Quantum asymmetry and noisy multi-mode interferometry

Superposition of quantum states is a fundamental principle of quantum mechanics and the ability to create and preserve coherent superpositions is the essential prerequisite for the realization of all kinds of quantum technologies. In recent years, the basic intuition that quantum superposition is a valuable and fragile resource has been formalized within the mathematical framework of quantum resource theories [1, 2, 3]. The mathematical notion of coherence relies on the decomposition of the Hilbert space of the system into subspaces; usually the orthogonal eigenspaces of an observable. This is actually a subcase of the more general resource theory of quantum asymmetry [4, 5], where the resource is the degree to which a quantum state breaks a certain symmetry, defined in terms of a Lie group. Quantum asymmetry has been recognized as the relevant physical resource in a variety of operational settings: reference frame alignment [6, 7, 8], quantum thermodynamic tasks [9, 10, 11], quantum speed limits [12, 13], assessing macroscopic quantumness [14, 15, 16], and, most importantly for this work, quantum metrology [17, 18, 5, 19]. In this framework, the coherence of a quantum state with respect to the eigenspaces of an observable $G$ corresponds to the asymmetry with respect to the one-parameter group of translations $e^{\mathrm{i}\theta G}$; in the following, the term quantum asymmetry will be used to refer to this specific notion.

In this work, we focus on a qutrit example, which is the simplest case where a non-trivial generator $G$ with a degenerate eigenvalue exists. Our main result is to show that the sensitivity to the phase $\theta$, which also quantifies the $G$-asymmetry, is increased for probe states that are more dephased in the degenerate subspace, thus effectively “noisier”, in a sense that will be made more precise. Furthermore, we also show that a similar, yet less pronounced, behavior appears when the dephasing channel $\mathcal{E}$ is used for entanglement distribution.

We study the phenomena in terms of a physical realization of the system as a three-arm interferometer and we complement our theoretical analysis with a single-photon optical experiment that confirms our predictions.

While quantum coherence theory is intimately related with multiple phase interferometry [20, 21, 22, 23, 24, 25, 26], in this work we focus on the estimation of a single phase $\theta$, imprinted onto a signal mode, while the phases of the other reference modes fluctuate, generalizing the paradigmatic phase diffusion model [27, 28, 29, 30, 31, 32, 33]. Interestingly, this interferometric point of view will shed light on the apparently unintuitive effect observed in the abstract description: we will show that an increased dephasing in the degenerate subspace is caused by less correlated fluctuations of the reference phases, which allows to decrease the measurement noise without altering the signal.

Asymmetry and the Quantum Fisher Information. Quantum resource theories, with the entanglement theory as the most famous example [34], arise naturally when a set of quantum states can be regarded as free and any state outside of it as a resource [2]. This immediately defines also free operations, i.e. quantum channels that cannot transform free states into resources. In the resource theory of asymmetry the starting point is the representation of a symmetry group. In particular, we consider a finite-dimensional representation of $U(1)$: $U_{\theta}=e^{-\mathrm{i}\theta G}$, where $G=\sum_{k}E_{k}|k\rangle\langle k|$ is the Hermitian generator, $\ket{k}$ are its eigenstates and $E_{k}$ its eigenvalues, which can be degenerate. Resource states $\mathcal{G}$ are those not invariant under the action of $U_{\theta}$ and free operations those commuting with $U_{\theta}$. If $G$ is nondegenerate, $\mathcal{G}$ is the set of incoherent states with no off-diagonal elements in the eigenbasis; otherwise, all superpositions of eigenstates of the same degenerate eigenvalue are also free.

There are inequivalent ways to quantify the asymmetry of a quantum state, called asymmetry monotones, which must not increase under free operations ¹¹1More precise definitions can be found elsewhere [1, 2] and are not needed for our discussion.. When $U_{\theta}$ is regarded as a $\theta$-parameter encoding operation on a quantum state, the quantum Fisher information (QFI) of the state is an asymmetry monotone  [19]. It is defined as $\mathcal{F}_{\theta}(\rho)=\operatorname{Tr}[\rho_{\theta}L_{\theta}^{2}]$ where $\rho_{\theta}=U_{\theta}\rho U_{\theta}^{\dagger}$ and the symmetric logarithmic derivative (SLD) operator $L_{\theta}$ is defined as $2\frac{d\rho_{\theta}}{d\theta}=L_{\theta}\rho_{\theta}+\rho_{\theta}L_{\theta}$. The QFI quantifies the sensitivity of $\rho$ to the imprinted phase $\theta$, since the quantum Cramér-Rao bound (CRB) states that the variance of any unbiased estimator $\tilde{\theta}$ is lower bounded as [36, 37, 38] $\Delta^{2}\tilde{\theta}\geq\frac{1}{\nu\mathcal{F}_{\theta}(\rho)}$, where $\nu$ is the number of repetitions and the bound is saturable for large $\nu$.

Qutrit model. Consider a three-level system (qutrit) and the generator $G=E_{1}|1\rangle\langle 1|+E_{2}\mathopen{}\mathclose{{}\left(|2\rangle\langle 2|+|3\rangle\langle 3|}\right)$, where we have fixed an orthonormal basis $\{\ket{2},\ket{3}\}$ of the degenerate eigenspace, singled out by the dephasing channel $\mathcal{E}$, which describes a decrease of coherence between the two eigenspaces by a factor $\eta$ and between $\{\ket{2},\ket{3}\}$ by a factor $\kappa$:

The channel is physical (a completely positive map) provided

we restrict to real-valued $\eta$ and $\kappa$ without loss of generality.

We consider the phase parameter $\theta$ unitarily encoded by $G$, i.e. $\mathcal{E}_{\theta}(\rho)=e^{-\mathrm{i}\theta G}\mathcal{E}(\rho)e^{\mathrm{i}\theta G}$ and to study the effect of dephasing we focus on a pure probe state $\rho_{0}=|\psi_{0}\rangle\langle\psi_{0}|$ that is a superpostion of states from different eigenspaces: $\ket{\psi_{0}}=\sqrt{q}\ket{1}+\sqrt{\frac{1-q}{2}}(\ket{2}+\ket{3})$, with $0\leq q\leq 1$. The corresponding QFI is

and quantifies both the asymmetry of the state $\mathcal{E}(\rho)$ and its phase sensitivity. Details on the calculation are in [39].

As expected, the QFI is a monotonically increasing function of $\eta$. However, for any value of $\eta$ and $q$ it is a decreasing function of $\kappa$. This means that more dephasing in the degenerate subspace gives higher sensitivity to the imprinted phase $\theta$, despite the state becoming in some sense noisier—decreasing $\kappa$ from $1$ (no dephasing) to a value $\kappa\geq 0$ always decreases its purity $\operatorname{Tr}\rho^{2}$. While this result appears counter-intuitive, we stress that it does not violate the monotonicity property of the QFI. According to (2), a channel with $\eta=1$ and $\kappa<1$ is not physical. Thus, it is not possible to see the two decoherence effects parametrized by $\eta$ and $\kappa$ as the action of separate channels.

Since $\mathcal{E}$ and $e^{-\mathrm{i}\theta G}$ commute, the overall channel $\mathcal{E}_{\theta}$ also describes a physical process in which noise and parameter encoding happen simultaneously. While the QFI quantifies the asymmetry of the noisy state $\mathcal{E}(\rho)$, from a quantum metrology point of view one assumes that the channel $\mathcal{E}_{\theta}$ is given and optimizes the probe state. The so-called channel QFI $\max_{\rho}\mathcal{F}\mathopen{}\mathclose{{}\left(\mathcal{E}_{\theta}(\rho)}\right)=8\eta^{2}\mathopen{}\mathclose{{}\left(\kappa-2\sqrt{2}\sqrt{\kappa+1}+3}\right)/(\kappa-1)^{2}$ is obtained for $q=q_{\text{opt}}=1/\mathopen{}\mathclose{{}\left(\sqrt{2/(\kappa+1)}+1}\right)$; apart from the $\kappa=1$ case, the optimal state is not a balanced superposition.

Our results are not due to a peculiar behaviour of the QFI; we have also evaluated the relative entropy of asymmetry [40, 7, 8, 41] and observed a qualitatively analogous behaviour, with the maximal asymmetry achieved for a slightly different value of $q$. However, the simplest and in some sense the most intuitive measure, i.e, the sum of the trace norm of the modes of asymmetry [17], does not depend on $\kappa$ and hence would not reveal the effect.

Interferometric realization. We now show a simple physical explanation of the apparently counterintuitive result. Consider a three-mode interferometer, as depicted in Fig. 1. A single photon is prepared in a three mode superposition by the action of two beam splitters, creating the initial state $\ket{\psi_{0}}$. Then the phase $\theta$ is imprinted on the first signal mode, while the second and third reference modes do not have stable phases but are subject to random phase fluctuations $\phi_{1}$ and $\phi_{2}$, with zero mean and a joint probability distribution $P(\phi_{1},\phi_{2})$. The simplest choice is given by correlated “phase kicks”, i.e. $\phi_{1}$ and $\phi_{2}$ fluctuate to two equal and opposite values $\pm\phi_{0}\in[-\frac{\pi}{2},\frac{\pi}{2}]$ with probabilities $P(\phi_{0},\phi_{0})=P(-\phi_{0},-\phi_{0})=\frac{1}{4}(1+c)$ and $P(\phi_{0},-\phi_{0})=P(-\phi_{0},\phi_{0})=\frac{1}{4}(1-c)$, where $c=\mathbb{E}[\phi_{1}\phi_{2}]/\sqrt{\mathbb{E}[\phi_{1}^{2}]\mathbb{E}[\phi_{2}^{2}]}$ is the correlation coefficient, ranging from perfectly correlated $c=1$ to perfectly anticorrelated $c=-1$ and $\mathbb{E}$ denotes the expectation with respect to $P(\phi_{1},\phi_{2})$. These parameters are related to the previous ones as

as the action of the dephasing channel corresponds to $\mathcal{E}(\star)=\mathbb{E}\mathopen{}\mathclose{{}\left[e^{-\mathrm{i}(\phi_{2}|2\rangle\langle 2|+\phi_{3}|3\rangle\langle 3|)}\star e^{\mathrm{i}(\phi_{2}|2\rangle\langle 2|+\phi_{3}|3\rangle\langle 3|)}}\right]$. The parameter $\phi_{0}$ represents the magnitude of the phase kicks and is directly linked to the parameter $\eta$, which we will keep using for convenience. The parameter $\kappa$ on the other hand, depends both on $\phi_{0}$ and on the the correlation coefficient $c$, and the whole range of physical values of $\kappa$ in (2) can be obtained by varying $c$. It is also possible to obtain the same qutrit channel $\mathcal{E}$ if $P(\phi_{1},\phi_{2})$ is a bivariate Gaussian distribution, similarly to the standard phase diffusion model in a Mach-Zehnder (MZ) interferometer [27, 29, 42]. The qualitative picture remains the same: the less correlated (i.e. more anti-correlated) $\phi_{1}$ and $\phi_{2}$ are, the higher the phase sensitivity is. However for $c<0$ the effect is less pronounced, since this noise model does not reproduce the full range of physical $\kappa$ in Eq. (2), as detailed in Sec. B of [39], where we also present experimental results for this model.

In Fig. 1 we also show the optimal detection scheme that saturates the QFI, formally a projective measurement on the eigenstates of the SLD $L_{\theta_{0}}$: $\frac{1}{\sqrt{2}}\ket{1}\pm\mathrm{i}\frac{e^{-\mathrm{i}\theta_{0}}}{2}\mathopen{}\mathclose{{}\left(\ket{2}+\ket{3}}\right)$ and $\frac{1}{\sqrt{2}}\mathopen{}\mathclose{{}\left(\ket{2}-\ket{3}}\right)$. Here $\theta_{0}$ is the working point around which we estimate small variations of the parameter. Setting $\theta_{0}=0$ for concreteness, the probabilities of registering the photon at one of the three detectors read:

where we have also introduced an additional visibility parameter $0\leq v\leq 1$ that takes into account imperfect interference—see Sec. C of [39] for a complete derivation.

If $v=1$ and $\kappa=1$, detector ‘3’ never clicks ($p_{3}=0$). This corresponds to perfectly correlated phases fluctuations, $c=1$, and results in an effective standard qubit dephasing model (modes $2$ and $3$ can be regarded as a single mode). When $\kappa$ is decreased, which corresponds to moving from the perfectly correlated noise ($c=1$) via uncorrelated ($c=0$) to the perfectly anti-correalted one ($c=-1$), detector ‘3’ filters-out the ‘anti-correlated’ part of the phase noise. Doing so, it effectively increases the visibility of the interference pattern observed in detectors ‘1’ and ‘2’ (note that the $\theta$ dependent ‘fringe-terms’ $f_{\theta}$ in $p_{1}$ and $p_{2}$ do not depend on $\kappa$, see Eq. (5)):

and leads to a better phase sensitivity. The possibility of filtering out the anti-correlated part of the noise is the essence of the counter-intuitve behaviour of the QFI.

Experimental results. We have realized the proposed model in a photonic experiment that is schematically depicted in Fig. 1. The two reference arms of the three-arm interferometer are piezo-actuated which allow us to control both the phase to be measured $\theta$ and the zero-mean fluctuations $\phi_{1}$ and $\phi_{2}$, (see Sec. D of [39] for a detailed description of the physical implementation). The interferometer outputs are coupled to single mode fibers connected to superconducting nanowire single photon detectors (SNSPD) with coincidence counters. We feed the interferometer with heralded single-photons at 795 nm generated in a continuously driven non-degenerate (795 nm and 805 nm) SPDC source. We run the experiment in 80-second-long intervals and accumulate coincidences for various parameter settings $\{\theta,\phi_{0},c\}\in[0,2\pi]\times[0,1.16]\times[-1,1]$. To emulate the discrete $\pm\phi_{0}$ phase kicks we measure each $\{\phi_{1},\phi_{2}\}$ sub-setting separately and combine the results according to the given $c$. The compact form of the interferometer provides sufficient 15-minutes-long phase stability. To maintain long-time stability, each measurement interval is preceded by calibration step consisting of scanning the actuators and observing intensity fringes (without heralding) at the interferometer outputs. The calibration step not only allows us to stabilize $\theta$ value but also provides information about the intrinsic visibility $v\approx 0.97$ and relative efficiencies $\chi$ in the three measurement ports $\chi(p_{2}/p_{1})\approx 0.95$, $\chi(p_{3}/p_{1})\approx 1.61$.

We first compare the measured coincidences distributions with the theoretical predictions given by Eq. (5). To recover the probabilities from measured counts we normalize the number of detected photons accounting for different net efficiencies at the interferometer output ports. In Fig. 2(a-d) we show examples of normalized counts $\tilde{N}$ measured for range of $\theta$ phases along with the expected curves corresponding to the same $\eta$ and $c$ parameters. From those we recover the visibilities that are shown on a common plot in Fig. 2(e). The solid lines represent the expected behavior as described by Eq. (6).

In order to validate the improvement of the phase estimation precision in presence of increasingly anti-correlated noise, we have implemented estimation procedure based on the efficient locally unbiased estimator (at the working point $\theta_{0}=0$):

where $N_{i}$ denotes the detected photon counts in the given output port. We evaluate the estimator on $10^{4}$ randomly prepared sets of measurement outcomes each containing $10^{5}$ samples in total. Each set is prepared by randomly sampling (with repetitions) from the $10^{5}$ measured single-photon counts at the $\theta_{0}=0$ point. The resulting single-photon estimation precision, defined as the inverse of the estimator variance divided by the total number of detected photons, is presented in Fig. 3. The errorbars correspond to chi-squared 99% confidence interval. The rescaling to single-photon value allows us to compare the achieved precision with the classical FI (solid lines), calculated for the model probabilities (Eq. (5)) that take into account the finite interference visibility $v$. The dashed lines represent the ultimate bound given by the QFI. In Fig. 3 we clearly see that the estimation precision improves for the anti-correlated noise as predicted by the FI and QFI.

Entanglement. A similar effect can also be observed for entanglement. Let us apply the channel $\mathcal{E}$ to one part of the entangled state $\ket{\psi}^{AB}=\sqrt{q}\ket{11}+\sqrt{\frac{1-q}{2}}(\ket{22}+\ket{33})$. The resulting state takes the form of a maximally correlated state [43]:

with parameters $\alpha_{11}=q$, $\alpha_{22}=\alpha_{33}=(1-q)/2$, $\alpha_{12}=\alpha_{21}=\alpha_{13}=\alpha_{31}=\eta\sqrt{q(1-q)/2}$, and $\alpha_{23}=\alpha_{32}=\kappa(1-q)/2$. In fact, these states can be regarded as the maximally correlated states associated to the qutrit states in Eq. (1). It is known that coherence properties of a quantum state are closely related to entanglement properties of the corresponding maximally correlated state [1, 44]. Following this analogy, it is reasonable to expect that the results presented above will also extend to entanglement of the states in Eq. (8). For this, we investigate distillable entanglement of $\rho^{AB}$, characterizing how many singlets can be extracted from the state $\rho^{AB}$ via local operations and classical communication in the asymptotic limit [45, 46]. While the distillable entanglement is hard to evaluate in general, a closed expression exists for states of the form (8): $E_{\mathrm{d}}(\rho^{AB})=S(\rho^{B})-S(\rho^{AB})$ with the von Neumann entropy $S(\rho)=-\operatorname{Tr}(\rho\log_{2}\rho)$. For this class of states only the the global entropy $S(\rho)$ depends on $\kappa$ and the distillable entanglement $E_{\mathrm{d}}(\rho^{AB})$ is equal to the relative entropy of coherence of the state in Eq. (1) with respect to the orthonormal basis $\{\ket{1},\ket{2},\ket{3}\}$ [1, 44]. We can intuitively think that there are two competing effects at play: reducing $\kappa$ decreases the coherence in the subspace spanned by $\ket{2}$ and $\ket{3}$, but at the same time increases coherence between this subspace and $\ket{1}$. For this reason, the effect is far less pronounced than with the QFI, but there is still a region of parameters where the entanglement increases when moving from a positive value of $\kappa$ to a smaller (yet positive) value. More details on the entanglement analysis are presented in Sec. F of [39].

Classical light. It is natural to wonder what would happen if instead of a single photon we used classical light, i.e. a coherent input state $\ket{\alpha}$. In this case we use a Gaussian phase fluctuation model $P(\phi_{1},\phi_{2})$, see details in Sec. B of [39]. We consider an estimation strategy, analogous to the standard MZ interferometer, based on the photon number difference $I^{-}=n_{1}-n_{2}$, optimal in the single-photon regime, and we evaluate the variance using error propagation: $\Delta^{2}\theta=\Delta^{2}I^{-}/\mathopen{}\mathclose{{}\left|\frac{d}{d\theta}\mathbb{E}\mathopen{}\mathclose{{}\left[\langle I^{-}\rangle}\right]}\right|^{2}$, where the $\Delta^{2}I^{-}$ includes also the contribution of the distribution $P(\phi_{1},\phi_{2})$ of the fluctuating reference phases. In Sec. D of [39] we show that

where the first terms contains the single-photon QFI (3) (with the $\eta$ and $\kappa$ now related to the parameters $\sigma$ and $c$ of the Gaussian noise model, see Eq. (B2) in [39]). This result implies that this strategy with classical light performs worse than repeating a single photon experiment many times, but for a weak coherent state we have the same sensitivity per average photon, i.e. $(|\alpha|^{2}\Delta^{2}\theta)^{-1}\stackrel{{\scriptstyle\alpha\to 0}}{{\longrightarrow}}\mathcal{F}_{\theta}$. For an intense beam the variance saturates to a constant value, as in standard phase diffusion [29, 31], and only vanishes for $c=-1$.

Finally, we can also consider a ‘naive classical’ strategy, without any interference between the two reference modes, which is equivalent to two separate MZ interferometers sensing the phase shifts $\theta-\phi_{1}$ and $\theta-\phi_{2}$. In this case one finds the same structure as in $\eqref{eq:cohVarTheta}$, but the first term corresponds to the QFI evaluated at $c=1$, i.e. it does not improve by reducing the correlations. Details are given in Sec. G.2 of Ref. [39].

Discussion. Starting with an abstract model of the loss of coherence within a degenerate subspace of the generator of a phase shift, we have discussed and demonstrated the nonintuitive effect of an increased sensing precision. By considering the optimal interferometric scheme we have related such an improvement to the possibility of filtering out the anti-correlated part of the dephasing noise. This effect may be of practical importance in interferometric schemes where one is able to cause the phase fluctuations affecting multiple reference modes to become uncorrelated (by e.g. separating them spatially) or even make them anticorrelated (if thermal fluctuations were responsible for dephasing, while the materials had opposite thermal and the temperature coefficient of the refractive index). Interestingly, the most significant gains from the effect can be obtained at the single photon level, and when the light from both the reference arms is interfered with each other before eventually interfering with the light from the signal arm. The results can be generalized to obtain qualitatively similar effects in case of $d$ reference modes instead of just $2$.

Acknowledgments. We thank Konrad Banaszek, Wojtek Górecki and Wojciech Wasilewski for fruitful discussions. F. A. and R. D. D. were supported by National Science Center (Poland) Grant No. 2016/22/E/ST2/00559. R. D. D. was additionally supported by the National Science Center (Poland) grant No. 2020/37/B/ST2/02134. M. M., M. L., A. S. and M. P. are supported by the Foundation for Polish Science “Quantum Optical Technologies” project (MAB/2018/4), carried out within the International Research Agendas programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund.

Supplemental Material

The Supplemental Material is organized as follows. In Sec. A we give a few more details on the calculations performed to obtain the results in the main text. In Sec. B we study the case of Gaussian phase fluctuations, mentioned in the main text, for which we have also performed an experiment. In Sec. C we derive detector clicks probabilities in the three-arm interferometer taking into account imperfect visibility. In Sec. D we give more details on the experimental implementation of the interferometer. In Sec.E we provide full statistics of obtained estimated values of the phase in case of both discrete and Gaussian noise models. In Sec. F we present the results mentioned in the main text about the behaviour of entanglement as a function of $\kappa$. In Sec. G we give more details on the calculations for classical light.