same pairs. The photons, after sending through a 3 nm bandwidth interference filter and transmitting through a single-mode fibre, were split between two paths using a calcite displacer. Equal splitting was ensured by using a fibre polarization controller. Removable half-wave plates H1 and H2 were used to prepare any combination of horizontal h and vertical υ polarization states for the two paths.

Figure 2: Noisy Mach–Zehnder interferometer.
figure 2

Experimental set-up. A heralded single photon with 810 nm central wavelength and 3 nm bandwidth from a down-conversion source is split between two paths using a calcite crystal. Polarizations in individual paths are prepared using optional half-wave plates (HWPs) H1 and H2. Noise Λ is introduced using four sets of orientations of HWPs H3 and H4 and quarter-wave plates Q1 and Q2 mounted on computer-controlled motorized rotation stages. Polarization components of the output state are filtered using a HWP H5 and a second calcite crystal. Interference is realized in the common-path configuration using HWP H6 oriented at 45° with respect to the set-up plane and a polarizing beam splitter P with output ports monitored by single-photon detectors D+ and D− . Two other detectors D0 and D1 are used to determine the overall count rate. The phase shift is introduced by rotating the second calcite crystal about an axis perpendicular to the set-up plane using a closed loop piezo actuator.

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Inside the interferometer the photons were subjected to path-dependent unitaries , where the pair was an equally weighted mixture of (,), , and , with , and denoting Pauli operators. The unitaries were realized by wave plates H3, H4, Q1, Q2 mounted on motorized rotation stages, with orientations specified in Table 1. This procedure scrambled the polarization state in a single arm to the completely mixed state. But the noise had a non-trivial effect on the joint path-polarization state, as

Table 1 Orientations of principal axes of wave plates used to simulate interaction with the environment.
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which provides a physical realization of equation (6) for d=2. We treat the realized noisy channel as a black box simulating interaction with an environment and analyse the maximum amount of which-way information that might have leaked to the environment in the worst-case scenario. We have seen that another realization of this channel, given in equation (7), can reveal certain which-way information depending on the preparation of the internal degree of freedom.

In order to estimate which-way information possibly deposited in the environment, we used four different combinations of preparations for the photon polarization in the upper and the lower paths, denoted jointly as μ=hh, hυ, υh, υυ. After the simulated interaction with the environment, we filtered out from each path either horizontal or vertical component using the half-wave plate H5, which directed selected polarizations to the same output port after the second calcite crystal. We used a common-path set-up based on the half-wave plate H6 and the polarizer P monitored by single-photon detectors D± to measure the corresponding fractional visibilities Vμν. The index ν labelling filters also assumes one of four values: hh, hυ, υh or υυ. The two non-interfering output ports after the second calcite crystal were monitored by detectors D0 and D1 in order to normalize overall count rates. The measured count rates were adjusted by binomial resampling to correct for non-uniform detection efficiencies of the detectors.

Because individual paths inside the interferometer are completely depolarized by the interaction with the environment, the theoretical probability of filtering out the photon for any selection of the preparation μ and the filter ν equals . Theoretical predictions for fractional visibilities are collected in Table 2. It is seen that only few combinations of preparations and filters are expected to produce interference fringes. Let us select for a moment a single preparation μ. After filtering a pair of polarization components, one could use the remaining components to measure also the second fractional visibility for complementary polarizations. For example, ν=υυ is complementary to ν=hh, and ν=υh is complementary to ν=hυ. This is an example of a scenario covered by the inequality (17). Although the photon is equally likely to choose either of the two complementary filters, interference fringes are visible only for one of them and the right-hand side of equation (17) is at most 1/2.

Table 2 Theoretical predictions of fractional visibilities Vμν for preparations μ and filters ν chosen in the rectilinear basis.
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Suppose now that before interaction with the environment the photon polarization is averaged to the completely mixed state and , for example by switching randomly between four preparations hh, hυ, υh, υυ. To estimate which-way information available in this case, let us take on the left-hand side of equation (14) four non-zero coefficients:

where θ1,…,θ4 are arbitrary phases. The corresponding operator

is unitary for any choice of phases θ1,…,θ4. Therefore, equation (15) provides a bound on generalized visibility in the form VG≥ |Vhh,hh+Vhυ,υh+Vυh,hυ+Vυυ,υυ|/2. Maximization over phases yields

This bound on generalized visibility can be applied directly to experimental data in order to estimate available which-way information for mixed polarization preparation.

In order to measure fractional visibilities, the phase in the interferometer was adjusted by rotating one of the calcite crystals about an axis perpendicular to the plane of the set-up using a closed loop piezo-electric actuator. For each selected phase value, we averaged over four unitaries simulating the interaction with the environment. Experimentally measured interference fringes for combinations of preparations μ and filters ν entering the inequality (21) are shown in Fig. 3 along with results obtained for complementary filters. Probabilities pμν and fractional visibilities Vμν determined from these measurements are collected in Table 3. Inserting these data in equation (21) gives VG ≥0.960±0.006. This translates into an upper bound on distinguishability in the form D ≤0.28±0.02 according to equation (16). Remarkably, this stringent bound is determined from just few measurements of fractional visibilities rather than complete quantum process tomography. In contrast, consider an individual pure preparation and complementary projective filters, corresponding to a pair of graphs shown in a single row of Fig. 3. In this case, experimental results yield at most VG ≥0.580±0.006 using equation (17), which implies D ≤0.815±0.005. This separation between estimated on distinguishability for mixed and pure polarization preparations strikingly demonstrates the convoluted effects of internal degrees of freedom in a which-way experiment.

Figure 3: Interference fringes in a noisy interferometer.
figure 3

Count rates on detectors D+ (+, black) and D− (× , red) conditioned upon detection of conjugate heralding photons from the down-conversion source as a function of the phase shift in the interferometer. Panels are labelled as μ,ν, where μ specifies polarization preparation and ν defines polarizations filtered from the two paths. Solid lines are sinusoidal fits used to determine fractional visibilities. Experimental uncertainties of count rates are smaller than the size of the symbols.

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Table 3 Experimental values of filtering probabilities pμν and fractional visibilities Vμν.
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Discussion

The scenario considered in this work assumed a clear distinction between the environment and the spin that are both external to the path subsystem. The amount of available which-way information was determined solely from the quantum state of the environment after the interaction with the particle in the interferometer and no classical information about either spin preparation or measurement results could be used for that purpose. This makes our approach distinct from previous studies of the quantum erasure phenomenon8,20,21,22,23, where postselection carried out on the environment can restore conditional interference fringes. Here, the environment was treated as an adversary whose information about the path taken by the particle we try to control and estimate by manipulating the spin subsystem at the preparation and the detection stages.

In this context, the analysis of a which-way experiment with an internal degree of freedom points at non-trivial issues in prepare-and-measure protocols for quantum key distribution over complex noisy channels. Suppose that a cryptographic key is to be established by sending a particle along a randomly selected one of two paths, and the key security is to be verified by preparing occasionally the particle in a superposition of the two paths and measuring its coherence at the output by sampling interference fringes. If the particle is equipped with spin, the trade-off between distinguishability and visibility derived in this work implies that certain eavesdropping strategies require the sender to use randomness in spin preparation in order to ensure security. Further, to verify the security, the sender and the receiver may need to combine data collected for an array of spin preparations and filterings.

An entanglement-based analogue of the scenario analysed above would be to prepare jointly two particles in a maximally entangled path–spin state and to subject one of them to interaction with the environment modelling an eavesdropping attempt. In this case, spin can have the role of a shield subsystem fully protecting the privacy of a cryptographic key generated by detecting particles in individual paths, even though the noise present in the bipartite state may prevent entanglement distillation at the same rate as key generation24,25,26,27. To complete the parallel, estimates for generalized visibility determined from fractional visibility measurements can be viewed as a dynamical analogue of recently introduced privacy witnesses28.

Methods

Our basic tool will be the Choi–Jamiołkowski isomorphism29,30 between linear maps and operators acting on the duplicated Hilbert space. Let us define a maximally entangled state of two replicas of the system QS as |Φ+QSQ′S′=|Φ+QQ′⊗|Φ+SS′, where and . The Choi–Jamiołkowski state corresponding to a channel Λ is given by , where the identity map I acts on the subsystem QS, whereas Λ acts on Q′S′. The transformation of a state can be written as

where the transposition T is performed in the same basis in which the state |Φ+QSQ′S′ has been defined. The input state is encoded in the subsystem QS, whereas the output state is inscribed in the subsystem Q′S′.

In order to take into account information deposited in the environment in course of the interaction, we will consider a purification of the state :

Because the particle is not transferred between the two paths, the purified state has the form

where the states |ΛiSS′E, i=0, 1, are normalized.

The joint state of the particle and the environment after the interaction can be written as follows:

In the first expression, transposition is performed on subsystems QS, while in the second expression only on S. Thus, the environment states correlated with the particle present in one or another path are given after normalization by , where are the normalized internal states of the particle in the upper and the lower arms of the interferometer. Distinguishability can only increase by access to additional subsystems that purify the states. We will take as purifications of states where are arbitrary unitaries. For any choice of these unitaries, we will have

The most stringent bound is obtained by minimizing the right-hand side over and . This can be equivalently written as , where

and is an arbitrary unitary, or using the trace operation

The maximum can be expressed in terms of the trace norm as follows:

The channel characteristics enters the above formula through

where Λ01 has been defined in equation (3). Using the fact that yields equation (5).

The expression for fractional visibilities Vμν given in equation (13) can be rewritten in the tensor form

Maximization in equation (28) can be extended to all operators satisfying . Therefore, for any such operator we have

where we used equation (30). Consequently, tracing both sides of equation (14) multiplied by and comparing resulting expressions with equations (31) and (32) immediately yields a lower bound on the generalized visibility presented in equation (15).

Additional information

How to cite this article: Banaszek, K. et al. Quantum mechanical which-way experiment with an internal degree of freedom. Nat. Commun. 4:2594 doi: 10.1038/ncomms3594 (2013).