# Coherence, Path-Predictability and I-Concurrence: A Triality

Abhinash Kumar Roy1<EMAIL_ADDRESS>Neha Pathania2 neha@ctp-
jamia.res.in Nitish Kumar Chandra1<EMAIL_ADDRESS>Prasanta K.
Panigrahi1<EMAIL_ADDRESS>Tabish Qureshi2<EMAIL_ADDRESS>1Department of Physical Sciences, Indian Institute of Science Education and
Research Kolkata, Mohanpur, West Bengal-741246, India  2Centre for Theoretical
Physics, Jamia Millia Islamia, New Delhi-110025, India

###### Abstract

It is well known that fringe contrast is not a good quantifier of the wave
nature of a quanton in multipath interference. A new interference visibility,
based on the Hilbert-Schmidt coherence is introduced. It is demonstrated that
this visibility is a good quantifier of wave nature, and can be experimentally
measured. A generalized path predictability is introduced, which reduces to
the predictability of Greenberger and Yasin, for the case of two paths. In a
multipath, which-way interference experiment, the new visibility, the
predictability and the I-concurrence (quantifying the entanglement between the
quanton and the path-detector), are shown to follow a tight _triality_
relation. It _quantifies_ the essential role that entanglement plays in
multipath quantum complementarity, for the first time.

##  I Introduction

We have come a long way since Neils Bohr proposed his principle of
complementarity in 1928 bohr , but instead of fading away, the topic has
increasingly become more interesting and involved. Wootters and Zurek were the
first ones to quantitatively analyze the phenomenon of complementarity or
wave-particle duality, as it is more commonly called wootters . Greenberger
and Yasin analyzed the problem with the assumption that unequal beams in a
two-path interferometer allows for predicting, to a certain degree, which of
the two paths the quanton followed. They established a duality relation GY

${P}^{2}+{V}^{2}\leq 1,$ (1)

which involved a _path predictability_ ${P}$ and an interference visibility,
which is just the fringe contrast $V=(I_{max}-I_{min})/(I_{max}+I_{min})$,
$I_{max},I_{min}$ being the maximum and minimum intensities in a region on the
screen. Here, equal beams would imply no path information. It was further
refined by Jaeger, Shimony and Vaidman jaeger . This approach, however, does
not take into account the fact that even when the two beams are equal, one may
introduce a probe to find out which of the two paths the quanton followed.
Such a scenario was considered by Englert englert , where he studied a two-
path interferometer in the presence of a path-detector. He arrived at a
duality relation

${D}^{2}+{V}^{2}\leq 1,$ (2)

involving a _path distinguishability_ ${D}$ and the fringe contrast ${V}$.
After a lull of nearly two decades, the issue of wave-particle duality saw a
resurgence when the concept of complementarity was quantitatively extended to
mulitpath interference cd15 ; nduality ; bagan . The multipath complementarity
relations used a _coherence_ introduced in the context of quantum information
theory plenio , instead of conventional interference visibility. This
coherence measure has been shown to be a good quantifier of the wave nature of
a quanton coherence . The duality relation, based on path-predictability, was
also generalized to the multipath case using coherence roy . However, the two
kinds of duality relations, one based on predictability, and the other on
distinguishability, remained separated.

Figure 1: A schematic representation of a multipath interference experiment,
with which-way detection.

Jakob and Bergou triality1  were the first to point out that, in two-path
interference, the predictability and distinguishability were connected through
entanglement, and the duality relation (2) can be written as a triality
relation between predictability, visibility and concurrence concurrence ,
which is a measure of entanglement. This was followed by a similar realization
in classical optics triality2 . This idea was later investigated in a
multipath quantum setting, and a multipath predictability, coherence plenio
and entanglement were shown to follow a triality relation triality3 . While
that analysis used a robust measure of coherence, the quantifier of
entanglement could not be shown to be a proper entanglement measure. Although
the analysis brought out the role of entanglement in multipath
complementarity, it could not be rigorously quantified. There have been other
interesting works trying to find an operational entanglement measure from
complementarity in the general case of multipartite and multi-dimensional
systems huber .

In the following, we investigate the issue of complementarity in multipath
interference and show that a robust entanglement measure, namely I-concurrence
rungta , plays an essential role in it.

##  II Preliminaries

###  II.1 Visibility measure

Two decades back it was experimentally demonstrated that in multipath
interference, conventional visibility, defined as fringe contrast, can
_increase_ under path-selective decoherence or selective path information mei
. Subsequent debate concluded that conventional visibility does not quantify
the wave nature of a quanton in multipath interference, and cannot be used to
analyze the notion of Bohr’s complementarity luis ; bimonte1 ; prillwitz . In
the experiment of Mei and Weitz mei , in a multipath interference, one
particular path was given an additional 180∘ phase flip. A selective
decoherence was applied on the particular path, by scattering photons off the
quanton. Counter-intutively, this resulted in an increase in the fringe
contrast of the interference pattern.

We use the normalized Hilbert-Schmidt coherence to define a generalized
visibility for a quanton passing through a multipath interferometer (see FIG.
1)

$\mathcal{V}^{2}\equiv\frac{n}{n-1}\sum_{i\neq
j}|\rho_{ij}|^{2}=\mathcal{C}_{HS},$ (3)

where $\mathcal{C}_{HS}$ is the normalised Hilbert-Schmidt coherence. The
system is assumed to have a $n$-dimensional Hilbert space, which arises out of
the quanton passing through $n$ paths. The states corresponding to the quanton
following each path are mutually orthogonal, and can be used to construct a
Hilbert space basis. The density matrix element $\rho_{ij}$ is taken between
two such states. This visibility satisfies all the criteria proposed by Dürr
for a valid measure of interference and wave properties durr . It is worth
noting that, for the two slit interference case, the generalized visibility
reduces to $\mathcal{V}=2|\rho_{12}|$, which can be easily shown to be equal
to the usual visibility measure ${V}=(I_{max}-I_{min})/(I_{max}+I_{min})$.

It might be useful to mention at this point that in classical optics, the
interference visibility in a two-slit interference is equal to the degree of
mutual coherence between the two slits, for equal intensity in the two beams.
Thus for a two-slit experiment, $2\rho_{12}$ can be identified with mutual
coherence function $\gamma_{12}$, at least for equal intensity in the two
beams. The classical optical equivalent of the Hilbert-Schmidt coherence then
would be proportional to the square of mutual coherence function between two
slits, summed over all slit pairs, $\sum_{i\neq j}|\gamma_{ij}|^{2}$.

Consider a quanton passing through $n$-path interferometer, which could have a
multi-slit or some kind of beam splitter, as the first element. The state
corresponding to the $i^{th}$ path can be represented by the pure state
$|\psi_{i}\rangle$. In general the density operator of the quanton may be
mixed, and will have the general form

$\rho=\sum_{j=1}^{n}\sum_{k=1}^{n}\rho_{jk}|\psi_{j}\rangle\langle\psi_{k}|.$
(4)

The diagonal elements represent the fractional populations of the various
beams. We allow the possibility that the phase in $i^{th}$ beam may be shifted
by $\theta_{i}$, in order to analyze the experimental scenario of Mei and
Weitz mei . On coming out, the state of the quanton is

$\rho=\sum_{j=1}^{n}\sum_{k=1}^{n}\rho_{jk}|\psi_{j}\rangle\langle\psi_{k}|e^{i(\theta_{j}-\theta_{k})},$
(5)

where $\rho_{jk}$ are assumed to be real without loss of generality, as any
phases can be absorbed in the exponential factor. In the multi-slit
interference experiment, after coming out, the beams spread and overlap and
are registered at different positions on the screen. In another kind of
multipath interference, the beams may be split and recombined, as happens in a
Mach-Zehnder interferometer. In general, we may assume that each beam is split
into new channels, whose states may be represented by $|\xi_{i}\rangle$. The
simplest case would be where all the original beams have equal overlap with a
particular output channel, say $|\xi_{i}\rangle$. This translates to an
assumption that $\langle\xi_{i}|\psi_{k}\rangle=A_{i}$ is same for all $k$.
The probability $I$ of finding the quanton in the $i^{th}$ output channel (not
the $i^{th}$ path) is then given by $I=\langle\xi_{i}|\rho|\xi_{i}\rangle$.
For the present case, this probability is given by

$\displaystyle I$ $\displaystyle=$
$\displaystyle|A_{i}|^{2}\Big{[}\sum_{j=1}^{n}\rho_{jj}+\sum_{j\neq
k}\rho_{jk}e^{i(\theta_{j}-\theta_{k})}\Big{]}$ (6) $\displaystyle=$
$\displaystyle|A_{i}|^{2}\Big{[}1+\sum_{j\neq
k}\rho_{jk}e^{i(\theta_{j}-\theta_{k})}\Big{]}$ $\displaystyle=$
$\displaystyle|A_{i}|^{2}\Big{[}1+\sum_{j\neq
k}|\rho_{jk}|\cos(\theta_{j}-\theta_{k})\Big{]}.$

If one were to simulate Mei and Weitz experiment in this situation, one would
first change the phase of the $\ell$’th path as
$\theta_{\ell}\rightarrow\theta_{\ell}+\pi$. It is to be noted that our
visibility $\mathcal{V}$, given by (3), is unaffected by this phase change,
whereas the fringe contrast does get affected.

If one wants to know which path the quanton followed, one needs to have the
quanton paths entangled with a path-detector. The combined state of the
quanton and the path-detector, after the two have interacted, will be of the
form

$\rho^{\prime}=\sum_{j=1}^{n}\sum_{k=1}^{n}\rho_{jk}|\psi_{j}\rangle\langle\psi_{k}|e^{i(\theta_{j}-\theta_{k})}\otimes|d_{j}\rangle\langle
d_{k}|,$ (7)

where $\\{|d_{i}\rangle\\}$ are the states of the path detector, assumed to be
normalized, but not necessarily mutually orthogonal. The reduced density
operator of the quanton is obtained by tracing over the path-detector states:

$\rho_{r}^{\prime}=\sum_{j=1}^{n}\sum_{k=1}^{n}\rho_{jk}|\psi_{j}\rangle\langle\psi_{k}|e^{i(\theta_{j}-\theta_{k})}\langle
d_{k}|d_{j}\rangle.$ (8)

In order to reproduce the Mei and Weitz experimental situation, we assume that
the path-detector can only tell if the quanton passed through path $\ell$ or
not, it cannot discriminate between any other paths. This is achieved by
having all the path detector states being identical, (say) equal to
$|d_{1}\rangle$, and only $|d_{\ell}\rangle$ being different from
$|d_{1}\rangle$. Now if one were to calculate the visibility, as defined by
(3), for this state of the quanton, it will turn out to be

$\mathcal{V}^{\prime 2}=\frac{n}{n-1}\sum_{i\neq j}|\rho_{ij}|^{2}|\langle
d_{k}|d_{j}\rangle|^{2}.$ (9)

Since $|\langle d_{i}|d_{j}\rangle|=1$ for all $i,j$ except those involving
$\ell$, one can easily see that

$\mathcal{V}^{\prime 2}\neq\frac{n}{n-1}\sum_{i\leq j}|\rho_{ij}|^{2}$ (10)

or

$\mathcal{V}^{\prime 2}\leq\mathcal{V}^{2},$ (11)

which means that the visibility will never increase (equality is achieved for
the trivial case when visibility is zero i.e., for diagonal quanton states) if
one tries to find out if the quanton passed through path $\ell$ or not . Thus
the new visibility works satisfactorily even for Mei and Weitz experiment,
where conventional visibility fails.

The next question we address is, how to measure this visibility in an
interference experiment. The probability given by (6), can also be interpreted
as the intensity at a position on the screen, or in an output channel. In
general it depends on the phases of the paths. Dürr has shown that the second
moment of this interference pattern durr ,

$\langle(\Delta I)^{2}\rangle_{\theta}=\tfrac{1}{n^{2}}\sum_{i\neq
j}|\rho_{ij}|^{2},$ (12)

where $\Delta I=I-\langle I\rangle_{\theta}$, and
$\langle\dots\rangle_{\theta}$ denotes the average over all phases. Thus the
new visibility can be written as

$\mathcal{V}^{2}=\tfrac{n^{3}}{n-1}\langle(\Delta I)^{2}\rangle_{\theta},$
(13)

which means, it can be measured if one can design an experimental arrangement
to vary the phases of all paths independently, and do an averaging over them.
This is one method of experimentally obtaining the new visibility.

Notice that (3) represents a sum over pairs of paths. If one were to modify
the $n$-path interference experiment by blocking all paths except one pair
$i,j$, the _conventional visibility_ of the interference of that pairs of
paths will be given by

${V}_{ij}=\frac{2|\rho_{ij}|}{\rho_{ii}+\rho_{jj}}.$ (14)

Now one can repeat the experiment with another pair of paths, and so on, until
one has gone over all path pairs. If all paths are equally probable,
$\rho_{ii}=\rho_{jj}=\frac{1}{n}$, ${V}_{ij}=n|\rho_{ij}|$. The _mean square_
of the visibilities, of all path pairs, is given by

$\tfrac{2}{n(n-1)}\sum_{\text{pairs}}{V}_{ij}^{2}=\tfrac{n}{n-1}\sum_{i\neq
j}|\rho_{ij}|^{2}=\mathcal{V}^{2}.$ (15)

This is a very interesting result, which shows that the multipath visibility
$\mathcal{V}$ is the _root mean square_ of the fringe contrasts of
interference from all path pairs. So, another way $\mathcal{V}$ can be
measured is by opening only a pair of paths at a time, and measuring
conventional visibility, and then taking root mean square over all path pairs.
Such a procedure is very much feasible, and selectively opening path pairs has
already been experimentally demonstrated padua . If the paths are not equally
probable, each two-path visibility has to be multiplied with
$\frac{n}{2}(\rho_{ii}+\rho_{jj})$ to normalize the weights before taking the
average. It can be shown that
$\tfrac{n}{2(n-1)}\sum_{\text{pairs}}(\rho_{ii}+\rho_{jj})^{2}{V}_{ij}^{2}=\
\mathcal{V}^{2}$.

###  II.2 Predictability measure

A generalized predictability can be defined as,

$\mathcal{P}^{2}=\sum_{i=1}^{n}\rho_{ii}^{2}-\frac{1}{n-1}\sum_{i\neq
j}\rho_{ii}\rho_{jj}=1-\frac{n}{n-1}\sum_{i\neq j}\rho_{ii}\rho_{jj}$ (16)

where $\rho$ is the density matrix of the quanton and $n$ is the dimension of
the density matrix (number of paths). As is evident from the above expression
$\mathcal{P}$ varies from zero to one, therefore normalized. Although defined
in a very different manner, it turns out that (16) is identical to the
predictability defined by Dürr durr . To show that (16) is a good measure of
generalized predictability, we show that it follows all of Dürr’s criteria.
(i) $\mathcal{P}$ is a polynomial function of $\rho_{ii}$, therefore a
continuous function involving only the diagonal elements.
(ii) When the quanton’s path is known for sure, i.e., $\rho_{ii}=1$ and
$\rho_{jj}=0~{}~{}\forall~{}~{}j\neq i$ (only $i^{th}$ slit is open),
predictability reaches its global maximum i.e., $\mathcal{P}=1$.
(iii) When quanton’s probability of going through all paths are equally
likely, i.e., $\rho_{ii}=\frac{1}{n}~{}~{}\forall~{}~{}i$, generalized
predictability is zero, since $\mathcal{P}^{2}=1-\frac{n}{n-1}\sum_{i\neq
j}\frac{1}{n^{2}}=0$.
(iv) A change of probabilities (diagonal elements $\rho_{ii}$) towards
equalization leads to decrease in the generalized predictability. To show
that, consider $\rho_{11}<\rho_{22}$ and a small parameter
$0<\varepsilon<\rho_{22}-\rho_{11}$. A shift in probabilities,
$\tilde{\rho}_{11}=\rho_{11}+\varepsilon$ and
$\tilde{\rho}_{22}=\rho_{22}-\varepsilon$, with other elements unchanged
yields the following,

$\mathcal{P}^{2}-\tilde{\mathcal{P}}^{2}=\frac{2n\varepsilon}{n-1}\left(\rho_{22}-\rho_{11}-\varepsilon\right).$
(17)

Since, $\varepsilon>0$ and $\varepsilon<\rho_{22}-\rho_{11}$, one obtains
$\mathcal{P}>\tilde{\mathcal{P}}$, therefore, a decrease in predictability.

Therefore, generalized predictability as defined in (16) satisfies Dürr’s
criteria. Furthermore, for two beam interferometer case ($n=2$), it reduces
to,

$\mathcal{P}=|\rho_{11}-\rho_{22}|$ (18)

which matches with the predictability proposed by Greenberger and Yasin GY .

Interestingly, this multipath predictability can also be written as root-mean-
square of the two-path predictabilities from all path pairs:

$\mathcal{P}^{2}=\tfrac{1}{n-1}\sum_{\text{pairs}}(\rho_{ii}+\rho_{jj})^{2}{P}_{ij}^{2},$
(19)

where ${P}_{ij}$ is the path predictability for the i’th and j’th path, _a la_
Greenberger and Yasin. The fact that our $n$-path predictability can also be
expressed in terms of the conventional two-slit predictability for path pairs,
has not been recognized earlier, although the predictabilities proposed by
Greenberger and Yasin GY  and Dürr durr  have been around for a long time.

###  II.3 Path distinguishability

A multipath distinguishability has been proposed earlier as nduality

$\mathcal{D}^{2}=1-\left(\frac{1}{n-1}\sum_{i\neq
j}\sqrt{\rho_{ii}\rho_{jj}}|\langle d_{i}|d_{j}\rangle|\right)^{2},$ (20)

which can be related to unambiguous quantum state discrimination. We define
the path-distinguishability $\mathcal{D}$ in a different way:

$\mathcal{D}^{2}=1-\frac{n}{n-1}\sum_{i\neq j}\rho_{ii}\rho_{jj}|\langle
d_{i}|d_{j}\rangle|^{2}.$ (21)

In the spirit of complementarity, our proposed distinguishability measure
satisfies the following basic criteria for being a reliable path quantifier
durr :
(i) When the detector states $\\{{|d_{i}\rangle}\\}$ form an orthonormal
basis, i.e., $\langle d_{i}|d_{j}\rangle=\delta_{ij}$, the path
distinguishability $\mathcal{D}$ is maximum ($\mathcal{D}=1$), since in this
case all the paths can be unambiguously discriminated.
(ii) When all the detector states are parallel, i.e., state of quanton and the
detector is separable, and probability through all the slits are equal, i.e.,
$\rho_{ii}=1/n~{}\forall i$, then the path distinguishability $\mathcal{D}$ is
minimum ($\mathcal{D}=0$).
(iii) Whenever the detector states $|d_{i}\rangle$ and $|d_{j}\rangle$ are
made more orthogonal i.e., with an increment in $|\langle
d_{i}|d_{j}\rangle|$, the distinguishability increases.
(iv) Distinguishability and visibility both can not increase simultaneously
under any operation, which is evident through the fact
$\mathcal{D}^{2}+\mathcal{V}^{2}=1$.

For the case of two paths, (21) reduces to the distinguishability defined by
Englert englert . Next we address the issue of measuring this
distinguishability. Taking cue from the visibility, we note that if one were
to close all paths, except paths $i,j$, the path distinguishability, based on
minimum error state discrimination, is given by

${D}_{ij}=\sqrt{1-\frac{4\rho_{ii}\rho_{jj}}{(\rho_{ii}+\rho_{jj})^{2}}|\langle
d_{i}|d_{j}\rangle|^{2}},$ (22)

which is essentially Englert’s distinguishability, generalized for unequal
path probabilities by using the Helstrom bound helstrom . As done in the case
of visibility, one can repeat the interference experiment by selectively
opening different pairs of slits, while measuring the distinguishability. When
all the path are equally probable, the _mean square_ of the
distinguishabilities, from all pairs of paths, is given by

$\tfrac{2}{n(n-1)}\sum_{\text{pairs}}{D}_{ij}^{2}=1-\frac{n}{n-1}\sum_{i\neq
j}\rho_{ii}\rho_{jj}|\langle d_{i}|d_{j}\rangle|^{2}=\mathcal{D}^{2},$ (23)

which is the same as (21). Thus the distinguishability too can be measured
experimentally, as the _root mean square_ of conventional
distinguishabilities, from all path pairs. When the paths are not equally
probable, the distinguishability can be measured through the conventional two-
slit distinguishability from path pairs $D_{ij}$ and the path probabilities
$\rho_{ii}$ as follows

$\displaystyle\mathcal{D}^{2}$ $\displaystyle=$
$\displaystyle\frac{n}{2(n-1)}\sum_{pairs}(\rho_{ii}+\rho_{jj})^{2}D_{ij}^{2}$
(24) $\displaystyle+1-\frac{n}{2(n-1)}\sum_{pairs}(\rho_{ii}+\rho_{jj})^{2}.$

The last two terms in the above relation may be interpreted as a measure of
the inequality in the probability of various path pairs, by rewriting them as
$1-\frac{2}{n(n-1)}\sum_{pairs}\frac{n^{2}}{4}(\rho_{ii}+\rho_{jj})^{2}$. For
equally probable paths $\rho_{ii}+\rho_{jj}=2/n$ for all $i,j$. One can check
that for equally probable paths, only the first term survives and reduces to
(23). For $n=2$, there is only one path pair, and there is no question of
inequality between different pairs. In that case, the last two terms will be
zero even if the two paths are unequally probable.

##  III Complementarity

###  III.1 Role of entanglement

Consider a quanton passing through a $n$-path interferometer, such that its
quantum state is given by the following pure state,

$|\psi\rangle=\sum_{i=1}^{n}c_{i}|\psi_{i}\rangle,$ (25)

where, $c_{i}\in\mathbf{C}$. The density operator for the quanton can be
written as $\rho=\sum_{i,j}c_{i}c_{j}^{*}|\psi_{i}\rangle\langle\psi_{j}|$.
Using (3) and (16), one obtains a $n$-path duality relation as follows. The
sum of the squares of the predictability and visibility is given by

$\displaystyle\mathcal{P}^{2}+\mathcal{V}^{2}$ $\displaystyle=$ $\displaystyle
1-\frac{n}{n-1}\sum_{i\neq j}\left(\rho_{ii}\rho_{jj}-|\rho_{ij}|^{2}\right).$
(26)

Since for the pure state given by (25) $\rho_{ij}=c_{i}c_{j}^{*}$ resulting in
$\rho_{ii}\rho_{jj}=|\rho_{ij}|^{2}$, the duality relation saturates for the
pure state case. When a quanton is given by an ensemble
$\\{(p^{\alpha},~{}|\psi^{\alpha}\rangle)\\}$, where $p^{\alpha}\geq 0$ with
$\sum p^{\alpha}=1$, and
$|\psi^{\alpha}\rangle=\sum_{i}c^{\alpha}_{i}|\psi_{i}\rangle$. The resulting
density matrix is mixed with the elements given by
$\rho_{ij}=\sum_{\alpha}p^{\alpha}c_{i}^{\alpha}c_{j}^{\alpha*}$ :

$\rho_{ii}\rho_{jj}-|\rho_{ij}|^{2}=\sum_{\alpha}\left\lvert
c_{i}^{\alpha}\right\rvert^{2}\sum_{\beta}\lvert
c_{j}^{\beta}\rvert^{2}-\absolutevalue{\sum_{\alpha}c_{i}^{\alpha}c_{j}^{\beta*}}^{2}\geq
0.$ (27)

Here, the last inequality follows from the Cauchy-Schwartz inequality.
Therefore, in general we have the following $n$-path duality relation,

$\mathcal{P}^{2}+\mathcal{V}^{2}\leq 1,$ (28)

which saturates for pure $\rho$. This result was already derived by Dürr durr
, but now sets the stage for investigating the case where there is a path-
detector in place, which can perform a which-path or which-way detection.

Consider an interferometer with an added path-detecting device, such that the
quanton going through the $n$ paths is described by the state

$|\Psi\rangle=\sum_{i=1}^{n}c_{i}|\psi_{i}\rangle|d_{i}\rangle,$ (29)

where, $|\psi_{i}\rangle$ is the state of the quanton corresponding to its
passing through the $i^{th}$ path, and $|d_{i}\rangle$ is the state of the
path-detector, corresponding to that possibility. The states
$\\{|d_{i}\rangle\\}$ are assumed to be normalized, but not necessarily
orthogonal.

The reduced density operator of the quanton, after tracing over the path-
detector states is,

$\rho_{r}=\sum_{i,j}c_{i}c_{j}^{*}\langle
d_{j}|d_{i}\rangle~{}|\psi_{i}\rangle\langle\psi_{j}|.$ (30)

In terms of the density operator for the quanton in the absence of the path-
detector, one can write (30) as,

$\rho_{r}=\sum_{i,j}\rho_{ij}\langle
d_{j}|d_{i}\rangle~{}|\psi_{i}\rangle\langle\psi_{j}|$ (31)

For this reduced density operator, one can write (26) as,

$\displaystyle\mathcal{P}^{2}+\mathcal{V}^{2}=$ $\displaystyle
1-\frac{n}{n-1}\sum_{i\neq j}\rho_{ii}\rho_{jj}+\frac{n}{n-1}\sum_{i\neq
j}|\rho_{ij}|^{2}|\langle d_{j}|d_{i}\rangle|^{2}$ (32) $\displaystyle=$
$\displaystyle 1-\frac{n}{n-1}\sum_{i\neq
j}\rho_{r_{ii}}\rho_{r_{jj}}+\frac{n}{n-1}\sum_{i\neq j}|\rho_{r_{ij}}|^{2}$

One can define a normalized entanglement measure $\mathcal{E}$ such that,

$\mathcal{E}^{2}=\frac{n}{n-1}\sum_{i\neq
j}\left(\rho_{r_{ii}}\rho_{r_{jj}}-|\rho_{r_{ij}}|^{2}\right)=\frac{n}{2(n-1)}E^{2}$
(33)

where $E$ is the generalized concurrence gconc , and coincides with
I-concurrence rungta , when the bipartite state under consideration is a pure
state. Eqn. (32) can then be written as,

$\mathcal{P}^{2}+\mathcal{V}^{2}=1-\mathcal{E}^{2},$ (34)

or as a triality relation,

$\mathcal{P}^{2}+\mathcal{V}^{2}+\mathcal{E}^{2}=1.$ (35)

Therefore, the generalized predictability, Hilbert-Schmidt coherence, and
I-concurrence obey a tight triality relation. It is interesting to note that
the path-distinguishability measure, as defined in the previous section, is
related to the generalized predictability through I-concurrence as follows,

$\mathcal{D}^{2}=\mathcal{P}^{2}+\mathcal{E}^{2}.$ (36)

This above relation shows that the two different kinds of path knowledge,
predictability and distinguishability, are quantitatively connected through a
measure of entanglement, the I-concurrence. It very naturally generalizes
Englert’s duality relation (2) to multipath interference:

$\mathcal{D}^{2}+\mathcal{V}^{2}=1.$ (37)

Needless to say, Englert’s duality relation is recovered for $n=2$. Since the
above relation is an equality, it is obvious that this multipath
distinguishability does not suffer from the shortcoming of Dürr’s multipath
distinguishability, pointed out by Bimonte and Musto bimonte2 .

We have shown that a proper entanglement measure, the I-concurrence, is an
integral part of complementarity in multipath interference. For $n=2$, (35)
reduces to

${P}^{2}+{V}^{2}+\mathcal{E}_{2}^{2}=1,$ (38)

where is ${P}$ is the predictability of Greenberger and Yasin GY , ${V}$ is
the fringe contrast, and $\mathcal{E}_{2}$ is the concurrence concurrence .
This two-path result is the same as that derived earlier by Jakob and Bergou
triality1 , and it is satisfying to see that our general multipath relation
reduces to it in the appropriate limit. In addition, for $n=2$ the relation
(36) reduces to

${D}^{2}={P}^{2}+\mathcal{E}_{2}^{2},$ (39)

where ${D}^{2}$ is Englert’s distinguishability englert , ${P}$ is the
predictability GY , and $\mathcal{E}_{2}$ is the concurrence, a known result
due to Jakob and Bergou triality1 .

Since (35) is an equality for pure quanton path-detector states, it opens up
the possibility of measuring I-concurrence. In an unbalanced $n$-path
interferometer, if the quanton gets entangled with an ancilla system, its
interference visibility will be affected. Since the path predictability is
known from the experimental setting, and can also be measured, and the
interference visibility can be measured, as discussed earlier, one can
actually infer the degree of entanglelment of the quanton with the ancilla,
without probing the ancilla in any way.

The last interesting point is that our normalized entanglement measure
$\mathcal{E}$ can be expressed as the root-mean-square of _concurrences_
concurrence  corresponding to all pairs of paths

$\mathcal{E}^{2}=\tfrac{2}{n(n-1)}\sum_{\text{pairs}}{E}_{ij}^{2},$ (40)

where ${E}_{ij}$ is the concurrence of entanglement between the quanton and
the path-detector when all except path $i,j$ are blocked. This is true if all
the paths are equally probable. For unequally probable paths, the above
expression modifies to
$\mathcal{E}^{2}=\tfrac{n}{2(n-1)}\sum_{\text{pairs}}(\rho_{ii}+\rho_{jj})^{2}{E}_{ij}^{2}$.
This is an interesting connection between concurrence and I-concurrence which
becomes meaningful in the kind of experiment we have been discussing, where
pairs of paths are opened at a time.

###  III.2 How quantum is the connection?

In the light of the results obtained here, we look back at an earlier work
where a triality relation has been derived for two-slit interference, in the
classical optical domain triality2 . The triality relation obtained in that
work is precisely of the form (35). It is then natural to wonder how quantum
is the relation obtained here. In classical optics, there are light fields
arriving on the screen from the two slits, which may differ in intensity and
may have a varying degree of mutual coherence. Since the two fields differ in
intensity, one can talk about a measure of particle nature which is referred
to as predictability. The word distinguishability has often been used to refer
to predictability in the classical optics literature, and has caused some
confusion. However, there is no sense in which one can talk of
distinguishability in classical optics. Distinguishability pertains to the
question, which slit did the particle go through? For classical fields, it is
not meaningful to ask such a question. Then if in the presence of
polarization, or some other mode, the duality relation of the form (1) becomes
an inequality, it is natural to think that the duality relation is
_incomplete_ , and can be completed by including the entanglement with
polarization or some other mode. That has been the focus of the work in Ref.
triality2 .

On the other hand, in the quantum regime, one can talk of path
distinguishability in the context of experimentally determining through which
slit did the quanton pass. This distinguishability is a quantum feature, and
is intimately tied to entanglement. Thus the duality relation of the form (2),
involving distinguishability is always an equality, for a pure state. So the
triality relation derived here is purely quantum in this sense. It not only
quantifies the connection between predictability and interference through the
entanglement which may have been generated because of the quanton interacting
with an external quantum system, before it arrives on the screen, it also
quantitatively connects distinguishability with predictability.

Interestingly, when one moves away from single particle build up of
interference, and is in the regime where the various paths may have many
particles, the distinction between true entanglement and mere non-separability
becomes blurred. The triality relation then continues to describe the
inteference and predictability faithfully, but via classical entanglement
(non-separability). In fact, for $n=2$, the triality relation obtained here is
exactly the same as the one obtained in the classical optical regime triality2
, provided one identifies the absolute value of the off-diagonal density
matrix element with the mutual coherence function in classical optics
coherence .

##  IV Perspective

As mentioned before, a triality relation has been obtained earlier in the
context of multi-slit quantum interference triality3 . Let us look at the
results obtained here in that perspective. The relation obtained earlier
involved the coherence measure introduced in the context of quantum
information theory plenio . That coherence has been shown to be a proper
measure of coherence, in the sense that it does not increase under any
incoherent operation. On the other hand, the Hilbert-Schmidt coherence used by
us, although very commonly used, has been shown to not be a robust coherence
monotone, as it may increase under some incoherent operations. Nevertheless,
it has been recently highlighted that in addition to having a simpler
structure, the Hilbert-Schmidt coherence has information theoretic
significance, in particular, through its manifestation as quantum uncertainty
Luo . Furthermore, the shortcoming of monotonicity is absent in the genuine
quantum coherence framework Vicente . Therefore, Hilbert-Schmidt coherence may
still be good enough to quantify visibility in multipath interference. From a
practical consideration too, the coherence measure of Baumgratz et al. has a
simpler connection to multislit interference coherence , which has also been
experimentally demonstrated chen ; padua .

The predictability used in both the works has been shown to be a proper
measure of path predictability. Both can be easily measured in a multipath
interference experiment, as it only involves measuring the intensities at each
slit, or in each path.

The measure of entanglement is an aspect in which the earlier work falls
short. The quantifier used there has not yet been shown to be a proper
entanglement measure. On the other hand, the quantifier used for entanglement
in the present work is I-concurrence, a well-known entanglement monotone. This
advantage allows the possibility of experimentally inferring the degree of
entanglement between the quanton and an ancilla system.

Thus the two formulations connecting predictability, entanglement, and
interference, are equivalent, but both have their strengths and weaknesses.
For two-path interference, they become identical.

##  V conclusion

In conclusion, we have introduced an interference visibility based on the
Hilbert-Schmidt coherence, and shown that it constitutes a good measure of
wave nature, and does not suffer from the shortcomings of the conventional
fringe contrast, which emerges in certain experiments mei . Interestingly,
this visibility can be obtained as a root mean square of the fringe contrasts
from all _pairs of paths_ , if the experiment is done by opening only a pair
of paths at a time. Selectively opening pairs of slit in a multi-slit
interference experiment has already been realized padua .

A multipath predictability has been introduced, which is a natural
generalization of the two-path predictability of Greenberger and Yasin GY .
Interestingly, this multipath predictability can also be measured using the
two-path predictabilities of pairs of paths. A multipath distinguishability
has also been introduced which, again, can be expressed in terms of root mean
square of the two-path distinguishability (introduced by Englert) of all pairs
of paths. The predictability and visibility introduced here, together with the
I-concurrence (related to entanglement between the quanton and a path-
detector), are shown to follow a tight triality relation. This brings out the
essential role of entanglement in the phenomenon of complementarity, in the
general context of multipath interference. As an interesting offshoot, the
normalized I-concurrence turns out to be the root mean square of concurrences
relating to all path pairs.

###### Acknowledgements.

AKR, NKC and PKP gratefully acknowledge the financial support from DST, India
through Grant No. DST/ICPS/QuST/Theme1/2019/2020-21/01. NP acknowledges
financial support from DST, India through the Inspire Fellowship (registration
number IF180414).

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