1.4.1. Interference of identical particles from entanglement to boson-sampling[9]

This tutorial introduces the physics of many-boson and many-fermion interference required for the description of current experiments and for the understanding of novel approaches to quantum computing.

This tutorial presents an illustrative introduction to the physics of many-boson and many-fermion interference, which supplies the reader the necessary tools to describe photonic entanglement manipulation and novel approaches to optical quantum computing.

1.4.1.1. Interference of two particles

1.4.1.1.1. Distinguishable particles: uncorrelated binomial behaviour

Any incoming particle is reflected with probability R and transmitted with probability $T = 1 -R$. An event with $s_1$ and $s_2$ particles in the first and second output mode, respectively, occurs with probability $P_D (s_1 , s_2 )$. In particular, for a balanced setup, $T = R = 1/2$, the distribution of distinguishable particles in the output modes follows a binomial, and $$P_D (2, 0) = \frac{1}{4} , P_D (1, 1) = \frac{1}{2} , P_D (0, 2) = \frac{1}{4}$$

1.4.1.1.2. Identical particles: correlation without interaction

The creation operator of a particle in input mode $j$ is denoted by $\hat{a}^\dagger_j$ , such that the initial state reads $$|\chi_{\text{ini}}^{(2)}\rangle =\hat{a}_1^\dagger\hat{a}_2^\dagger|\text{vac}\rangle$$

where $|\text{vac}\rangle$ denotes the vacuum state. Throughout this tutorial,the letter $\chi$ denotes Fock-states, i.e. $|\chi\rangle = |n_1 , n_2 , \cdots\rangle _{1,2,\cdots}$ is interpreted as mode occupation $n_1 , n_2$ , etc. The letters $\psi$ and $\phi$ are reserved for qudits controlled by observing parties; $|\phi\rangle= |\phi_1 , \phi_2 ,\cdots\rangle$ then denotes the state in which $|\phi_1\rangle$ is observed by the first party, $|\phi_1\rangle$ by the second, and so on.

The time-evolution of particle creation operators in the Heisenberg picture reads accordingly $$\hat{a}_j^\dagger\to\hat{U}\hat{a}_j^\dagger\hat{U}^{-1}=i\sqrt{R}\hat{b}_j^\dagger+\sqrt{T}\hat{b}_{3-j}^\dagger$$ the single-particle wavefunction acquires a phase-shift of $i = e ^{i\pi/2}$ upon reflection. The single-particle time-evolution can be inserted into the initial state, and, for a balanced beam splitter $(R = T = 1/2)$, the final state becomes $$|\chi^{(2)}_{fin}\rangle=\frac{1}{2}(\text{i}(\hat{b}_1^\dagger)^2+\text{i}(\hat{b}_2^\dagger)^2-\hat{b}_1^\dagger\hat{b}_2^\dagger+\hat{b}_2^\dagger\hat{b}_1^\dagger)|\text{vac}\rangle$$ Fermions $(\hat{b}_j^\dagger)^2=0$ for fermionic creation operators. The final state becomes $$|\chi^{(2)}_{fin,F}\rangle=\hat{b}_2^\dagger\hat{b}_1^\dagger|\text{vac}\rangle=|1,1\rangle_{1,2}$$ from which we immediately read off $$P_F (2, 0) = 0, P_F (1, 1) = 1, P_F (0, 2) = 0$$

The Pauli principle prohibits the double population of any output mode, which is compatible with the constructive collective interference that enhances the probability to find the particles in different modes.

Bosons $$P_F (2, 0) = \frac{1}{2}, P_F (1, 1) = 0, P_F (0, 2) = \frac{1}{2}$$ Bosonic bunching and the Pauli principle are not the consequence of coherently superimposed many-particle paths, but they are rather kinematic boundary conditions on the state-space of identical particles.

Collective many-particle interference can only affect many-particle observables

1.4.1.1.3. Two-particle distinguishability transition

Two partially distinguishable bosons

As a realistic example for such transition, we consider photons and choose the time of arrival at the beam splitter as the degree of freedom through which the indistinguishability of the light quanta may be affected. In quantum optics experiments , the relative delay in the arrival time is manipulated by the spatial displacement of the wave-packets to the beam splitter.

The resulting event probability for partially distinguish- able particles becomes the average of the probabilities assigned to bosons and to distinguishable particles, weighted by $|c_{2,1}|^2$and $|c_{2,2}|^2$ , respectively: $$P_T(\vec{s}=(s_1,s_2),x)=|c_{2,1}|^2P_B(s_1,s_2)+|c_{2,2}|^2P_D(s_1,s_2)$$ To model typical experiments with photons , a Gaussian frequency distribution around a central frequency $\omega_0$ with width $\Delta \omega$ is assumed, $$|t_j\rangle=\int\mathop{}\!\mathrm{d}\omega\frac{1}{\pi^{1/4}\Delta\omega^{1/2}}e^{-\frac{(\omega-\omega_0)^2}{2\Delta\omega^2}}e^{i\omega t_j}|\omega\rangle$$ The temporal overlap of two photons then becomes $$\begin{align} \langle t_j|t_k\rangle & =\frac{1}{\pi^{1/2}\Delta\omega}\iint\mathop{}\!\mathrm{d}\omega_1\mathop{}\!\mathrm{d}\omega_2 e^{-\frac{(\omega_1-\omega_0)^2+(\omega_2-\omega_0)^2}{2\Delta\omega^2}}e^{i\omega_1 t_j+i\omega_2 t_k}\langle\omega_1|\omega_2\rangle \\ & =\frac{1}{\pi^{1/2}\Delta\omega}\int\mathop{}\!\mathrm{d}\omega_2 e^{-\frac{(\omega_2-\omega_0)^2}{\Delta\omega^2}}e^{i\omega_2 (t_j+t_k)} \\ & =e^{i(t_k-t_j)\omega_0}e^{-\frac{(t_j-t_k)^2}{4}\Delta\omega^2} \end{align}$$

The visibility of the HOM-dip constitutes today the defacto standard for ensuring the indistinguishability of two bosonic particles in the experiment.

By post-selecting events with a well-defined total number of photons N, i.e. by neglecting all final events in which the total number of detected photons does not match the desired N, one can effectively project with high fidelity onto the component with a total of N light quanta in the initial state.

1.4.1.1.4. Bipartite entanglement generation

Generation of classical correlations

the outcomes of the measurements are determined by the distribution of the objects among the observers, i.e. before the detection actually takes place.

Quantum correlations via propagation and detection.

$$|\chi^{2ent}_{fin}\rangle=\frac{1}{2}(\text{i}(\hat{b}_{2,0}^\dagger\hat{b}_{1,1}^\dagger+\hat{b}_{1,0}^\dagger\hat{b}^\dagger_{2,1})+\hat{b}_{2,0}^\dagger\hat{b}_{1,1}^\dagger-\hat{b}_{1,0}^\dagger\hat{b}_{2,1}^\dagger)|\text{vac}\rangle$$

By post-selecting those events with exactly one particle per spatial output mode, i.e. coincident events, we can effectively project the state onto $$|\chi^{2ent}_{fin}\rangle=\frac{\sqrt{2}}{2}(\hat{b}_{2,0}^\dagger\hat{b}_{1,1}^\dagger-\delta\hat{b}_{1,0}^\dagger\hat{b}_{2,1}^\dagger)|\text{vac}\rangle$$ where $\delta = +1$ for bosons and $\delta= −1$ takes into account the anti-commutation relation of fermionic creation operators.

Partial distinguishability and degradation of entanglement.

If the particles prepared in the input modes do not differ only in the degree of freedom in which entanglement is to be created, but also in another physical property, the correlations measured at the output modes will eventually degrade to purely classical ones. In practice, this deteriorating degree of freedom is often the time of arrival.

The indistinguishability $|c_{2,1}|^2$ directly quantifies the many-body coherences of the emerging state and, thus, the quantum nature of the induced correlations. Indeed, a quantitative indicator for entanglement, the concurrence $$C=|c_{2,1}|$$ directly reflects the indistinguishability

Relying on propagation and detection for entanglement manipulation also leads to natural constraints on operations. For example, a measurement that fully discriminates all four Bell states is impossible, which inhibits deterministic teleportation, for which nonlinear interactions are necessary.

1.4.1.2. Many-particle interference

1.4.1.2.1. Many-particle evolution and transition probabilities

Distinguishable particles

$$P_D(\vec{r},\vec{s};U)=\sum_{\sigma\in S_{\vec{d}(\vec{s})}}\prod^N_{j=1}p_{d_j(\vec{r},\sigma(j))}$$

indistinguishable particles

For indistinguishable particles, the argument can be repeated. Now, however, the exchange of two particles in the output modes does not change the quantum state (up to a global sign change for fermions), such that all possibilities to distribute the particles among the output modes contribute coherently to the final state. As a consequence, many-body amplitudes of the form $U_{d_1 (\vec{r} ),d_1 (\vec{s} )}U_{d_2 (\vec{r} ),d_2 (\vec{s} )}\cdots U_{d_N (\vec{r} ),d_N (\vec{s} )}$ need to be summed $$P_B(\vec{r},\vec{s};U)=\frac{\prod_js_j!}{\prod_jr_j!}\left|\sum_{\sigma\in S_{\vec{d}(\vec{s})}}\prod^N_{j=1}U_{d_j(\vec{r},\sigma(j))}\right|^2$$

$$P_F(\vec{r},\vec{s};U)=\left|\sum_{\sigma\in S_{\vec{d}(\vec{s})}}\mathrm{sgn}(\sigma)\prod^N_{j=1}U_{d_j(\vec{r},\sigma(j))}\right|^2$$

where $\mathrm{sgn}(\sigma)$ takes into account the phase acquired upon exchange of two fermions in the output modes.

Coarse-grained statistical signatures

For Pauli arrangements, the difference between the average probability for bosons and fermions dies out when the density of particles N/n is decreased: in the dilute limit N/n → 0, bunched events with several particles per mode are a priori very unlikely, and their influence on the statistics vanishes

For fully bunched events for which one mode receives all particles, $\vec{s_b} = (0, . . . , 0, N, 0 . . . , 0)$, we can formulate an exact relation between the probabilities for distinguishable particles and bosons $$P_B(\vec{r},\vec{s_b};U)=\frac{N!}{\prod_jr_j!}P_D(\vec{r},\vec{s}_b;U)\geqslant P_D(\vec{r},\vec{s_b};U)$$ On the other hand, if all bosons are prepared in the same mode, the probabilities for bosons and distinguishable particles do not differ at all.

Determinants, permanents and computational complexity

In order to find a more compact expression for the probabilities, a characteristic $N \times N$-matrix can be defined as $$M_{j,k}=U_{d_j(\vec{r}),d_k(\vec{s})}$$ i.e. M contains those rows and columns of U that correspond to (possibly multiply) occupied in- and output modes, respectively. With M defined as such, we can write $$\begin{align} P_D(\vec{r},\vec{s};U)&=\frac{1}{\prod_{j=1}^ns_j!}\sum_{\sigma\in S(\{1,\dots,N\})}\prod^N_{k=1}|M_{k,\sigma(k)}|^2\\ &=\frac{1}{\prod_{j=1}^ns_j!}\text{perm}(|M|^2) \end{align}$$

where the function $\text{perm} (|M| ^2 )$is the permanent of the matrix $|M| ^2$ , for which the absolute-square is taken component-wise. For bosons, we find $$P_B(\vec{r},\vec{s};U)=\frac{1}{\prod_{j=1}^ns_j!}|\text{perm}(M)|^2 \tag{PB}$$ i.e. the permanent of the complex matrix $M$ needs to be computed before taking the absolute-square to obtain the probability. For fermions, the sign-change allows us to identify the total amplitude as a determinant, $$P_F(\vec{r},\vec{s};U)=|\text{det}(M)|^2$$ The evaluation of the determinant can be performed in time polynomial in the matrix size $N$.

For distinguishable particles, by sampling over many randomly chosen permutations $\sigma$ , the permanent can be estimated.

Boson-sampling

It is tempting to state that quantum mechanics makes it possible to 'compute the permanent of a complex matrix with exponential speedup with respect to classical computers' using bosons that propagate through a multimode setup. However, even though the amplitudes of final states characterized by $\vec{s}$ correspond to the permanent of a certain matrix $M$, these amplitudes are very small, which makes them difficult to extract. Although it is possible to define an observable whose expectation value coincides with the permanent, its large variance requires an exponential number of measurements to extract the permanent experimentally . This caveat jeopardizes any strategy to directly compute the permanent via the scattering of bosons, although there are current attempts in this direction.

Can we trust a boson-sampling device?

An alternative approach to certifying boson-sampling devices is to leave the space of computationally complex problems: instead of choosing the scattering matrix randomly according to the unitary Haar-measure (which is a crucial assumption for the very hardness of the problem ), one can artificially design test-cases for which pertinent properties of the resulting probability distribution are obtained efficiently on a classical computer. In other words, the boson-sampling device is assessed by a task whose exact or approximate solution is known beforehand.

One approach is based on processes with well-understood average behaviour, such as many-particle quantum walks.In other words, although bosonic statistical effects witness features that go beyond the uncorrelated behaviour of distinguishable particles, the observation of these effects is nevertheless insufficient to strictly guarantee a behaviour in accordance to equation (PB).

A falsifiable instance of boson-sampling: the Fourier suppression law

The superior computational complexity of many-boson scattering in comparison to the analogous problem with fermions is rooted in the absence of symmetries of the permanent compared to the more benevolent determinant.

Our strategy to devise a solvable instance of boson-sampling therefore relies on imposing artificial symmetries on the scattering matrix $U$, which then allows us to exactly compute the permanent of the pertinent submatrices in a large number of cases.