Hong–Ou–Mandel Effect

The HOM effect is a landmark in quantum optics as it is the most spectacular manifestation of boson bunching. It is a two-photon intrinsically quantum interference effect where the probability amplitude of both photons being transmitted cancels out the probability amplitude of both photons being reflected. A 50:50 BS effects the single-photon transformations (for details, see Materials and Methods, Gaussian Unitaries for a BS and PDC)

Partial Time Reversal

Bogoliubov transformations on two bosonic modes comprise passive and active transformations. The BS is the fundamental passive transformation, while parametric down conversion (PDC) gives rise to the class of active transformations (also called nondegenerate parametric amplification). Although the involved physics is quite different (a simple piece of glass makes a BS, while an optically pumped nonlinear crystal is needed to effect PDC), the Hamiltonians generating these two unitaries are amazingly close, namely

The underlying concept of PTR will be formalized in Eq. 7, but we first illustrate this duality between a BS and PDC with the simple example of Fig. 2A, where $$ n$$ photons impinge on port $$ \widehat{b}$$ of a BS (with vacuum on port $$ â$$), resulting in the binomial output state

These examples reflect the existence of a general duality between a BS and PDC. Indeed, as demonstrated in Materials and Methods, Proof of PTR Duality, partial transposition in Fock basis gives rise to PTR duality

The notion of time reversal can be conveniently interpreted using the so-called “retrodictive” picture of quantum mechanics (21). Along this line, PTR must be understood here as the fact that the “retrodicted” state of mode $$ \widehat{b}$$ propagates backward in time, while the state of mode $$ â$$ normally propagates forward in time (this is made precise in Materials and Methods, Retrodictive Picture of Quantum Mechanics). As shown in Fig. 2B, the PTR duality can be made operational by sending half of a so-called Einstein–Podoslky–Rosen (EPR) entangled state on mode $$ \widehat{b}$$, so that we access the output retrodicted state on the second entangled mode $$ {\widehat{b}}^{\prime \prime }$$.

Two-Boson Interference in an Amplifier

Due to this duality, the HOM effect for a BS of transmittance 1/2 immediately suggests the possible existence of a related interferometric suppression effect in a PDC of gain 2, namely $$ \left.⟨\mathrm{1,1}\right|U_2^{\mathrm{P}\mathrm{D}\mathrm{C}}\left|\mathrm{1,1}\right.⟩=0$$. This striking prediction can indeed be verified by examining the state at the output of a PDC of gain $$ g=2$$ (see Two-Photon Interference in a BS and PDC):

Fig. 3.

(A) Probability $$ P_n$$ of observing $$ n$$ photon pairs at the output of a PDC of gain $$ g=2$$ when a single photon impinges on both the signal and idler input ports. The output state $$ U_2^{\mathrm{P}\mathrm{D}\mathrm{C}}\left|\mathrm{1,1}\right.⟩=\frac{1}{2}\left(-\left|\mathrm{0,0}\right.⟩+\sqrt{\frac{1}{4}}\hspace{0.17em}\left|\mathrm{2,2}\right.⟩+\sqrt{\frac{1}{2}}\hspace{0.17em}\left|\mathrm{3,3}\right.⟩\right.$$ $$ \left.+\sqrt{\frac{9}{16}}\hspace{0.17em}\left|\mathrm{4,4}\right.⟩+\sqrt{\frac{1}{2}}\hspace{0.17em}\left|\mathrm{5,5}\right.⟩+\cdots \right)$$ has a vanishing $$ \left|\mathrm{1,1}\right.⟩$$ component, owing from two-photon interferometric suppression. The significant components are the vacuum as well as the terms with 2 to $$ \sim $$10 pairs (the next terms quickly decay to zero). (B) Corresponding distributions of the photon pair number for a gain $$ g=3$$ (showing a dip at $$ n=2$$) and $$ g=4$$ (showing a dip at $$ n=3$$). The distributions are shown as continuous curves in order to guide the eye, but only integer values of $$ n$$ are relevant. The curve for $$ g=2$$ is also plotted for comparison.

The dependence of the probability of detecting a single pair ($$ n=1$$) on the gain $$ g$$ of PDC is given by (see Two-Photon Interference in a BS and PDC)

Space-Like vs. Time-Like Indistinguishability

The origin of the two-boson quantum interference effect that we predict can be traced back to boson indistinguishability, similarly as for the HOM effect albeit in a time-like version (involving bosons from the past and future). We first recall that the HOM effect originates from what can be viewed as space-like indistinguishability (Fig. 4, Upper). When two photons impinge on a BS of transmittance $$ \eta $$, each photon has a probability amplitude $$ \sqrt{\eta }$$ of being transmitted, so the double-transmission amplitude is $$ A_{\mathrm{d}\mathrm{t}}=\eta $$. In contrast, the probability amplitude of reflection is $$ \sqrt{1-\eta }$$ but with opposite signs for the two photons, so the double-reflection amplitude is $$ A_{\mathrm{d}\mathrm{r}}=-\left(1-\eta \right)$$. Since a double-transmission event is indistinguishable from a double-reflection event, we must add probability amplitudes, leading to $$ |A_{\mathrm{d}\mathrm{t}}+A_{\mathrm{d}\mathrm{r}}{|}^2=P_{\mathrm{c}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{c}}\left(\eta \right)$$. The double-transmission and double-reflection amplitudes exactly cancel out for $$ \eta =1/2$$, which originates from the fact that an exchange of the two indistinguishable photons in space (which turns a double-transmission into a double-reflection event) cannot lead to any observable consequence.

Fig. 4.

The HOM effect (Upper) is due to space-like indistinguishability: the double-transmission path (of amplitude $$ A_{\mathrm{d}\mathrm{t}}$$) where the two photons are transmitted interferes destructively with the double-reflection path (of amplitude $$ A_{\mathrm{d}\mathrm{r}}$$) where the two photons are reflected. Exchanging the two photons in space leads to the HOM effect in a BS of transmittance $$ \eta =1/2$$. According to PTR duality (by reversing the arrow of time in the second mode; Lower), the two interfering paths in a PDC correspond to the double-transmission event associated with amplitude $$ A_{\mathrm{d}\mathrm{t}}^{\prime }$$ (the two photons simply cross the PDC) and double-stimulated event of amplitude $$ A_{\mathrm{s}\mathrm{t}}^{\prime }$$ (the input photon pair is up converted into the pump beam, and another pump photon is down converted into a new photon pair). Exchanging the photon pairs in time induces a time-like interferometric suppression in a PDC of gain $$ g=2$$.

We now argue that it is the exchange of indistinguishable photons in time that is responsible for the interference effect in an amplifier (Fig. 4, Lower). When two photons impinge on a PDC with gain $$ g$$, they can be both transmitted without triggering a stimulated event, which is dual to the double transmission in a BS (where $$ \eta $$ is substituted by $$ 1/g$$). Hence, the double-transmission amplitude in a PDC is $$ A_{\mathrm{d}\mathrm{t}}^{\prime }=g^{-1/2}{A}_{\mathrm{d}\mathrm{t}}=1/g^{3/2}$$. Another possible path giving rise to the coincident detection of two single photons is the combination of the stimulated annihilation and emission of a pair of photons, which is dual to the double reflection in a BS. This double-stimulated event admits a probability amplitude $$ A_{\mathrm{s}\mathrm{t}}^{\prime }=g^{-1/2}{A}_{\mathrm{d}\mathrm{r}}=-\left(g-1\right)/g^{3/2}$$, where the minus sign results from the fact that the probability amplitude that the input pair disappears by stimulated annihilation and the probability amplitude that a new pair is created by stimulated emission have opposite signs. Again, since the double-transmission and double-stimulated events are indistinguishable, we must add their probability amplitudes and get $$ |A_{\mathrm{d}\mathrm{t}}^{\prime }+A_{\mathrm{s}\mathrm{t}}^{\prime }{|}^2=P_{\mathrm{c}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{c}}^{\prime }\left(g\right)$$, which vanishes when $$ g=2$$. Roughly speaking, we cannot know whether the “old” photons have been replaced by “new” photons or have been left unchanged, which we dub time-like indistinguishability.

Discussion and Conclusion

The role of time reversal in quantum physics has long been a fascinating subject of questioning (see, e.g., ref. 22 and references therein), but the key idea of the present work is to consider a bipartite quantum system (two bosonic modes) with counterpropagating times. Incidentally, we note that the notion of time reversal has been exploited in the context of defining separability criteria (23, 24), but this seems to be unrelated to PTR duality. Further, the link between time reversal and optical-phase conjugation has been mentioned in the quantum optics literature (see, e.g., ref. 25), but it exploits the fact that the complex conjugate of an electromagnetic wave is the time-reversed solution of the wave equation (the phase conjugation time-reversal mirror concerns one mode only). The PTR duality introduced here bears some resemblance with an early model of lasers (26) based on the coupling of an “inverted” harmonic oscillator (having a negative frequency $$ \omega $$) with a heat bath. The inverted harmonic oscillator ($$ {\mathrm{e}}^{-i\omega t}\to {\mathrm{e}}^{i\omega t}$$) can indeed be viewed as a time-reversed harmonic oscillator. Quantum amplification in this model occurs from a PDC-like coupling of this inverted harmonic oscillator, whereas quantum damping follows from the BS-like coupling of a usual harmonic oscillator with the bath. The PTR duality is also reminiscent of Klyshko’s so-called “advanced-wave picture” in PDC (27), which provides an interpretation of coincidence-based two-photon experiments: the wave that is detected by one of the detectors behind PDC can be viewed as resulting from an “advanced wave” emitted by the second detector, so that PDC acts as a mirror (28). This picture may indeed be interpreted as a special case of Eq. 25, namely $$ \left.⟨\mathrm{0,0}\right|U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}\left|\psi ,\varphi \right.⟩=g^{-1/2}\left.⟨0,{\varphi }^{*}\right|U_{1/g}^{\mathrm{B}\mathrm{S}}\left|\psi ,0\right.⟩$$. In the limit where the gain $$ g\to \infty $$, the two outputs of PDC can be viewed as the input $$ \psi $$ and output $$ {\varphi }^{*}$$ of a fully reflecting (phase-conjugating) mirror with $$ \eta \to 0$$.

In this work, we have promoted PTR as the proper way to approach the duality between passive and active bosonic transformations. As a compelling application of PTR duality, we have unveiled a hitherto unknown quantum interference effect, which is a manifestation of quantum indistinguishability for identical bosons in active transformations (space-like indistinguishability, which is at the root of the HOM effect, transforms under PTR into time-like indistinguishability). The interferometric suppression of the coincident $$ \left|\mathrm{1,1}\right.⟩$$ term is induced by the indistinguishability between a photon pair originating from the past and a photon pair going to the future. Stated more dramatically, while the two photons may cross the amplification medium and be detected, the sole fact that they could instead be annihilated and replaced by two other photons makes the detection probability drop to zero when $$ g=2$$.

The experimental verification of this effect can be envisioned with present technologies (see Experimental Scheme). A coincidence probability lower than 25% would be sufficient to rule out a classical interpretation, which could in principle be reached with a moderate gain of 1.28 (see Classical Baseline). Observing time-like two-photon interference in experiments involving active optical components would then be a highly valuable metrology tool given that the HOM dip is commonly used today as a method to benchmark the reliability of single-particle sources and mode matching. More generally, the interference of many photons scattered over many modes in a linear optical network has generated a tremendous interest in the recent years, given the connection with the “boson sampling” problem [i.e., the hardness of computing the permanent of a random matrix (29)], and technological progress in integrated optics now makes it possible to access large optical circuits (see, e.g., ref. 30). In this context, it would be exciting to uncover new consequences of PTR duality and time-like interference.

Finally, we emphasize that our analysis encompasses all bosonic Bogoliubov transformations, which are widespread in physics, appearing in quantum optics, quantum field theory, or solid-state physics, but also in black hole physics or even in the Unruh effect (describing an accelerating reference frame). This suggests that time-like quantum interference may occur in various physical situations where identical bosons participate in such a transformation. Beyond bosons, let us point out an intriguing connection with the notion of “crossing” in quantum electrodynamics (31, 32). Crossing symmetry refers to a substitution rule connecting two scattering matrix elements that are related by a Wick rotation (antiparticles being turned into particles going backward in time). For example, the scattering of a photon by an electron (Compton scattering) and the creation of an electron–positron pair by two photons are processes that are related to each other by such a substitution rule (see, e.g., ref. 33). This is in many senses analogous to the PTR duality described here: since a photon (or truly neutral boson) is its own antiparticle, we may view PTR duality as a substitution rule connecting the BS diagram to the PDC diagram. We hope that this connection with quantum electrodynamics may open up even broader perspectives.

Materials and Methods

Example of PTR.

We illustrate the PTR duality between a BS and PDC by considering the additional example of a BS with $$ m$$ photons impinging on input port $$ â$$ and $$ n$$ photons impinging on input port $$ \widehat{b}$$ (Fig. 5A). If we condition on the vacuum on mode $$ {\widehat{b}}^{\prime }$$ at the output of the BS, the (unnormalized) conditional output state of mode $$ {â}^{\prime }$$ is

$\left|{\varphi }_n\right.⟩={\left(\genfrac{}{}{0.0pt}{}{n+m}{m}\right)}^{1/2}{\sin }^n\hspace{0.17em}\theta \hspace{0.17em}{\cos }^m\hspace{0.17em}\theta \hspace{0.17em}\left|n+m\right.⟩\hspace{0.17em}\hspace{0.17em}.$ [20]

Hence, we have the probability amplitude

$\left.⟨n+m,0\right|U_{\eta }^{\mathrm{B}\mathrm{S}}\left|m,n\right.⟩={\left(\genfrac{}{}{0.0pt}{}{n+m}{m}\right)}^{1/2}{\sin }^n\hspace{0.17em}\theta \hspace{0.17em}{\cos }^m\hspace{0.17em}\theta .$ [21]

The special case $$ m=0$$ is considered in the text (Fig. 2) and corresponds to $$ \left|{\varphi }_n\right.⟩={\sin }^n\hspace{0.17em}\theta \hspace{0.17em}\left|n\right.⟩$$. By reversing the arrow of time on mode $$ \widehat{b}$$, we compare the probability amplitude of Eq. 21 with the probability amplitude

$\left.⟨n+m,n\right|U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}\left|m,0\right.⟩={\left(\genfrac{}{}{0.0pt}{}{n+m}{m}\right)}^{1/2}\frac{{\tanh }^{n}r}{{\left(\cosh r\right)}^{m+1}}$ [22]

for a PDC of gain $$ g$$ (Eq. 19). Now, if we make the substitution $$ \sin \theta =\tanh r$$ and $$ \cos \theta ={\left(\cosh r\right)}^{-1}$$, which is equivalent to $$ g=1/\eta $$, we confirm that Eqs. 21 and 22 are dual under PTR, namely

$\left.⟨n+m,n\right|U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}\left|m,0\right.⟩=\frac{1}{\sqrt{g}}\hspace{0.17em}\left.⟨n+m,0\right|U_{1/g}^{\mathrm{B}\mathrm{S}}\left|m,n\right.⟩\hspace{0.17em}\hspace{0.17em}.$ [23]

As shown in Fig. 5B, if the input mode $$ \widehat{b}$$ is entangled (that is, if we use the entangled state $$ {\left|\mathrm{\Psi }\right.⟩}_{b,b^{\prime \prime }}\propto {\sum }_{n=0}^{\infty }\left|n,n\right.⟩$$), we get the output state $$ {\left|\mathrm{\Psi }\right.⟩}_{a^{\prime },b^{\prime \prime }}\propto {\sum }_{n=0}^{\infty }\left|{\varphi }_n,n\right.⟩$$, which is precisely proportional to the output of a PDC when the signal and idler modes are initially in states $$ \left|m\right.⟩$$ and $$ \left|0\right.⟩$$, respectively (Eq. 19).

Fig. 5.

(A) PTR duality in a more general case with $$ m$$ and $$ n$$ photons impinging on modes $$ â$$ and $$ \widehat{b}$$ of a BS. By conditioning the output mode $$ {\widehat{b}}^{\prime }$$ on vacuum and reversing the arrow of time, we get the same transition probability amplitude (up to a constant) as for a PDC of gain $$ g=1/\eta $$ with input state $$ \left|m,0\right.⟩$$ and output state $$ \left|n+m,n\right.⟩$$. (B) Corresponding operational scheme using an entangled (EPR) state at the input of mode $$ \widehat{b}$$. This makes it possible to access the output retrodicted state on mode $$ {\widehat{b}}^{\prime \prime }$$.

Proof of PTR Duality.

The PTR duality is illustrated in Table 1 for few photons. As expressed by Eq. 7, it can be viewed as a consequence of partial transposition of the state of mode $$ \widehat{b}$$ (leaving mode $$ â$$ unchanged), namely the fundamental relation

${\left(U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}\right)}^{T_b}=\frac{1}{\sqrt{g}}\hspace{0.17em}{U}_{1/g}^{\mathrm{B}\mathrm{S}},$ [24]

where $$ T_b$$ stands for transposition in the Fock basis of the Hilbert space associated with mode $$ \widehat{b}$$. This is sketched in Fig. 6A and can also be interpreted by comparing the unitaries $$ U_{\eta }^{\mathrm{B}\mathrm{S}}$$ and $$ U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}$$ in Eqs. 10 and 12 or their corresponding decompositions, Eqs. 14 and 17. In general terms, we may say that the (passive) linear coupling of two bosonic modes is dual, under PTR, to an (active) Bogoliubov transformation, which is expressed as

$\left.⟨{\varphi }_a,{\varphi }_b\right|U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}\left|{\psi }_a,{\psi }_b\right.⟩=\frac{1}{\sqrt{g}}\hspace{0.17em}\left.⟨{\varphi }_a,{\psi }_b*\right|U_{1/g}^{\mathrm{B}\mathrm{S}}\left|{\psi }_a,{\varphi }_b*\right.⟩$ [25]

for any states $$ {\psi }_a$$, $$ {\psi }_b$$, $$ {\varphi }_a$$, and $$ {\varphi }_b$$, where $$ *$$ denotes the complex conjugation in the Fock basis. In Fig. 6B, the corresponding operational scheme is depicted, relying on the preparation of an entangled (EPR) state at the input of mode $$ \widehat{b}$$ and the projection onto an entangled (EPR) state at the output of mode $$ {\widehat{b}}^{\prime }$$.

Table 1.

Illustration of PTR duality with few photons

BS	PDC
$$ \left.⟨\mathrm{0,0}\right|U_{\eta }^{\mathrm{B}\mathrm{S}}\left|\mathrm{0,0}\right.⟩=1$$	$$ \left.⟨\mathrm{0,0}\right|U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}\left|\mathrm{0,0}\right.⟩=1/\sqrt{g}$$
$$ \left.⟨\mathrm{1,0}\right|U_{\eta }^{\mathrm{B}\mathrm{S}}\left|\mathrm{1,0}\right.⟩=\sqrt{\eta }$$	$$ \left.⟨\mathrm{1,0}\right|U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}\left|\mathrm{1,0}\right.⟩=1/g$$
$$ \left.⟨\mathrm{0,1}\right|U_{\eta }^{\mathrm{B}\mathrm{S}}\left|\mathrm{0,1}\right.⟩=\sqrt{\eta }$$	$$ \left.⟨\mathrm{0,1}\right|U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}\left|\mathrm{0,1}\right.⟩=1/g$$
$$ \left.⟨\mathrm{0,1}\right|U_{\eta }^{\mathrm{B}\mathrm{S}}\left|\mathrm{1,0}\right.⟩=-\sqrt{1-\eta }$$	$$ \left.⟨\mathrm{0,0}\right|U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}\left|\mathrm{1,1}\right.⟩=-\sqrt{g-1}/g$$
$$ \left.⟨\mathrm{1,0}\right|U_{\eta }^{\mathrm{B}\mathrm{S}}\left|\mathrm{0,1}\right.⟩=\sqrt{1-\eta }$$	$$ \left.⟨\mathrm{1,1}\right|U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}\left|\mathrm{0,0}\right.⟩=\sqrt{g-1}/g$$

The second column (PDC) is obtained from the first column (BS) by timereversing mode $$ \widehat{b}$$, substituting $$ \eta $$ with $$ 1/g$$ and dividing by the factor$$ \sqrt{g}$$. The first row explains the latter factor: vacuum is obviously conserved in a BS, while PDC implies the stimulated emission of photon pairs (hence, the probability of keeping vacuum is strictly lower than one). The second and third rows correspond to the transmission of a single photon through the BS or PDC. The fourth and fifth rows correspond to the reflection of a single photon by the BS or the stimulated annihilation (fourth row) or emission (fifth row) of a photon pair by PDC.

Fig. 6.

(A) General statement of the PTR duality as expressed in Eq. 7, when $$ i$$ and $$ j$$ photons impinge on input modes $$ â$$ and $$ \widehat{b}$$ of a BS, while $$ n$$ and $$ m$$ photons are detected in output modes $$ {â}^{\prime }$$ and $$ {\widehat{b}}^{\prime }$$. We recover a PDC with input $$ \left|i,m\right.⟩$$ and output $$ \left|n,j\right.⟩$$. (B) Corresponding operational scheme, where a PDC is emulated with a BS. The input state $$ \left|{\psi }_a\right.⟩$$ of mode $$ â$$ simply propagates through the BS and is projected onto $$ \left|{\varphi }_a\right.⟩$$. The input state $$ \left|{\psi }_b\right.⟩$$ of mode $$ {\widehat{b}}^{‴}$$ is converted, by projection on an EPR state, into the retrodicted state $$ \left|{}_{\psi }^b\right.⟩$$ of mode $$ {\widehat{b}}^{\prime }$$, which back propagates through the BS and is projected onto $$ \left|{}_{\varphi }^b\right.⟩$$ in mode $$ \widehat{b}$$. We obtain the corresponding output state $$ \left|{\varphi }_b\right.⟩$$ on mode $$ {\widehat{b}}^{\prime \prime }$$ by using an EPR pair. We thus recover a PDC with input $$ \left|{\psi }_a,{\psi }_b\right.⟩$$ and output $$ \left|{\varphi }_a,{\varphi }_b\right.⟩$$, as encapsulated by Eq. 25.

We now prove PTR duality by reexpressing Eq. 24 in the Heisenberg picture, namely

$\begin{array}{cc}\hfill V^{T_a}\hspace{0.17em}â\hspace{0.17em}{V}^{T_b}& =\left(â\hspace{0.17em}\cos \theta +\widehat{b}\hspace{0.17em}\sin \theta \right)/g,\hfill \\ \hfill V^{T_a}\hspace{0.17em}\widehat{b}\hspace{0.17em}{V}^{T_b}& =\left(\widehat{b}\hspace{0.17em}\cos \theta -â\hspace{0.17em}\sin \theta \right)/g,\hfill \end{array}$ [26]

where $$ V\equiv U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}$$. Note that $$ â$$ and $$ \widehat{b}$$ are real operators in Fock basis; hence, the same is true for $$ V$$. Therefore, we have $$ {â}^{†}=a^T$$, $$ {\widehat{b}}^{†}=b^T$$, and $$ V^{†}=V^T$$, where $$ T$$ stands for transposition in Fock basis. Obviously, we have $$ {\left(V^{T_b}\right)}^T=V^{T_a}$$. By using the identity $$ {\left(\left(A\otimes B\right)M\right)}^{T_b}=\left(A\otimes \mathbb{1}\right)M^{T_b}\left(\mathbb{1}\otimes B^T\right)$$, where $$ \mathbb{1}$$ represents the identity operator, we express $$ {â}_{\mathrm{P}\mathrm{T}\mathrm{R}}\equiv V^{T_a}\hspace{0.17em}â\hspace{0.17em}{V}^{T_b}$$ as

$\begin{array}{ll}\hfill V^{T_a}\hspace{0.17em}{\left(â\hspace{0.17em}V\right)}^{T_b}& =V^{T_a}\hspace{0.17em}{\left(V{V}^{†}â\hspace{0.17em}V\right)}^{T_b}\hfill \\ \hfill & =V^{T_a}\hspace{0.17em}{\left[V\left(â\hspace{0.17em}\cosh r+{\widehat{b}}^{†}\sinh r\right)\right]}^{T_b}\hfill \\ \hfill & =V^{T_a}{V}^{T_b}\hspace{0.17em}â\hspace{0.17em}\cosh r+V^{T_a}\widehat{b}\hspace{0.17em}{V}^{T_b}\hspace{0.17em}\sinh r.\hfill \end{array}$ [27]

Equivalently, using $$ {\left(\left(A\otimes B\right)M\right)}^{T_a}=\left(\mathbb{1}\otimes B\right)M^{T_a}\left(A^T\otimes \mathbb{1}\right)$$, we may also reexpress $$ {â}_{\mathrm{P}\mathrm{T}\mathrm{R}}$$ as

$\begin{array}{ll}\hfill {\left({â}^{†}V\right)}^{T_a}\hspace{0.17em}{V}^{T_b}& ={\left(V{V}^{†}{â}^{†}V\right)}^{T_a}\hspace{0.17em}{V}^{T_b}\hfill \\ \hfill & ={\left[V\left({â}^{†}\cosh r+\widehat{b}\hspace{0.17em}\sinh r\right)\right]}^{T_a}\hspace{0.17em}{V}^{T_b}\hfill \\ \hfill & =â\hspace{0.17em}{V}^{T_a}{V}^{T_b}\cosh r+V^{T_a}\widehat{b}\hspace{0.17em}{V}^{T_b}\hspace{0.17em}\sinh r.\hfill \end{array}$ [28]

Defining the operators $$ {\widehat{b}}_{\mathrm{P}\mathrm{T}\mathrm{R}}\equiv V^{T_a}\hspace{0.17em}\widehat{b}\hspace{0.17em}{V}^{T_b}$$, $$ W\equiv V^{T_a}{V}^{T_b}$$, and identifying Eq. 27 with Eq. 28, we see that

${â}_{\mathrm{P}\mathrm{T}\mathrm{R}}=â\hspace{0.17em}W\cosh r+{\widehat{b}}_{\mathrm{P}\mathrm{T}\mathrm{R}}\hspace{0.17em}\sinh r,\left[W,â\right]=0.$ [29]

We can perform a similar development starting from $$ {\widehat{b}}_{\mathrm{P}\mathrm{T}\mathrm{R}}$$, resulting in

${\widehat{b}}_{\mathrm{P}\mathrm{T}\mathrm{R}}=\widehat{b}\hspace{0.17em}W\cosh r-{â}_{\mathrm{P}\mathrm{T}\mathrm{R}}\hspace{0.17em}\sinh r,\left[W,\widehat{b}\right]=0.$ [30]

Now, solving Eqs. 29 and 30 for $$ {â}_{\mathrm{P}\mathrm{T}\mathrm{R}}$$ and $$ {\widehat{b}}_{\mathrm{P}\mathrm{T}\mathrm{R}}$$, we get

$\begin{array}{c}\hfill {â}_{\mathrm{P}\mathrm{T}\mathrm{R}}=â\hspace{0.17em}W\cos \theta +\widehat{b}\hspace{0.17em}W\sin \theta ,\\ \hfill {\widehat{b}}_{\mathrm{P}\mathrm{T}\mathrm{R}}=\widehat{b}\hspace{0.17em}W\cos \theta -â\hspace{0.17em}W\sin \theta ,\end{array}$ [31]

where we have made the substitutions $$ \cosh r={\left(\cos \theta \right)}^{-1}$$ and $$ \sinh r=\tan \theta $$.

Similar equations can be derived starting from $$ V^{T_a}\hspace{0.17em}{â}^{†}\hspace{0.17em}{V}^{T_b}$$ and $$ V^{T_a}\hspace{0.17em}{\widehat{b}}^{†}\hspace{0.17em}{V}^{T_b}$$, which imply that the operator $$ W$$ also commutes with $$ {â}^{†}$$ and $$ {\widehat{b}}^{†}$$. Hence, $$ W$$ is a scalar (proportional to $$ \mathbb{1}$$), and it is sufficient to compute its diagonal matrix element in an arbitrary state (e.g., $$ \left|\mathrm{0,0}\right.⟩$$). Recalling that $$ V^{†}=V^T$$, we have $$ W={\left(V^{†}\right)}^{T_b}{V}^{T_b}$$, so that

$\begin{array}{ll}\hfill \left.⟨\mathrm{0,0}\right|W\left|\mathrm{0,0}\right.⟩& =\sum _{k,l=0}^{\infty }\left.⟨\mathrm{0,0}\right|{\left(V^{†}\right)}^{T_b}\left|k,l\right.⟩\hspace{0.17em}\left.⟨k,l\right|V^{T_b}\left|\mathrm{0,0}\right.⟩\hfill \\ \hfill & =\sum _{k,l=0}^{\infty }\left.⟨0,l\right|V^{†}\left|k,0\right.⟩\hspace{0.17em}\left.⟨k,0\right|V\left|0,l\right.⟩\hfill \\ \hfill & =|\hspace{0.17em}\hspace{0.17em}\left.⟨\mathrm{0,0}\right|V\left|\mathrm{0,0}\right.⟩{|}^2\hfill \\ \hfill & =1/g.\hfill \end{array}$ [32]

Substituting $$ W$$ with $$ 1/g$$ in Eq. 31, we get Eq. 26, which concludes the proof of Eq. 24.

Note that PTR duality can be reexpressed by using the identity

$\begin{array}{cc}\hfill & \mathrm{T}\mathrm{r}\left[U_{ab}\left({\widehat{X}}_a\otimes {\widehat{X}}_b\right)U_{ab}^{†}\left({Ŷ}_a\otimes {Ŷ}_b\right)\right]\hfill \\ \hfill & =\mathrm{T}\mathrm{r}\left[U_{ab}^{T_b}\left({\widehat{X}}_a\otimes {Ŷ}_b^T\right){\left(U_{ab}^{T_b}\right)}^{†}\left({Ŷ}_a\otimes {\widehat{X}}_b^T\right)\right],\hfill \end{array}$ [33]

which is valid for any joint unitary $$ U_{ab}$$ and for any operators $$ {\widehat{X}}_{a\left(b\right)}$$ and $$ {Ŷ}_{a\left(b\right)}$$ acting on mode $$ a$$ ($$ b$$). Plugging $$ U_{ab}=U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}$$ into Eq. 33 and using Eq. 24 implies the general relation

$\begin{array}{cc}\hfill & \mathrm{T}\mathrm{r}\left[U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}\left({\widehat{X}}_a\otimes {\widehat{X}}_b\right)U_g^{\mathrm{P}\mathrm{D}\mathrm{C}†}\left({Ŷ}_a\otimes {Ŷ}_b\right)\right]\hfill \\ \hfill & =\frac{1}{g}\hspace{0.17em}\mathrm{T}\mathrm{r}\left[U_{1/g}^{\mathrm{B}\mathrm{S}}\left({\widehat{X}}_a\otimes {Ŷ}_b^T\right)U_{1/g}^{\mathrm{B}\mathrm{S}†}\left({Ŷ}_a\otimes {\widehat{X}}_b^T\right)\right].\hfill \end{array}$ [34]

This equation is needed to interpret PTR in the context of the retrodictive picture of quantum mechanics.

Retrodictive Picture of Quantum Mechanics.

In the usual, predictive approach of quantum mechanics, one deals with the preparation of a quantum system followed by its time evolution and ultimately, its measurement. Specifically, one uses the prior knowledge on the state $$ {\widehat{\rho }}_j$$ (prepared with probability $$ p_j$$) in order to make predictions about the outcomes of a measurement $$ \left\{{\widehat{\mathrm{\Pi }}}_m\right\}$$. If the state evolves according to unitary $$ U$$ before being measured, Born’s rule provides the conditional probabilities $$ \mathbb{P}\left(m|j\right)=\mathrm{T}\mathrm{r}\left[U{\widehat{\rho }}_j{U}^{†}{\widehat{\mathrm{\Pi }}}_m\right]$$. In contrast, in the retrodictive approach of quantum mechanics (21), one postselects the instances where a particular measurement outcome $$ m$$ was observed, and one focuses on the probability of the preparation variable $$ j$$ conditionally on this measurement outcome. This can be interpreted as if the actually measured state had propagated backward in time to the preparer (Fig. 7A). Specifically, one associates a retrodicted state $$ {\widehat{\sigma }}_m$$ with the observed outcome $$ m$$ and makes retrodictions about the preparation by evolving $$ {\widehat{\sigma }}_m$$ according to $$ U^{†}$$ and applying a measurement $$ \left\{{\widehat{\mathrm{\Theta }}}_j\right\}$$, whose outcome $$ j$$ discriminates the prepared state $$ {\widehat{\rho }}_j$$. In the simplest case (to which we restrict here) where there is no a priori information about the source, one sets $$ {\sum }_j{p}_j\hspace{0.17em}{\widehat{\rho }}_j\propto \mathbb{1}$$. Then, by defining

${\widehat{\sigma }}_m=\frac{{\widehat{\mathrm{\Pi }}}_m}{\mathrm{T}\mathrm{r}\left[{\widehat{\mathrm{\Pi }}}_m\right]},{\widehat{\mathrm{\Theta }}}_j=p_j\hspace{0.17em}{\widehat{\rho }}_j,$ [35]

one recovers the expected properties of a state ($$ {\widehat{\sigma }}_m\ge 0$$, $$ \mathrm{T}\mathrm{r}\hspace{0.17em}{\widehat{\sigma }}_m=1$$) and of a measurement ($$ {\widehat{\mathrm{\Theta }}}_j\ge 0$$, $$ {\sum }_j{\widehat{\mathrm{\Theta }}}_j=\mathbb{1}$$). Now, applying Born’s rule to the retrodicted state $$ {\widehat{\sigma }}_m$$ having evolved according to $$ U^{†}$$ followed by a measurement $$ {\widehat{\mathrm{\Theta }}}_j$$, we get

$\mathrm{T}\mathrm{r}\left[U^{†}{\widehat{\sigma }}_{m}U{\widehat{\mathrm{\Theta }}}_j\right]=\frac{p_j\mathrm{T}\mathrm{r}\left[{\widehat{\rho }}_j{U}^{†}{\widehat{\mathrm{\Pi }}}_{m}U\right]}{{\sum }_j{p}_j\mathrm{T}\mathrm{r}\left[U{\widehat{\rho }}_j{U}^{†}{\widehat{\mathrm{\Pi }}}_m\right]}=\mathbb{P}\left(j|m\right),$ [36]

which is consistent with Born’s rule in the forward direction combined with Bayes’ rule.

Fig. 7.

(A) Predictive (Left) and retrodictive (Right) pictures, describing the same experiment where state $$ {\widehat{\rho }}_j$$ is prepared, evolves according to $$ U$$, and leads to measurement outcome associated with $$ {\widehat{\mathrm{\Pi }}}_m$$. The probability $$ \mathbb{P}\left(j|m\right)$$ can be written as resulting from the retrodictive state $$ {\widehat{\sigma }}_m$$ back propagating according to $$ U^{†}$$, followed by measurement $$ {\widehat{\mathrm{\Theta }}}_j$$. (B) Predictive (Left) and intermediate (Right) pictures for a bipartite system. In the latter picture, associated with PTR, the retrodictive state of subsystem $$ b$$ propagates backward in time, while the predictive state of subsystem $$ a$$ propagates forward in time. If $$ U_{ab}$$ is a unitary, $$ V_{ab}=U_{ab}^{T_b}$$ is not necessarily proportional to a unitary; however, it is the case when considering the BS vs. PDC duality.

The retrodictive picture can be successfully exploited in different situations [for example, to characterize the quantum properties of an optical measurement device (35)], but it is always used in lieu of the predictive picture. Here, we instead combine it with the predictive picture in order to properly define PTR duality and describe a composite system that is propagated partly forward and partly backward in time, as represented in Fig. 7 B, Right. Specifically, we consider a composite system prepared in a product state $$ {\widehat{\rho }}_i^a\otimes {\widehat{\rho }}_j^b$$, then undergoing a unitary evolution $$ U_{ab}$$ followed by a product measurement $$ \left\{{\widehat{\mathrm{\Pi }}}_n^a\otimes {\widehat{\mathrm{\Pi }}}_m^b\right\}$$. In the fully predictive picture (Fig. 7 B, Left), the conditional probabilities are given by

$\mathbb{P}\left(n,m|i,j\right)=\mathrm{T}\mathrm{r}\left[U_{ab}\left({\widehat{\rho }}_i^a\otimes {\widehat{\rho }}_j^b\right)U_{ab}^{†}\left({\widehat{\mathrm{\Pi }}}_n^a\otimes {\widehat{\mathrm{\Pi }}}_m^b\right)\right].$ [37]

In our intermediate picture, we postselect the instances where a particular measurement outcome $$ m$$ was observed in subsystem $$ b$$ when subsystem $$ a$$ was prepared in state $$ {\widehat{\rho }}_i^a$$ and consider the joint probability of the preparation variable $$ j$$ of subsystem $$ b$$ together with the measurement outcome $$ n$$ of subsystem $$ a$$. Bayes’ rule yields

$\begin{array}{ll}\hfill \mathbb{P}\left(n,j|i,m\right)& =\frac{\mathbb{P}\left(i\right)\hspace{0.17em}\mathbb{P}\left(j\right)\hspace{0.17em}\mathbb{P}\left(n,m|i,j\right)}{\mathbb{P}\left(i\right){\sum }_{n^{\prime }{j}^{\prime }}\mathbb{P}\left(j^{\prime }\right)\hspace{0.17em}\mathbb{P}\left(n^{\prime },m|i,j^{\prime }\right)}\hfill \\ \hfill & =\frac{p_j^b\hspace{0.17em}\mathbb{P}\left(n,m|i,j\right)}{{\sum }_{n^{\prime }{j}^{\prime }}{p}_{j^{\prime }}^b\hspace{0.17em}\mathbb{P}\left(n^{\prime },m|i,j^{\prime }\right)},\hfill \end{array}$ [38]

where $$ p_j^b$$ is the probability that subsystem $$ b$$ is prepared in state $$ {\widehat{\rho }}_j^b$$. Similarly as before, without information about the source, we set $$ {\sum }_j{p}_j^b\hspace{0.17em}{\widehat{\rho }}_j^b\propto \mathbb{1}$$. We associate a retrodicted state $$ {\widehat{\sigma }}_m^b$$ with the observed outcome $$ m$$ on subsystem $$ b$$, while subsystem $$ a$$ is still considered in the initial state $$ {\widehat{\rho }}_i^a$$. Similarly, we make retrodictions about the preparation of subsystem $$ b$$ by applying measurement $$ \left\{{\widehat{\mathrm{\Theta }}}_j^b\right\}$$, whose outcome $$ j$$ discriminates the prepared state $$ {\widehat{\rho }}_j^b$$, while still measuring subsystem $$ a$$ according to $$ \left\{{\widehat{\mathrm{\Pi }}}_n^a\right\}$$. Using identity (33) and defining

${\widehat{\sigma }}_m^b=\frac{{\left({\widehat{\mathrm{\Pi }}}_m^b\right)}^T}{\mathrm{T}\mathrm{r}\left[{\widehat{\mathrm{\Pi }}}_m^b\right]},{\widehat{\mathrm{\Theta }}}_j^b=p_j^b\hspace{0.17em}{\left({\widehat{\rho }}_j^b\right)}^T,$ [39]

which are easily seen to behave as a proper state and measurement, we may reexpress Eq. 38 as

$\mathbb{P}\left(n,j|i,m\right)=\mathrm{T}\mathrm{r}\left[V_{ab}\left({\widehat{\rho }}_i^a\otimes {\widehat{\sigma }}_m^b\right)V_{ab}^{†}\left({\widehat{\mathrm{\Pi }}}_n^a\otimes {\widehat{\mathrm{\Theta }}}_j^b\right)\right],$ [40]

where $$ V_{ab}\equiv U_{ab}^{T_b}$$. This can be viewed as the evolution of state $$ {\widehat{\rho }}_i^a\otimes {\widehat{\sigma }}_m^b$$ according to $$ V_{ab}$$, followed by the measurement of $$ {\widehat{\mathrm{\Pi }}}_n^a\otimes {\widehat{\mathrm{\Theta }}}_j^b$$. In other words, in Eq. 40, the predictive picture is used for subsystem $$ a$$, while the retrodictive picture is used for subsystem $$ b$$.

In our analysis of a BS under PTR, we have $$ U_{ab}=U_g^{\mathrm{P}\mathrm{D}\mathrm{C}}$$ and $$ V_{ab}=U_{1/g}^{\mathrm{B}\mathrm{S}}/\sqrt{g}$$, so that Eq. 33 reduces to Eq. 34. Hence, $$ V_{ab}$$ is unitary (up to a constant) and can be interpreted as the propagation of the retrodicted state of mode $$ \widehat{b}$$ backward in time through the BS, while the predictive state of mode $$ â$$ normally propagates forward in time through the BS. According to Eq. 40, the joint state is then shown to evolve according to a PDC. Note that it is not always possible to construct an operator $$ V_{ab}$$ that is proportional to a unitary operator, as it is the case here.