# Spacetime Path Integrals for Entangled States

Narayani Tyagi, Ken Wharton
Department of Physics and Astronomy
San José State University, San José, CA 95192-0106

###### Abstract

Although the path-integral formalism is known to be equivalent to conventional
quantum mechanics, it is not generally obvious how to implement path-based
calculations for multi-qubit entangled states. Whether one takes the formal
view of entangled states as entities in a high-dimensional Hilbert space, or
the intuitive view of these states as a connection between distant spatial
configurations, it may not even be obvious that a path-based calculation can
be achieved using only paths in ordinary space and time. Previous work has
shown how to do this for certain special states; this paper extends those
results to all pure two-qubit states, where each qubit can be measured in an
arbitrary basis. Certain three-qubit states are also developed, and path
integrals again reproduce the usual correlations. These results should allow
for a substantial amount of conventional quantum analysis to be translated
over into a path-integral perspective, simplifying certain calculations, and
more generally informing research in quantum foundations.

##  1 Introduction

Entangled $N$-particle quantum states live in a high-dimensional Hilbert space
or configuration space, with no evident representation on ordinary space or
spacetime. Even when limiting analysis to which-way entanglement – a case
seemingly aligned with a spatial representation – quantum theory formally uses
a superposition of spatial configurations, with the total wavefunction
therefore defined as a function on a $3N$-dimensional configuration space
$\psi(\bm{x}_{1},\bm{x}_{2},\dots,\bm{x}_{N})$ rather than a set of $N$
single-particle functions on ordinary 3D space, $\psi_{n}(\bm{x})$. In fact,
the hallmark of entangled states is that they cannot be represented as
functions on ordinary spacetime. This disconnect between wavefunctions and
conventional spacetime not only makes it difficult to visualize quantum
states, but is also arguably a fundamental source of conflict between quantum
theory and general relativity. [1]

There is a mistaken but commonly-held view that the disconnect between quantum
theory and spacetime also applies to the Feynman path integral [2]. In this
view, analyzing entangled systems via the path integral requires one to
consider all possible “paths” through Hilbert space, including “paths” that
consist of pure entangled states. Because a single entangled state has no
spatial representation, any of these “paths” would also have no possible
spacetime representation. Approaches using such Hilbert space “paths” are
common in the literature [3, 4], and have even been extended to quantum field
theory [5].

The central result of this paper is to demonstrate how the path integral can
be applied to general entangled states without requiring a departure from a
conventional spatial representation. In other words, each relevant step in the
calculation can correspond to physical paths through ordinary spacetime, with
no need for scare-quotes around “path”. The success of this procedure is most
evident for the case of which-way entanglement. One need simply use the basis
of the spatial configuration space
$\psi(\bm{x}_{1},\bm{x}_{2},\dots,\bm{x}_{N})$, and perform the path integral
in this basis. Any point on one of these paths is simply a configuration of
spatial particle positions, representing each particle at a position in
ordinary 3D space (the same way configuration space is used in classical
statistical mechanics). Any single path through points in 3N-dimensional
configuration space, therefore, corresponds to N paths through conventional
spacetime. We will follow standard terminology in calling this collection of
$N$ paths a single “history” [6]. This preference for the position basis was
advocated by Feynman’s original paper [2], and has been utilized in other
attempts to have a realistic interpretation of the path integral [7, 8, 9].

A few explicit examples of spacetime-based path integrals for entangled states
have already appeared in the literature [6, 10], but these only cover very
special states (maximally entangled 2-qubit Bell states). No paper has yet
addressed how to perform this analysis for partially-entangled states or
3-qubit GHZ states; these cases are developed in detail below. As in the
previous work, we indicate how to simplify this analysis by only needing to
consider a few classical paths, rather than also summing over all non-
classical paths. (In many known cases, all non-classical paths end up either
cancelling exactly or simply reinforcing classical paths [11]; fortunately the
same simplification works correctly in the cases considered here.)

The techniques developed below are evidently generalizable to more complicated
entangled states. Quantum information theory has a substantial number of tools
which allow one to convert any conceivable entangled state into a physical
experimental geometry – translating any sort of entanglement into which-way
entanglement – for which this general approach could be utilized.111There are
reasonable arguments as to why the approach here could also be extended to
other types of entanglement, but for the purposes of this paper, we shall
focus on path integral accounts of which-way entanglement.

There are a number of possible motivations for taking an abstract high-
dimensional entangled state and analyzing it via paths in conventional 4D
spacetime. For many physicists, the main advantage might simply be a matter of
ease of visualization. A generic three-qubit state lives in an 8-dimensional
complex space, or, if normalized, as a point on a real 15-dimensional sphere.
Analyzing such states via paths through ordinary space is evidently a
preferable representation.222Granted, many physicists commonly think of
entangled states as “superpositions” of configurations in ordinary space,
which is arguably comparable to a set of possible paths. But apart from the
disconnect between this viewpoint and the formal mathematics, this model of
entangled states further requires the visualization of a mysterious direct
connection between spacelike-separated events, a connection not required in
the path integral.

Another set of motivations will be evident to those in the field of quantum
foundations. Many key theorems and discussions involve entangled states, but
it is usually not clear how to translate those results into a path-integral
framework, missing out on a potentially useful additional perspective.
Evidently, such a translation would also be critical for research programs
which aim to develop a realistic account of path integrals [7, 8, 9]. The mere
fact that the path integral mathematics is framed on a spacetime background
makes it more suitable for any analysis of entangled states in curved
spacetime. And since the path integral solves systems “all at once” rather
than via dynamical evolution, it ties directly into the main remaining
approach for finding spacetime-based accounts of quantum phenomena [12, 13].
Any such analysis would necessarily require some way to translate between
entangled states and physical paths through spacetime. The results in this
paper indicate how such a translation might be accomplished.

The next section focuses on a which-way representation of a single qubit and
develops a single qubit measurement device. This device is a key part of the
experimental geometries used in rest of the paper. Section 3 delves into the
observable correlations of the entangled two-qubit state. A generic two qubit
state is mapped to an experimental setup that is amenable to a path-integral
analysis. The basic rules for the path integral calculations are given in
Section 3.3, which are then applied to this experiment. The results exactly
compare to the usual quantum probabilities. Then, in Section 4, we develop the
specific case of the three-qubit GHZ state [14]. Again, the path integral
calculation is successful. Finally, these results are discussed in Section 5,
including future applications.

##  2 Single Qubit Measurements

While there are many experimental realizations of any given entangled state,
the approach in this paper is to utilize photons rather than massive
particles, interacting exclusively with beamsplitters, phase-shifting devices
(phase plates), and single-photon detectors. (The sources of these photons
will be developed in section 3.)

In this section we focus on a which-way representation of a single qubit, and
demonstrate how to build a device which can serve as a generic single qubit
measurement. Specifically, we would like a device that could be adjusted to
measure a qubit on any chosen basis (aligned in any chosen direction on the
Bloch sphere). As we shall see, once the measurement basis is imposed via
beamsplitters and phase plates, paths through this measurement device will
enter into the path integral calculations in the next section.

###  2.1 Quantum Probabilities

The most straightforward path-based implementation of a generic qubit
$\ket{\psi}=\bm{a}\ket{0}+\bm{b}\ket{1}$ is to have $\ket{0}$ correspond to
one path and $\ket{1}$ correspond to another. The goal is to design a
measurement device that can effectively project this qubit onto an arbitrary
basis using beamsplitters and phase plates. Using the usual spherical
coordinates, this chosen basis is defined on the Bloch sphere via adjustable
variables $\phi$ and $\theta$. Specifically, this detector should project the
qubit into the chosen basis $\ket{\large{\bm{e_{+}}}}$ and
$\ket{\large{\bm{e_{-}}}}$, where

$\ket{\large{\bm{e_{+}}}}=\begin{pmatrix}cos\frac{\theta}{2}\\\
sin\frac{\theta}{2}\>e^{i\phi}\end{pmatrix},$ (1)

$\ket{\large{\bm{e_{-}}}}=\begin{pmatrix}sin\frac{\theta}{2}\>e^{-i\phi}\\\
-\>cos\frac{\theta}{2}\end{pmatrix}.$ (2)

This is written in the usual spinor notation, where $\ket{0}$ is the top term
and $\ket{1}$ is the bottom. From the Born rule, the probability for a
$\bm{+}$ outcome (corresponding to $\ket{\large{\bm{e_{+}}}}$) should be

$\displaystyle\left|\braket{\large{\bm{e_{+}}}}{\large\psi}\right|^{2}=\left|\>\bm{a}\>cos\frac{\theta}{2}+\bm{b}\>sin\frac{\theta}{2}\>e^{-i\phi}\>\right|^{2}.$
(3)

Similarly, the probability for the $\bm{-}$ outcome should be

$\displaystyle\left|\braket{\large{\bm{e_{-}}}}{\large\psi}\right|^{2}=\left|\>\bm{a}\>sin\frac{\theta}{2}\>e^{i\phi}\>-\>\bm{b}\>cos\frac{\theta}{2}\>\right|^{2}.$
(4)

###  2.2 Measurement Device

To design a device which will yield the above results, it is useful to apply
the correspondence principle that relates classical electromagnetic (EM)
intensities with photon detection probabilities. With this in mind, consider
replacing the single photon with two classical EM plane waves, one on path
$\ket{0}$ and one on path $\ket{1}$. The electric field of each of these EM
waves has an amplitude and a phase, and they can be expressed as complex field
amplitudes $E_{a}$ and $E_{b}$. Normalizing the total intensity
$|{E_{a}}|^{2}+|{E_{b}}|^{2}=1$ allows one to relate these waves to the
original qubit $\ket{\psi}=\bm{a}\ket{0}+\bm{b}\ket{1}$. In other words, if
one wave is assigned a complex amplitude of $E_{a}=\bm{a}$, and the other one
$E_{b}=\bm{b}$, they encode the same information as the original qubit. (These
classical EM waves are assumed to have the same frequency and polarization, to
allow for perfect interference.)

Now consider the measurement device shown in Figure 1 with two inputs
corresponding to paths $\ket{0}$ and path $\ket{1}$. As explained above, these
inputs can correspond to classical EM waves with complex field amplitudes
$\bm{a}$ and $\bm{b}$. We can now calculate the intensities at the two
detectors ( $\bm{+}$ and $\bm{-}$), corresponding to the two possible outcomes
in the chosen measurement basis). If we design a device where the intensities
properly correspond to the quantum probabilities, it follows that this device
is guaranteed to perform the desired measurement for the corresponding single
photon.

Figure 1: A generic measurement device for a qubit, where the superposition of
the two input paths corresponds to a single photon. An adjustable beamsplitter
and phase plates project the qubit onto an arbitrary direction on the Bloch
Sphere.

The desired measurement device consists of an adjustable phase plate adding a
phase angle $\phi$ to the input $\ket{0}$, and a fixed phase plate adding a
phase angle $\pi/2$ to the input $\ket{1}$. The latter phase plate is
indicated by “$i$”, since this is the net complex phase factor
$exp(i{\pi}/{2})=i$. After the phase shifts, these two waves encounter a
lossless beamsplitter, with a reflectivity factor corresponding to the
adjustable variable $\theta$. The reflected intensity factor is $R$, but the
reflected field amplitude is scaled by $\sqrt{R}$, where

$\displaystyle\sqrt{R}=\cos\frac{\theta}{2},$ (5)
$\displaystyle\sqrt{T}=\sin\frac{\theta}{2}.$ (6)

such that $R+T=1$.

First we will calculate the total intensity at the $\bm{+}$ detector. As the
input wave along path $\ket{0}$ passes through the adjustable phase plate
$\bm{\phi}$, the amplitude $\bm{a}$ is multiplied by the phase factor
$exp(i\phi)$ . At the beamsplitter, this wave gets both transmitted and
reflected. However, we’re interested in the reflected part that reaches the
$\bm{+}$ detector. At the beamsplitter, the reflected component picks up a
$\pi/2$ phase shift relative to the transmission [15, 16], corresponding to a
net factor $i\sqrt{R}$ due to the reflection. Thus this component of the wave
arriving at the $\bm{+}$ detector has an amplitude of
$i\bm{a}\sqrt{R}exp(i\phi)$.

For the input wave travelling along $\ket{1}$, the amplitude $\bm{b}$ is
multiplied by a phase factor $i$ due to the phase plate, and a transmission
factor $\sqrt{T}$ from the beamsplitter (to reach the $\bm{+}$ detector)
yielding an amplitude of $i\bm{b}\sqrt{T}$ at the detector.

Thus, the total intensity at the $\bm{+}$ detector is the square of the sum of
field amplitudes travelling along the $\ket{0}$ and $\ket{1}$ paths and can be
given as:

$\displaystyle
I_{+}=\left|i\bm{a}\>cos\frac{\theta}{2}\>e^{i\phi}+i\bm{b}\>sin\frac{\theta}{2}\right|^{2}.$
(7)

Recall that the sum of intensities has been normalized to be equal to 1. With
the normalized intensity, this result is equivalent to Eqn. (3).

A similar calculation can be done to find the intensity at the $\bm{-}$
detector. The input along path $\ket{0}$ passes through the phase plate $\phi$
and reaches the beamsplitter. In this case we are interested only in the
transmitted wave that reaches the $\bm{-}$ detector, for another factor of
$\sqrt{T}$. The input wave along path $\ket{1}$ passes through the phase plate
$i$, but only the reflected wave at the beamsplitter reaches the $\bm{-}$
detector, for a factor of $i\sqrt{R}$. Together, the intensity at the $\bm{-}$
detector is therefore:

$\displaystyle
I_{-}=\left|\bm{a}\>sin\frac{\theta}{2}\>e^{i\phi}\>+\>i^{2}\>\bm{b}\>cos\frac{\theta}{2}\right|^{2}.$
(8)

Once again, as the sum of intensities has been normalized to 1, this result is
equivalent to Eqn. (4).

From the above results, we can see that the device pictured in Figure 1 is
equivalent to a measurement of a qubit
$\ket{\psi}=\bm{a}\ket{0}+\bm{b}\ket{1}$, where the amplitude of the upper-
left input in Figure 1 corresponds to the $\ket{0}$ state and the amplitude of
the lower-left input corresponds to $\ket{1}$. In the single photon limit,
this is equivalent to measuring the qubit in a basis chosen by the angles
$(\theta,\phi)$.

##  3 The 2-Qubit Path Integral

###  3.1 Quantum Mechanical probabilities

The main result of this paper is to show that the observable correlations of
any entangled two-qubit state can be computed from a path-integral
perspective, using paths in physical spacetime. Thanks to the Schmidt
decomposition [17, 18, 19] – see Appendix A for explicit details – a generic
two-qubit (pure) state can always be written in the simple form

$\ket{\psi}=A\ket{0}\otimes\ket{1}+Be^{i\delta}\ket{1}\otimes\ket{0}=\begin{pmatrix}0\\\
A\\\ B\>e^{i\delta}\\\ 0\end{pmatrix}.$ (9)

Here the constants $A$ and $B$ are assumed to be real and positive.

The basis for each qubit in the above expression is fixed to the Schmidt basis
(defined in Appendix A), but the below results are still generally applicable
to any two-qubit state written in any basis, from which the Schmidt basis can
be calculated. The general case would merely require particular rotations of
the measurement coordinate systems, as precisely specified in Appendix A. The
main paper will now proceed assuming the translation of a generic state into
the form in Eqn. (9) has already been accomplished, such that the measurement
basis for each qubit is defined relative to the Schmidt basis used in (9).

Using similar notation to the previous section, the two independent
measurement bases are $\ket{\large{\bm{e_{+}}}}_{1}$ and
$\ket{\large{\bm{e_{-}}}}_{1}$ for the first qubit, and
$\ket{\large{\bm{e_{+}}}}_{2}$ and $\ket{\large{\bm{e_{-}}}}_{2}$ for the
second qubit. On the Bloch sphere, $\ket{\large{\bm{e_{+}}}}_{1}$ corresponds
to the freely chosen angles ($\theta_{1},\phi_{1}$), and
$\ket{\large{\bm{e_{+}}}}_{2}$ corresponds to ($\theta_{2},\phi_{2}$). The
tensor product of these single-qubit bases leads to the following definitions,
corresponding to the four possible measurement outcomes:

$\displaystyle\ket{\bm{++}}=\ket{\large{\bm{e_{+}}}}_{1}\otimes\ket{\large{\bm{e_{+}}}}_{2}=\begin{pmatrix}cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\\\
cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{i\phi_{2}}\\\
sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>e^{i\phi_{1}}\\\
sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{i(\phi_{1}+\phi_{2})}\end{pmatrix},$
(10)

$\displaystyle\ket{\bm{+\>-}}=\ket{\large{\bm{e_{+}}}}_{1}\otimes{\ket{\large{\bm{e_{-}}}}}_{2}=\begin{pmatrix}cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{-i\phi_{2}}\\\
-cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\\\
sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{i(\phi_{1}-\phi_{2})}\\\
-sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>e^{i\phi_{1}}\end{pmatrix},$
(11)

$\displaystyle\ket{\bm{-\>+}}=\ket{\large{\bm{e_{-}}}}_{1}\otimes{\ket{\large{\bm{e_{+}}}}}_{2}=\begin{pmatrix}sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>e^{-i\phi_{1}}\\\
sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{i(\phi_{2}-\phi_{1})}\\\
-cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\\\
-cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{i\phi_{2}}\end{pmatrix},$
(12)

$\displaystyle\ket{\bm{-\>-}}=\ket{\large{\bm{e_{-}}}}_{1}\otimes{\ket{\large{\bm{e_{-}}}}}_{2}=\begin{pmatrix}sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{-i(\phi_{2}+\phi_{1})}\\\
-sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>e^{i\phi_{1}}\\\
-cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{-i\phi_{2}}\\\
cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\end{pmatrix}.$ (13)

Suppose we have an entangled quantum state of the form $\ket{\psi}$ as shown
in Eqn. (9). Then, for the arbitrary measurement basis defined above, we can
find the corresponding QM probabilities for the 4 outcomes as follows:

$\displaystyle\left|\braket{{+\>+}}{\psi}\right|^{2}$ $\displaystyle=$
$\displaystyle\left|A\>cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{-i\phi_{2}}+B\>e^{i\delta}sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>e^{-i\phi_{1}}\right|^{2},$
(14) $\displaystyle\left|\braket{{+\>-}}{\psi}\right|^{2}$ $\displaystyle=$
$\displaystyle\left|-A\>cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}+B\>e^{i\delta}sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{-i(\phi_{1}-\phi_{2})}\right|^{2},$
(15) $\displaystyle\left|\braket{{-\>+}}{\psi}\right|^{2}$ $\displaystyle=$
$\displaystyle\left|A\>sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{-i(\phi_{2}-\phi_{1})}-B\>e^{i\delta}cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\right|^{2},$
(16) $\displaystyle\left|\braket{{-\>-}}{\psi}\right|^{2}$ $\displaystyle=$
$\displaystyle\left|-A\>sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>e^{-i\phi_{1}}-B\>e^{i\delta}cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{i\phi_{2}}\right|^{2}.$
(17)

The goal of this section is to show that these results can be easily
calculated in a spacetime-path integral framework.

Figure 2:  The setup consists of two generic measurement devices(refer Figure
1), one for each qubit. Source E emits two photons travelling in opposite
directions along solid or dashed paths. The first photon reaches the top set
of detectors while the second photon reaches the bottom set of detectors. An
adjustable beamsplitter and phase plates are associated with each measurement
device, corresponding to a measurement basis that can be chosen independently
for each qubit.

###  3.2 Two-Photon Experimental Geometry

From the results in section 2, we know that a which-way implementation of a
generic two-qubit measurement will require a two-photon source, with each
photon encountering one of the devices pictured in Figure 1. Such an
experiment is indicated in Figure 2. Here we have a two-photon source $\bm{E}$
that emits two photons in opposite directions.333While there are various
physical mechanisms that emit photons in opposite directions, further details
of this source are not important for any of the calculations below, except we
will need certain directions to be more probable than others. In the
traditional quantum viewpoint, this implies a superposition of photon
emissions on the solid and dashed lines of Figure 2, but such a viewpoint is
not needed for the path integral approach. Here, we take the photons to be
constrained to follow either the two solid lines or the two dashed lines.
Another difference is that no collapse is needed; in any given history, each
photon will either transmit or reflect from a beamsplitter, and will end up at
only one particular detector.

In order to have an analog to the most general two-qubit state where $A\neq
B$, we require a source that preferentially emits in certain directions.
Imagine that the source was surrounded by a screen. In this case, we require
that the probability of emissions along the solid lines is $A^{2}$, and the
probability along the dashed lines is $B^{2}$. We ignore all other emission
directions444If the photons are emitted in any other direction, they will not
be detected, and will not enter into to the measurement probabilities. such
that $A^{2}+B^{2}=1$. Notice that one of the photons emitted along the dashed
lines would pass through phase plate with phase angle $\delta$. We can imagine
that the set-up of the source $\bm{E}$ and the angle $\delta$ are under the
control of an experimenter with the goal of providing two photons to two
subsequent measurement devices.

The two detectors in the top device measure the $1^{st}$ qubit (or Particle 1)
while the two detectors in the bottom device measure the $2^{nd}$ qubit (or
Particle 2). For each device $j$, the controllable parameters $\theta_{j}$ and
$\phi_{j}$ can be adjusted via the beamsplitters and phase plates exactly as
defined in Section 2.

It should be noted that the optical path-lengths for each arm of both the
devices are exactly equal to each other, i.e. the photons always travel an
equal distance to each beamsplitter, no matter which way it is emitted by the
source $\bm{E}$. This ensures that no phase difference is introduced due to
the path length in each interferometer. Any phase factors being introduced are
represented by phase plates.

It may not be evident that the source pictured Figure 2 is equivalent to the
entangled state $\ket{\psi}$. The rest of this section will show how the
correct QM probabilities can be recovered from a path-integral analysis,
proving this equivalence.

###  3.3 Computing Probabilities with the Path Integral

The rules for combining path integrals (see, e.g., [6]) are very similar to
how one computes classical field amplitudes. When multiple classical fields
arrive at the same point, one adds the complex amplitudes and squares the
total to find the classical intensity. Similarly, when different histories end
at the same outcomes, one adds the complex amplitude corresponding to each
history and squares the sum to find the probability of those outcomes. (Recall
from the Introduction that a “history” corresponds to a single path through
configuration space, which in this two-photon case is equivalent to the
spacetime paths of the two photons, taken together as a pair.)

Feynman discovered that the amplitude for a history is proportional to
$exp(iS/\hbar)$ where $S$ is the classical action [2]. If that history is
comprised of multiple paths, as in this two-photon case, the total action is
$S_{1}+S_{2}$, the sum of the action of the two paths. Therefore, the total
amplitude of a history is $exp(iS_{1}/\hbar)exp(iS_{2}/\hbar)$, indicating
that one multiplies the amplitude factors from each path in the same history.
This is not to be confused with the rule in the previous paragraph, concerning
the case where multiple different histories end at the same outcome; in that
case one adds the amplitudes instead of multiplying. These rules will now be
precisely defined in the present context.

To calculate the amplitude for any history in the particular experimental
geometry shown in Figure 2, one undertakes the following procedure:

1\. Source Amplitude: Recall that the photons will be emitted on either the
two solid lines, or the two dashed lines. The amplitude of any history
beginning on the solid lines is $A$, and the amplitude of any history
beginning on the dashed lines is $B$. Note that this factor is not squared,
because the two photons are not independent. (If these photons were measured
by a screen, one would square these amplitudes to determine the joint path
probability $A^{2}$ or $B^{2}$.)

2\. Beamsplitter Factor: At a beamsplitter, $R$ and $T$ represent the single-
photon reflection and transmission probability with the relationship $R+T=1$.
When a path is reflected at a beamsplitter, the amplitude picks up factor of
$i\sqrt{R}$, in precise analogy with the classical field reflection discussed
in Section 2. Similarly, for a transmission we simply pick up the factor of
$\sqrt{T}$. (Note that in our case each history includes paths through both
measurement devices, so there are two beamsplitters which will enter into each
amplitude calculation.)

3\. Phase Plate Factor: Whenever a path in a history passes through a phase
plate of angle $\beta$, the amplitude picks up a factor of $e^{i\beta}$. (This
occurs at each circle in Figure 2.)

Once the amplitudes $\mathcal{E}$ for each history have been determined, we
proceed as described above. One can bin them into groups having the same final
outcomes $k$. With histories in each outcome-bin labelled by some index $n$,
every history has an amplitude of the form $\mathcal{E}^{n}_{k}$. We would
like to compute the probability of any particular outcome $k$. To find this
probability, one sums the amplitudes $\mathcal{E}^{n}_{k}$ which lead to
outcome $k$ and then squares the total;

$Prob(k)=\left|\sum_{n}\mathcal{E}^{n}_{k}\right|^{2}.$ (18)

In general, a normalization factor will be required, but in the specific case
analyzed here, the probabilities always sum to one.

###  3.4 Path Integral Calculations

We now will show that the path integral approach described above can be used
to calculate all the same probabilities as quantum theory for these arbitrary
two-qubit states. Notice that we will not need to use any non-classical paths
or histories; only eight histories need to be considered to get the proper
answers.

The reason that there are eight histories is as follows: since the two photons
are emitted in opposite directions (initially travelling along the solid lined
paths or the dashed lined paths), there are only two initial possibilities,
with amplitudes $A$ and $B$ for the solid and the dashed lines respectively.
For each of these two cases, the photons must either transmit or reflect off a
beamsplitter, ending at one detector in each device. Since there are 4
different outcomes, each one reachable from either of the two initial
possibilities, there are eight total histories.

Let’s consider in detail the case where both photons reach the $\bm{+}$
detectors. To find this probability, there are two histories that need to be
considered, with photons beginning on both the dashed lines or on both solid
lines in Figure 2.

Let’s consider the solid-line history first. Particle 1 reaches the $\bm{+}$
detector by getting reflected at the top beamsplitter, while Particle 2
reaches the $\bm{+}$ detector by getting transmitted through the bottom
beamsplitter. The amplitude of this history includes $exp(i\phi_{1})$ from the
top phase plate and $i\sqrt{R_{1}}$ from the top beamsplitter, and also $i$
from the lower phase plate and $\sqrt{T_{2}}$ from the bottom beamsplitter.
The rules from Section 3.3 therefore yield an amplitude

$\mathcal{E}^{A}_{++}={A}{\left(e^{i\phi_{1}}\>i\>cos\>\frac{\theta_{1}}{2}\right)\left(i\>sin\>\frac{\theta_{2}}{2}\right)}.$
(19)

For the other history with the same outcome, tracing the dashed lines to the
top and bottom $\bm{+}$ detectors, Particle 1 reaches the $\bm{+}$ detector by
getting transmitted through the top beamsplitter while Particle 2 must now be
reflected at the bottom beamsplitter. The top phase plate and top beamsplitter
yield a factor $i\sqrt{T_{1}}$, while Particle 2 passes through two phase
plates, thus picking up a factor $exp(i\delta)exp(i\phi_{2})$ and then another
$i\sqrt{R_{2}}$ from the reflection. Together these factors yield an amplitude

$\mathcal{E}^{B}_{++}={B}{\left(i\>sin\frac{\theta_{1}}{2}\right)\left(e^{i\delta}\>e^{i\phi_{2}}\>i\>cos\frac{\theta_{2}}{2}\right)}.$
(20)

Since histories $\mathcal{E}^{A}_{++}$ and $\mathcal{E}^{B}_{++}$ are leading
to the same outcome, we add them to get the total amplitude and use Eqn. (18)
to get the probability of the photons arriving at both of the $\bm{+}$
detectors:

$P(++)=\left|-{A}e^{i\phi_{1}}\>cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}-{B}\>e^{i\delta}\>e^{i\phi_{2}}\>sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\right|^{2}.$
(21)

Comparing this probability to Eqn. (14), we see an identical result. It is
evident that a measurement at both $\bm{+}$ detectors corresponds to two
qubits measured on the Bloch Sphere at arbitrary angles
$\theta_{1},\phi_{1},\theta_{2},\phi_{2}$ respectively.

This approach also yields the correct probabilities for the other three
outcomes. For the case where Particle 1 is detected at $\bm{+}$ and Particle 2
is detected at $\bm{-}$, the probability is calculated from amplitude
$\mathcal{E}^{A}_{+-}$ (reflecting at both beamsplitters) and
$\mathcal{E}^{B}_{+-}$ (transmitting through both beamsplitters). This yields

$P(+-)=\left|{A}i^{3}\>cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>e^{i\phi_{1}}+{B}i\>e^{i\delta}\>sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{i\phi_{2}}\right|^{2}$
(22)

which is equal to the quantum mechanical probability given by Eqn. (15).

Similarly, combining $\mathcal{E}^{B}_{-+}$ (reflecting at both the
beamsplitters) and $\mathcal{E}^{A}_{-+}$ (transmitting through both
beamsplitters) results in

$P(-+)=\left|{A}i\>sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{i\phi_{1}}+{B}i^{3}\>e^{i\delta}\>cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>e^{i\phi_{2}}\right|^{2},$
(23)

equal to the probability given by Eqn. (16).

Finally, combining $\mathcal{E}^{A}_{--}$ (transmitting through the top
beamsplitter and reflecting at the bottom beamsplitter), and
$\mathcal{E}^{B}_{--}$ (reflecting at the top beamsplitter and transmitting
through the bottom beamsplitter), results in

$P(--)=\left|{A}i^{2}\>sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>e^{i\phi_{1}}+{B}i^{2}\>e^{i\delta}\>cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>e^{i\phi_{2}}\right|^{2},$
(24)

equal to the probability Eqn. (17).

This concludes the demonstration that any observable correlation of any
entangled two-qubit state can be reproduced by a simple sum-over-paths
calculation, where each path can be said to have a spacetime representation.
No higher-dimensional configuration space has been required.

##  4 A 3-Qubit Example: GHZ

While Section 3 discussed a generic 2 qubit case, a similar representation is
also possible for an entangled 3 qubit case. Instead of a generic 3-qubit
state, here we discuss the special case of the GHZ state [14].

###  4.1 Quantum Mechanical probabilities

A 3-qubit state lives in a larger Hilbert space, the tensor product of three
single-qubit spaces. Using the same notation as before,
$\ket{\large{\bm{e_{+}}}}_{3}$ and $\ket{\large{\bm{e_{-}}}}_{3}$ indicate the
orthogonal basis chosen by angles $(\theta_{3},\phi_{3})$ on the Bloch sphere.
For example, a measurement of all three qubits aligned with each chosen basis
corresponds to the state

$\displaystyle\ket{\bm{+++}}=\ket{\large{\bm{e_{+}}}}_{1}\otimes\ket{\large{\bm{e_{+}}}}_{2}\otimes\ket{\large{\bm{e_{+}}}}_{3}=\begin{pmatrix}cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}\\\
cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}\>e^{i\phi_{2}}\\\
sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}\>e^{i\phi_{1}}\\\
sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}\>e^{i(\phi_{1}+\phi_{2})}\\\
cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{i\phi_{3}}\\\
cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{i(\phi_{2}+\phi_{3})}\\\
sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{i(\phi_{1}+\phi_{3})}\\\
sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{i(\phi_{1}+\phi_{2}+\phi_{3})}\end{pmatrix}.$
(25)

The other states ($\ket{\bm{++-}},\ket{\bm{+-+}}$, etc.) can be calculated by
taking the other tensor products in a similar manner. The GHZ state is
expressed as $\psi=A\ket{000}+\bm{B}\ket{111}$
$=\begin{pmatrix}\bm{A}&0&0&0&0&0&0&\bm{B}\end{pmatrix}^{T}$. We will focus on
the simple case A=B=$1/\sqrt{2}$, although other possibilities are easily
addressed using the strategy of unequal emission probabilities from the
previous section. The quantum mechanical probabilities for the 8 possible
outcome combinations can be calculated using the Born rule:

$\displaystyle\left|\braket{\psi}{{+++}}\right|^{2}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{2}\left|cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}+sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{i(\phi_{1}+\phi_{2}+\phi_{3})}\right|^{2}$
(26) $\displaystyle\left|\braket{\psi}{{++-}}\right|^{2}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{2}\left|cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{-i\phi_{3}}-sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}\>e^{i(\phi_{1}+\phi_{2})}\right|^{2}$
$\displaystyle\left|\braket{\psi}{{+-+}}\right|^{2}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{2}\left|cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}\>e^{-i\phi_{2}}-sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{i(\phi_{1}+\phi_{3})}\right|^{2}$
$\displaystyle\left|\braket{\psi}{{+--}}\right|^{2}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{2}\left|cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{-i(\phi_{2}+\phi_{3})}+sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}\>e^{i\phi_{1}}\right|^{2}$
$\displaystyle\left|\braket{\psi}{{-++}}\right|^{2}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{2}\left|sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}\>e^{-i\phi_{1}}-cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{i(\phi_{2}+\phi_{3})}\right|^{2}$
$\displaystyle\left|\braket{\psi}{{-+-}}\right|^{2}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{2}\left|sin\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{-i(\phi_{1}+\phi_{3})}+cos\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}\>e^{i\phi_{2}}\right|^{2}$
$\displaystyle\left|\braket{\psi}{{--+}}\right|^{2}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{2}\left|sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}\>e^{-i(\phi_{1}+\phi_{2})}+cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{i\phi_{3}}\right|^{2}$
$\displaystyle\left|\braket{\psi}{{---}}\right|^{2}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{2}\left|sin\frac{\theta_{1}}{2}\>sin\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\>e^{-i(\phi_{1}+\phi_{2}+\phi_{3})}-cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>sin\frac{\theta_{3}}{2}\right|^{2}.$

The goal of this section is to demonstrate that these probabilities can be
recovered in the path integral formalism.

###  4.2 Path Integral Calculation

For the GHZ case the experimental setup is expanded as shown in Figure 3, with
three measurement devices instead of two.555These measurement devices are
slightly different than those used in the previous sections, but this allows
for a smaller total number of phase plates, simplifying the system. For this
geometry we use two sources ($E_{1}$ and $E_{2}$), each emitting two photons
as before. Since we are interested in the case A=B=$1/\sqrt{2}$, we assume
these sources have equal probability for emission along either the dashed or
solid lines. To ensure that we are dealing with only three photons, one at
each measurement device, the fourth photon is registered by a central
detector, labeled by a "$1$" in the figure. If this central detector measures
exactly one photon, this signifies that the other three photons are now
effectively in the GHZ state, and the measurement can proceed. In other words,
the GHZ state is obtained only after a post-selection that depends on the
central detector. Given a successful post-selection, notice that all the
photons will either be on the solid lines or on the dashed lines, similar to
the two-qubit case. While it might not be evident that this indeed is the GHZ
state immediately, the equal quantum mechanical probabilities and path
integral calculations proves it.

Figure 3: Two sources $\bm{E_{1}}$ and $\bm{E_{2}}$ emit two photons each,
which travel along the solid or dashed lines. The central detector "$1$"
measures exactly one photon, while three measurement devices are used to
measure each of the other photons via a locally chosen basis ($\theta,\phi$).

Using the procedure described in Section 3.3, the path integral approach can
be applied for any combination of measurement outcomes. Here we show a
derivation of the amplitude for all three photons to be measured at the
$\bm{+}$ detectors. For the case where all photons are on the solid lines, the
amplitude is

$\mathcal{E}^{solid}_{+++}=\frac{1}{\sqrt{2}}\left(e^{i\phi_{1}}\>sin\frac{\theta_{1}}{2}\right)\left((e^{i\phi_{2}}\>sin\frac{\theta_{2}}{2})(e^{i\phi_{3}}\>sin\frac{\theta_{3}}{2})\right).$
(27)

This expression is derived as follows. The solid line path for ${E_{1}}$
passes through the phase plate $\phi_{1}$ and gets transmitted through
beamsplitter 1, yielding the factor $\sqrt{T_{1}}=sin(\theta_{1}/2)$ to reach
the $\bm{+}$ detector for particle 1. Two photons emerge from source
${E_{2}}$. One photon passes through phase plate $\phi_{2}$ and gets
transmitted through beamsplitter 2, for a factor
$\sqrt{T_{2}}=sin(\theta_{2}/2)$. The other passes through phase plate
$\phi_{3}$ and gets transmitted through beamsplitter 3 for a factor of
$\sqrt{T_{3}}=sin(\theta_{3}/2)$.

The other possible history is when all photons are on the dashed lines. In
this case, one photon reflects at beamsplitter 1 and passes through a fixed
phase plate $i$ yielding a factor $i^{2}\sqrt{R_{1}}$. Another reflects at
beamsplitter 3 with $i\sqrt{R_{3}}$ to reach the $\bm{+}$ detector. The final
photon simply reflects at beamsplitter 2 with a factor of $i\sqrt{R_{2}}$. The
total amplitude for these three paths is therefore

$\mathcal{E}^{dashed}_{+++}=\frac{1}{\sqrt{2}}\left((i\>cos\frac{\theta_{1}}{2})(i\>cos\frac{\theta_{2}}{2})\right)\left(i^{2}\>cos\frac{\theta_{3}}{2}\right).$
(28)

As both of these histories lead to the same outcome, they are added and then
squared to give the joint probability of all three particles being detected at
the $\bm{+}$ detectors. Specifically,

$\displaystyle P(+++)$ $\displaystyle=$
$\displaystyle\left|\mathcal{E}^{dashed}_{+++}+\mathcal{E}^{solid}_{+++}\right|^{2},$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left|\>cos\frac{\theta_{1}}{2}\>cos\frac{\theta_{2}}{2}\>cos\frac{\theta_{3}}{2}+e^{i\phi_{1}}sin\frac{\theta_{1}}{2}e^{i\phi_{2}}\>sin\frac{\theta_{2}}{2}e^{i\phi_{3}}\>sin\frac{\theta_{3}}{2}\right|^{2}.$

By rearranging the equation, this is exactly comparable to the first equation
in Eqn. (26). Similarly, using the amplitudes listed in Appendix B, the
probabilities of the other outcomes can also be calculated through path
integrals, which turn out to be exactly equal to the corresponding quantum
mechanical probabilities calculated in Eqn. (26). Therefore all of the
correlations evident from measurements of the GHZ state can be calculated from
an analysis of spacetime-based histories.

##  5 Discussion

###  5.1 Summary

We begin our discussion with a summary of the main results. The measurement of
a generic single qubit has been developed in Section 2. The measurement device
described in Section 2.2 consists of two adjustable parameters: the phase
plate of phase delay angle $\phi$ and the beamsplitter corresponding to an
angle $\theta$. These controllable factors are equivalent to choosing an
orthogonal basis for the qubit aligned in any direction $(\theta,\phi)$ on the
Bloch Sphere. In the single photon limit this device yields the same
probabilities as standard quantum mechanics.

Using N copies of this measurement device we can measure N-qubit systems. For
a generic 2 qubit case shown in Section 3 we would then require two such
devices, the geometry of which is discussed in detail in Section 3.2. A key
part of the setup was a conceptually simple source that emits two photons in
exactly opposite directions, but with different probabilities of emitting
along the solid or dashed paths as seen in Figure 2. The quantum mechanical
probabilities for each possible outcome calculated in Section 3.1 were shown
to be exactly equal to the probabilities calculated via the path integral
approach described in Section 3.4. It must be noted that these calculations
are carried out by following the path integral procedure described in Section
3.3, and correctly recovered all observable correlations for any entangled
state.

The special 3 qubit case of the GHZ state followed the same procedure. Two
sources were required to create this state, where each source emits two
photons in exactly opposite directions, as shown in Figure 3. Unlike the
generic two-qubit case, these sources emit with the same probability in all
directions. A version of the measurement device was used for each of the three
photons, while the fourth photon was registered by a special central detector.
(If this fourth detector does not see a single photon, the GHZ state is not
created.) The quantum probabilities calculated in Section 4.1 corresponded
exactly with the path integral calculations carried out in Section 4.2.

###  5.2 Analysis

Formally, a path integral calculation requires a sum over all possible
histories which lead to the same endpoint, but that is evidently not the case
in the above examples. As we saw, only a few classical paths were
required.666Here we are using the word ”classical” to indicate that only
straight-line paths are required. If a photon is thought of as an
electromagnetic wave, it is not literally classical for that wave to entirely
pass through a beamsplitter (or to entirely reflect), but the path integral
formalism does not evidently allow for including both beamsplitter outputs
into a single history if only one detector fires. Because of this
simplification, the difficulty of the calculation in the path integral
framework is not essentially more difficult than the traditional quantum
calculation. Indeed, since no tensor products or inner products need to be
calculated, the number of formal calculations seems somewhat reduced in the
path-integral approach, especially for the 3-qubit case. If the number of
qubits were to increase beyond three, we expect this advantage to be even more
pronounced. Of course, the difficulty of the calculation grows regardless;
even in the path integral framework one would need to increase the number of
measurement devices and create a setup with the appropriate number of photon
sources.

More important than the number of calculational steps, however, is the fact
that the histories used by the path integral framework reside in spacetime
rather than in an abstract Hilbert space. Even for entangled states, each
possible history can be mapped to physical paths of particles which travel
from the sources to the detectors. (These paths end at only the detectors
which actually register a particle, never at the detectors which do not.)
There is no comparable spacetime-based picture when analyzing entangled states
in the conventional QM formalism.

Some readers may be resistant to this conclusion, perhaps because of an
impression that Bell’s Theorem prevents a spacetime-based account of
entanglement experiments. Even setting aside the fact that this view of Bell’s
Theorem is not true in general [13], it is simple to see why the path integral
framework does not fall within Bell’s analysis, as Bell assigned probabilities
$Prob(\lambda)$ to hidden/unknown parameters $\lambda$. The path integral
framework does not assign probabilities to each spacetime-based history, but
rather complex amplitudes. These amplitudes are then combined in a way that is
incompatible with a probabilistic assignment. For example, if one history had
an amplitude of $iA$, and another history had an amplitude of $-iA$, the
square of the sum of these amplitudes is zero, corresponding to a zero
probability. This would be impossible for two hidden possibilities, each with
a non-zero probability, and it results from the curious way in which
amplitudes are combined; see [9] for more discussion of this issue.

But the above resolution to one concern might raise another. If the amplitudes
for each spacetime-based history are combined in a way that is inconsistent
with hidden spacetime-based parameters $\lambda$, then is it really fair to
say that this is a spacetime-based calculation? While these histories do not
directly link distant points in space, as conventional quantum superpositions,
it is indeed true that different histories are being combined into the same
probability calculation in some non-classical manner. (This must be the case,
or else it would not be possible to violate Bell Inequalities in the first
place.)

Still, it is certainly true that alternate perspectives of the same phenomena
have historically been found to generate useful advances and generalizations.
In this case, if one is concerned about the disconnect between entangled
states and conventional spacetime, the path integral is an obvious viewpoint
which might partially resolve this tension. In particular it does not require
any direct connection between spacelike-separated regions of spacetime, and is
therefore at least somewhat more compatible with classical spacetime than are
instantaneous entangled states. Additional applications of this alternate
perspective on entanglement will now be discussed.

###  5.3 Potential Applications

The results from Sections 3 and 4 provide a standard recipe for applying the
alternate path-integral viewpoint to many well-known results in quantum theory
and quantum foundations. Take, for example, the no-signalling theorem [20].
This theorem is formally developed in a conventional QM framework, but it must
also somehow be implied by the path-integral formalism. On its face, such a
theorem seems more surprising in the path integral framework, where adjusting
measurement settings ($\theta,\phi$) on one detector changes the amplitude for
entire histories that include distant measurement outcomes. But, since the
joint probabilities are identical to that of conventional QM, even the path
integral must respect this no-signalling condition; the marginal probabilities
for distant outcomes must always cancel when local devices are adjusted.
Exploring why this is necessarily the case seems like it might provide further
insight into how signal locality is necessarily enforced, even in an “all at
once” framework like the path integral.

A related application would be the question of “fine-tuning” [21], and whether
there is a natural explanation for the perfect cancellation of the marginal
probabilities. Analysis of this issue in a path-integral framework might
indicate whether this cancellation might be explained via symmetries [22],
global consistency [23], or some other factor that becomes evident from this
alternate perspective.

Another evident line of future research would follow from a basic extension of
these results: time-reversing the entire set-up in Figure 2, where the
detectors become single-photon sources and the central two-photon emitter
$\bm{E}$ becomes instead a possible two-photon absorber. Treating other two-
photon processes at $\bm{E}$ (scattering, etc.) as alternate possible
outcomes, one could likely build the analog of a two-photon measurement
device. With this in hand, it would be possible to apply the path integral
framework to other known results in quantum theory; Hanbury-Brown-Twiss
experiments, entanglement swapping, the geometry used in the Pusey-Barrett-
Rudolph theorem [24], etc. In every case, one expects the same results as QM,
but for evidently different reasons, and these reasons may very well shed
light on outstanding questions in quantum foundations.

Note that a further advantage to such a path integral framework is that the
accounts of these experiments would be far more evidently time-symmetric than
the conventional QM viewpoint [10]. In conventional QM, time-reversing a two-
particle Bell experiment transforms it into a geometry where the particles are
not even considered to be entangled. But a time-reversed history in the path-
integral viewpoint does not change its amplitude whatsoever. There are still
differences between these two experiments, even in the path-integral
viewpoint, but those differences come in via the experimenter (what the
experimenter can be said to control is different in these two cases) rather
than the mathematics that purportedly describes the quantum system.

Finally, it should again be noted that the path integral is the obvious
starting point for any reformulation of quantum theory built upon the “all at
once” analysis familiar from general relativity (where spacetime systems are
solved as coherent entities rather than computed sequentially). If one takes
entanglement phenomena to be one of the most difficult-to-reformulate aspects
of quantum theory, then a path-integral view of entanglement is a necessary
starting point for any “all at once” account of QM. Any resulting model of
these phenomena would almost certainly fall into the “Future Input Dependent”
class of models described in a recent review [13].

Even apart from the above potential applications, these results clearly
demonstrate not merely that a path integral approach can recover the usual
joint probabilities for measurements of entangled states, but precisely how to
calculate these probabilities for any pure two-qubit state. The analysis does
not require any computations in Hilbert space, and can be performed entirely
using histories built out of spacetime-based paths, with no direct connections
between spacelike-separated events. Using the above three-qubit GHZ state as a
guide, we expect that further generalizations to a path integral analysis of
any N-qubit entangled state should be conceptually straightforward, although
the details of how to accomplish this remains an open problem.

## Appendix A: The Schmidt Basis

For some applications of the two-qubit path integral, one may want to start
with a completely general two-qubit pure state

$\ket{\psi}=a\ket{\bar{0}\bar{0}}+b\ket{\bar{0}\bar{1}}+c\ket{\bar{1}\bar{0}}+d\ket{\bar{1}\bar{1}},$
(30)

where $a,b,c$ and $d$ are complex and $|a|^{2}+|b|^{2}+|c|^{2}+|d|^{2}=1$.
Here the overbars on $\ket{\bar{0}}$ and $\ket{\bar{1}}$ are to distinguish
these states from the Schmidt basis $\ket{{0}}$ and $\ket{{1}}$ used in
Section 3.

Each of the two qubits in $\ket{\psi}$ can then be measured in some arbitrary
direction on the Bloch sphere, corresponding to the angles
$(\bar{\theta}_{1},\bar{\phi}_{1})$ for the first qubit and
$(\bar{\theta}_{2},\bar{\phi}_{2})$ for the second. These angles are defined
relative to the corresponding basis on the right side of (30), and will
generally be different than the angles $(\theta_{1},\phi_{1})$ and
$(\theta_{2},\phi_{2})$ used in Section 3. (These latter angles are defined
relative to the Schmidt basis states $\ket{{0}}$ and $\ket{{1}}$.)

However, it is a relatively straightforward manner to relate these various
coordinate systems, if one can solve for the direction $(\alpha,\beta)$ on the
given Bloch sphere which lines up with the z-axis of the Schmidt basis Bloch
sphere. For the first qubit, this relationship is defined by

$\displaystyle\ket{0}_{1}$ $\displaystyle=$
$\displaystyle\cos\frac{\alpha_{1}}{2}\ket{\bar{0}}_{1}+\sin\frac{\alpha_{1}}{2}e^{i\beta_{1}}\ket{\bar{1}}_{1}$
(31) $\displaystyle\ket{1}_{1}$ $\displaystyle=$
$\displaystyle\sin\frac{\alpha_{1}}{2}e^{-i\beta_{1}}\ket{\bar{0}}_{1}-\cos\frac{\alpha_{1}}{2}\ket{\bar{1}}_{1}.$
(32)

And a similar (but reversed) expression for the second qubit is

$\displaystyle\ket{0}_{2}$ $\displaystyle=$
$\displaystyle\sin\frac{\alpha_{2}}{2}e^{-i\beta_{2}}\ket{\bar{0}}_{2}-\cos\frac{\alpha_{2}}{2}\ket{\bar{1}}_{2}$
(33) $\displaystyle\ket{1}_{2}$ $\displaystyle=$
$\displaystyle\cos\frac{\alpha_{2}}{2}\ket{\bar{0}}_{2}+\sin\frac{\alpha_{2}}{2}e^{i\beta_{2}}\ket{\bar{1}}_{2}.$
(34)

Recall, that if written in the Schmidt basis, the state $\ket{\psi}$ takes the
much simpler form given by Eqn. (9). If one knows the angles $(\alpha,\beta)$,
then one knows precisely how the measurement angles
$(\bar{\theta},\bar{\phi})$ in the given basis are related to the measurement
angles $(\theta,\phi)$ in the Schmidt basis. Geometrically, on the Bloch
sphere, they are related by a rotation which takes $(\alpha,\beta)$ to the
z-axis, but any careful analysis would certainly want to use the precise above
equations.

All that remains is to find the four angles $\alpha_{1}$, $\beta_{1}$,
$\alpha_{2}$ and $\beta_{2}$. We have not been able to find a derivation of
these angles in the published literature, but they can essentially be
recovered from the results in [19], except for the special case of maximally-
entangled states which will be worked out separately. In terms of the four
complex parameters in Eqn. (30), these angles are

$\cos\alpha_{1}=\frac{|a|^{2}+|b|^{2}-|c|^{2}-|d|^{2}}{\sqrt{1-4|ad-
bc|^{2}}},$ (35)

$e^{-i\beta_{1}}=\frac{ac^{*}+bd^{*}}{|ac^{*}+bd^{*}|},$ (36)

$\cos\alpha_{2}=\frac{|a|^{2}+|c|^{2}-|b|^{2}-|d|^{2}}{\sqrt{1-4|ad-
bc|^{2}}},$ (37)

$e^{-i\beta_{2}}=\frac{ab^{*}+cd^{*}}{|ab^{*}+cd^{*}|}.$ (38)

Via tedious algebra, all of the above equations in this appendix can be
combined to ascertain that both $\braket{\psi}{00}$ and $\braket{\psi}{11}$
are exactly zero. (We have also checked this numerically.) This algebra is
much easier if one uses the normalization of $\ket{\psi}$ to prove the
relationships

$\sin\alpha_{1}=\frac{2|ac^{*}+bd^{*}|^{2}}{\sqrt{1-4|ad-bc|^{2}}},$ (39)

$\sin\alpha_{2}=\frac{2|ab^{*}+cd^{*}|^{2}}{\sqrt{1-4|ad-bc|^{2}}}.$ (40)

These equations also indicate what is going on when one of the $\beta$ terms
become undefined – that is, when either $|ac^{*}+bd^{*}|$ or $|ab^{*}+cd^{*}|$
is zero. In these cases, the corresponding $\alpha$ also goes to zero or
$\pi$, such that the two bases are aligned for that qubit. (For $\alpha=0$ it
is already in the Schmidt basis, and for $\alpha=\pi$ one need merely set
$\ket{0}=\ket{\bar{1}}$ and $\ket{1}=\ket{\bar{0}}$.) The fact that $\beta$ is
undefined is therefore just the usual spherical-coordinate ambiguity for
$\phi$ on the z-axis; in this case $\beta$ can take any value, such as
$\beta=0$.

The only remaining problem is the special case of maximally-entangled states,
where ${{1-4|ad-bc|^{2}}}=0$; evidently in this case the above angles $\alpha$
are undefined. In that case, it is always possible to leave one of the qubits
unchanged, and then find a Schmidt basis using only the other qubit. These
maximally entangled states can always [25] be written in the form

$\displaystyle\bm{\ket{\psi}}=\frac{1}{\sqrt{2}}\left[e^{i\chi_{1}}cos\gamma\\!\ket{\bar{0}\bar{0}}+e^{i\chi_{2}}sin\gamma\\!\ket{\bar{0}\bar{1}}+e^{-i\chi_{2}}sin\gamma\\!\ket{\bar{1}\bar{0}}-e^{-i\chi_{1}}cos\gamma\\!\ket{\bar{1}\bar{1}}\right].$
(41)

For this general maximally-entangled state, if one leaves the first qubit
unchanged, $\bar{\ket{0}}_{1}=\ket{0}_{1}$, the second qubit can be written as
a superposition of a Schmidt basis $\ket{0}_{2}$ and $\ket{1}_{2}$

$\displaystyle\bar{\ket{0}}_{2}=f\ket{0}_{2}+g\ket{1}_{2},$ (42)
$\displaystyle\bar{\ket{1}}_{2}=g^{*}\ket{0}_{2}-f^{*}\ket{1}_{2}.$ (43)

For a Schmidt basis of the form Eqn.(9), we require $\braket{\psi}{00}=0$ and
$\braket{\psi}{11}=0$. Therefore

$\displaystyle(e^{i\chi_{1}}cos\gamma)^{*}f+(e^{i\chi_{2}}sin\gamma)^{*}g=0,$
(44)
$\displaystyle(e^{-i\chi_{2}}sin\gamma)^{*}g^{*}-(-e^{-i\chi_{1}}cos\gamma)^{*}f^{*}=0.$
(45)

As the coefficients $f$ and $g$ are normalized,we get:

$\displaystyle f$ $\displaystyle=$ $\displaystyle sin\gamma,$ (46)
$\displaystyle g$ $\displaystyle=$
$\displaystyle-e^{-i(\chi_{1}-\chi_{2})}\>cos\gamma.$ (47)

The above results can be combined to indicate precisely how to rotate the
coordinate systems of the measurement devices, such that the results from
Section 3 can directly apply to any possible pure two qubit state.

## Appendix B: GHZ Details

In this Appendix, we provide the details of the GHZ path integral calculations
for the other possible outcomes. (Calculations for the $\bm{+++}$ outcome have
been discussed in detail in Section 4.2.) Using Eqn.(18) one can recover the
probabilities in Eqn.(26)

For each outcome there are two histories that lead to it; the amplitudes for
these histories are listed below and have been calculated using the procedure
outlined by Section 3.3 for Figure (3). In each of these cases the first
parentheses corresponds to particle 1, the second corresponds to particle 2
and the third corresponds to particle 3.

$\displaystyle\begin{aligned}
\mathcal{E}^{solid}_{++-}&=\frac{1}{\sqrt{2}}\left(e^{i\phi_{1}}\>sin\frac{\theta_{1}}{2}\right)\left(e^{i\phi_{2}}\>sin\frac{\theta_{2}}{2}\right)\left(i\>e^{i\phi_{3}}\>cos\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{dashed}_{++-}&=\frac{1}{\sqrt{2}}\left(i\>cos\frac{\theta_{1}}{2}\right)\left(i\>cos\frac{\theta_{2}}{2}\right)\left(i\>sin\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{solid}_{+-+}&=\frac{1}{\sqrt{2}}\left(e^{i\phi_{1}}\>sin\frac{\theta_{1}}{2}\right)\left(e^{i\phi_{2}}\>i\>cos\frac{\theta_{2}}{2}\right)\left(e^{i\phi_{3}}\>sin\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{dashed}_{+-+}&=\frac{1}{\sqrt{2}}\left(i\>cos\frac{\theta_{1}}{2}\right)\left(sin\frac{\theta_{2}}{2})\right)\left(i^{2}\>cos\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{solid}_{+--}&=\frac{1}{\sqrt{2}}\left(e^{i\phi_{1}}\>sin\frac{\theta_{1}}{2}\right)\left(e^{i\phi_{2}}\>i\>cos\frac{\theta_{2}}{2}\right)\left(e^{i\phi_{3}}\>i\>cos\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{dashed}_{+--}&=\frac{1}{\sqrt{2}}\left(i\>cos\frac{\theta_{1}}{2}\right)\left(sin\frac{\theta_{2}}{2}\right)\left(i\>sin\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{solid}_{-++}&=\frac{1}{\sqrt{2}}\left(e^{i\phi_{1}}\>i\>cos\frac{\theta_{1}}{2}\right)\left(e^{i\phi_{2}}\>sin\frac{\theta_{2}}{2}\right)\left(e^{i\phi_{3}}\>sin\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{dashed}_{-++}&=\frac{1}{\sqrt{2}}\left(sin\frac{\theta_{1}}{2}\right)\left(i\>cos\frac{\theta_{2}}{2}\right)\left(i^{2}\>cos\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{solid}_{-+-}&=\frac{1}{\sqrt{2}}\left(e^{i\phi_{1}}\>i\>cos\frac{\theta_{1}}{2}\right)\left(e^{i\phi_{2}}\>sin\frac{\theta_{2}}{2}\right)\left(e^{i\phi_{3}}\>i\>cos\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{dashed}_{-+-}&=\frac{1}{\sqrt{2}}\left(sin\frac{\theta_{1}}{2}\right)\left(i\>cos\frac{\theta_{2}}{2}\right)\left(i\>sin\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{solid}_{--+}&=\frac{1}{\sqrt{2}}\left(e^{i\phi_{1}}\>i\>cos\frac{\theta_{1}}{2}\right)\left(e^{i\phi_{2}}\>i\>cos\frac{\theta_{2}}{2}\right)\left(e^{i\phi_{3}}\>sin\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{dashed}_{--+}&=\frac{1}{\sqrt{2}}\left(sin\frac{\theta_{1}}{2}\right)\left(i\>cos\frac{\theta_{2}}{2}\right)\left(i\>sin\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{solid}_{---}&=\frac{1}{\sqrt{2}}\left(e^{i\phi_{1}}\>i\>cos\frac{\theta_{1}}{2}\right)\left(e^{i\phi_{2}}\>i\>cos\frac{\theta_{2}}{2}\right)\left(e^{i\phi_{3}}i\>cos\frac{\theta_{3}}{2}\right),\\\
\mathcal{E}^{dashed}_{---}&=\frac{1}{\sqrt{2}}\left(sin\frac{\theta_{1}}{2}\right)\left(sin\frac{\theta_{2}}{2}\right)\left(i\>sin\frac{\theta_{3}}{2}\right).\end{aligned}$
(48)

Plugging these amplitudes into Eqn.(18) yields the probabilities given by
Eqn.(26).

## Acknowledgements

The authors gratefully thank Emily Adlam and Hilary Hurst for helpful
suggestions.

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