Direct estimation of quantum coherence by collective measurements

The recently established resource theory of quantum coherence allows for a quantitative understanding of the superposition principle, with applications reaching from quantum computing to quantum biology. While different quantifiers of coherence have been proposed in the literature, their efficient estimation in today's experiments remains a challenge. Here, we introduce a collective measurement scheme for estimating the amount of coherence in quantum states, which requires entangled measurements on two copies of the state. As we show by numerical simulations, our scheme outperforms other estimation methods based on tomography or adaptive measurements, leading to a higher precision in a large parameter range for estimating established coherence quantifiers of qubit and qutrit states. We show that our method is accessible with today's technology by implementing it experimentally with photons, finding a good agreement between experiment and theory.

Having identified quantum coherence as a valuable feature of quantum systems, it is important to develop methods for its rigorous quantification. First attempts for resource quantification were made in the resource theory of entanglement [52,53], leading to various entanglement measures based on physical or mathematical considerations. The common feature of all resource quantifiers is the postulate that they should not increase under free operations of the theory, which in entanglement theory are known as "local operations and classical communication". In the resource theory of coherence, the free operations are incoherent operations, corresponding to quantum measurements which cannot create coherence for individual measurement outcomes [1].
While various coherence measures have been proposed [6], an important issue is how to efficiently estimate them in experiments. Clearly, one possibility is to perform quantum state tomography [54] and then use the derived state density matrix to estimate the amount of coherence. However, estimation of coherence measures in general does not require the complete information about the state of the system, a fact which has been exploited in various approaches for detecting and estimating coherence of unknown quantum states [55][56][57][58].
In this paper, we put forward a general method to directly measure quantum coherence of an unknown quantum state using two-copy collective measurement scheme (CMS) [59][60][61][62]. We simulate the performance of this method for qubits and qutrits and compare the precision of CMS with other methods for coherence estimation, including tomography. The simulations show that in certain setups CMS outperforms all other schemes discussed in this work. We also report an experimental demonstration of CMS for qubit states. The collective measurements are performed on two identically prepared qubits which are encoded in two degrees of freedom of a single photon. In this way, we experimentally obtain two widely studied coherence measures, finding a good agreement between the numerical simulations and the experimental results.

Theoretical framework
We aim to estimate coherence of a quantum state ρ by performing measurements on two copies of the state. As arXiv:2001.01384v1 [quant-ph] 6 Jan 2020 quantifiers of coherence we use the 1 -norm of coherence C 1 and the relative entropy of coherence C r , defined as [1] Here, S(ρ) = −Tr[ρ log 2 ρ] is the von Neumann entropy, ρ diag = i |i i|ρ|i i|, and we consider coherence with respect to the computational basis {|i }. For single-qubit states with Bloch vector r = (r x , r y , r z ), both quantities can be expressed as [63] with the binary entropy h(x) = −x log 2 x−(1−x) log 2 (1−x) and the Bloch vector length r = (r 2 x + r 2 y + r 2 z ) 1/2 . In the next step, we will express both C 1 and C r in terms of the outcome probabilities of a collective measurement in the maximally entangled basis, performed on two copies ρ ⊗ ρ. We denote the corresponding outcome probabilities as are projectors onto maximally entangled states |ψ ± = (|01 ± |10 )/ √ 2 and |ϕ ± = (|00 ± |11 )/ √ 2. As we show in Supplement A, the outcome probabilities fulfill the relations r 2 x + r 2 y = 2(P 1 − P 2 ), (6a) Thus, both coherence measures C 1 and C r can be expressed as simple functions of P i . We further note that in general CMS can be used to estimate absolute values of the Bloch vector components of a single-qubit state ρ. This implies that CMS allows to evaluate any coherence measure of single-qubit states, as any such measure is a function of the absolute values of the Bloch coordinates, see Supplement A for more details.
In the following, we use numerical simulation to compare the collective measurement scheme (CMS) discussed above to three alternative schemes for measuring C 1 for single-qubit states. The first alternative scheme is to directly measure observables σ x and σ y , and estimate C 1 via Eq. (3). The second scheme is a two-step adaptive measurement: step one is to measure observables σ x and σ y ; based on the feedback results of the first step, step two is to choose optimal observable aσ x + bσ y to obtain | 0|ρ|1 |. The third alternative scheme is to perform  and relative entropy coherence in (b) for a family of qubit states with different measurement schemes. The states have the form |Ψ = sin θ|0 + cos θ|1 with θ ranging from 0 to π/2. In (a), the performances of CMS (numerical simulation and experiment); σ x , σ y measurement (simulation); two-step adaptive measurements (simulation); and tomography (simulation) are shown for comparison. In (b), the performances of CMS (numerical simulation and experiment) and tomography (simulation) are shown for comparison. The sample size is N = 1200. Each data point is the average of 1000 repetitions, and the error bar denotes the standard deviation. state tomography and then, subject to the derived density matrix, to estimate the value of C 1 . We further use the tomography results to estimate the relative entropy of coherence C r via Eq. (4), and compare the performance of the estimation with CMS.
For the numerical simulation we use single-qubit states with θ ranging from 0 to π/2. All simulations are performed on N = 1200 copies of |Ψ . We further repeat each simulation 1000 times and average the numerical data over all repetitions. We are in particular interested in the error of the estimation: where C est and C are the estimated and the actual coherence measures, respectively. Fig. 1(a) shows the results of numerical simulation for C 1 , together with experimental data; the experiment will be discussed in more detail below. Each data point in the figure is the average of T = 1000 repetitions, i.e., 1 where ε i is the error of the ith measurement. The error bar denotes the standard deviation of ε i . Fig. 1(b) shows the corresponding comparison between CMS and tomography for estimating the relative entropy of coherence C r .
As we see from the data shown in Fig. 1(a,b), there is a range of θ where CMS outperforms all other schemes, leading to the smallest error. Moreover, while the error in general depends on θ, this dependence is comparably weak for CMS. To compare the accuracy achieved by different estimation methods more intuitively and clearly, we average the mean error for all input states shown in Fig. 1(a), and the average results are shown in Fig. 2. For the estimation of C 1 the adaptive measurement scheme outperforms CMS on average, which again outperforms all other estimation schemes presented above. In Supplement B we further report theoretical and experimental results for estimating coherence of formation [3,64] for qubits. Also in this case CMS outperforms all other schemes discussed in this paper in a certain range of θ.
While the above discussion was restricted to qubit systems, the CMS method can also be applied to estimate C 1 for states of higher dimensions. We consider an arbitrary quantum state ρ = i, j ρ i j |i j|, where i, j = 0, 1, ..., d−1 and d is the dimension of Hilbert space. After an appropriate set of collective measurements are performed on the two-copy state ρ ⊗ ρ, we find that the absolute value of  the off-diagonal element |ρ i j | for i j can be expressed as where |ψ ± i j = (|i j ± |ji )/ √ 2. Therefore, the 1 -norm coherence can be written as We use numerical simulation to compare the performance of the CMS method to the qutrit state tomography (see Supplement C) for the family of qutrit states with α ranging from 0 to π/2. The results of the simulation are shown in Fig. 3. As before, we use N = 1200 copies of the state |Φ for both CMS and state tomography, and average over 1000 repetitions. The results show that CMS outperforms the tomography method for a large range of α. Apart from a higher accuracy, the CMS method requires only a single measurement setup, while four measurement setups are required for qutrit tomography.

Experimental implementation
The experimental setup for realizing CMS to estimate coherence of qubit states is presented in Fig. 4. The setup is composed of three modules designed for single-photon source, two-copy state preparation, and collective measurements, respectively. In the single-photon source module, a 80-mW cw laser with a 404-nm wavelength (linewidth=5 MHz) pumps a type-II beamlike phasematching beta-barium-borate (BBO, 6.0×6.0×2.0 mm 3 , θ = 40.98 • ) crystal to produce a pair of photons with wavelength λ = 808 nm. The two photons pass through two interference filters (IF) whose FWHM (full width at half maximum) is 3 nm. The photon pairs generated in spontaneous parametric down-conversion (SPDC) are coupled into single-mode fibers separately. One photon is detected by a single-photon detector acting as a trigger. In the two-copy state preparation module, we first prepare copy 1 in the path degree of freedom of single photon, i.e., the first qubit encoded in positions 1 and 0 (see a in Fig. 4). After passing a half-wave plate (HWP) and a quarter-wave plate (QWP) with deviation angles H 1 , Q 1 , the photon is prepared in the desired state ρ.
To encode the polarization state into the path degree of freedom, beam displacer (BD 1 ) is used to displace the horizontal polarization (H) component into path 0, which is 4-mm away from the vertical polarization (V) component in path 1; then a HWP (H 3 ) with deviation angle 45 • is placed in path 0. The resulting photon is described by the state ρ ⊗ |V V|. Then we encode the second copy of ρ into the polarization degree of freedom of single photon using a HWP and a QWP with devia-tion angles H 2 , Q 2 (see b in Fig. 4). In this way, we can prepare the desired two-copy state ρ ⊗ ρ. The collective measurement module realizes a measurement on ρ ⊗ ρ in the maximally entangled basis, where M i are given in Eq. (5). When estimating the 1norm coherence, only the probabilities of the outcomes M 1 and M 2 are used, see the discussion below Eq. (3). The probabilities of all outcomes are used for estimating the relative entropy of coherence, see Eq. (4). To verify the experimental implementation of the collective measurement, we take the conventional method of measuring the probability distributions after preparing the input states |ψ + , |ψ − , |ϕ + and |ϕ − . These input states can be prepared by choosing proper rotation angles H 1 , Q 1 , H 2 , H 3 as specified in Supplement D. Each input state is prepared and measured 5000 times, and the probability of obtaining the outcomes M 1 , M 2 , M 3 and M 4 are 0.9981 ± 0.0006, 0.9973 ± 0.0007, 0.9962 ± 0.0009 and 0.9961 ± 0.0009, respectively (ideal value is 1). The theoretical values of other probability distributions for the input states are all 0, experimentally the maximum error of other probability is 0.0037 ± 0.0009.
The experimental deterministic realization of the collective measurement allows us to estimate the amount of coherence with a single measurement setup. We experimentally investigate the error achieved by CMS when the input states |Ψ have the form (7) with θ ranging from 0 to π/2. The sample size of the experiment is N = 1200 copies of |Ψ ; same sample size has been used in the numerical simulations reported above. As in the numerical simulation, we average over 1000 repetitions of the experiment. The experimental results for the estimation precision of C 1 and C r are shown in Fig. 1(a) and Fig. 1(b), respectively. The experimental data is in good agreement with the theoretical prediction.

Discussion
We introduce a general method to directly measure quantum coherence of an unknown quantum state using two-copy collective measurement, focusing on two established coherence quantifiers: 1 -norm coherence and relative entropy coherence. As we demonstrate by numerical simulation for qubit and qutrit states, in a certain parameter region the collective measurement scheme outperforms other estimation techniques, including methods based on adaptive σ x , σ y measurement for qubits, and tomography-based coherence estimation for qubits and qutrits. We test our results by experimentally estimating the 1 -norm coherence and relative entropy coherence of qubit states by collective measurements in optical setup, finding good agreement between theory and experiment. For single-qubit states our method allows to estimate absolute values of the Bloch coordinates, implying that any coherence quantifier of a qubit can be estimated with the collective measurement scheme.
Although the precision achieved by our method is not always better than by adaptive measurement, our scheme has several advantages with respect to other techniques. In particular, our method does not need any optimization procedures or feedback, which are required for coherence estimation via adaptive measurements. Moreover, the entire experiment can be performed in a single measurement setup. Thus, our work provides a simple method to measure coherence, and highlights the application of collective measurement in quantum information processing.

A. Estimating general coherence measures for qubits with collective measurements
For a single-qubit state ρ with Bloch vector r = (r x , r y , r z ) the probabilities P i = Tr[M i ρ ⊗ ρ] are given explicitly as It thus follows that collective measurements can be used to evaluate absolute values of the Bloch coordinates: From these results, it is straightforward to verify Eqs. (6) of the main text. In the following, C(r x , r y , r z ) will denote a coherence measure for a qubit state ρ with Bloch vector r = (r x , r y , r z ). As we will now show, for single-qubit states any coherence measure C depends only on the absolute values of the Bloch vector coordinates. For this, it is enough to show that C(r x , r y , r z ) = C(−r x , r y , r z ) = C(r x , −r y , r z ) = C(r x , r y , −r z ) (S3) for any coherence measure C and any Bloch vector. This can be seen by noting that the vector (r x , r y , r z ) can be transformed into the vector (−r x , r y , r z ) via a rotation around the z-axis, which corresponds to an incoherent unitary operation. Since any coherence measure is invariant under incoherent unitaries, it follows that C(r x , r y , r z ) = C(−r x , r y , r z ). By similar arguments we obtain C(r x , r y , r z ) = C(r x , −r y , r z ). Moreover, note that σ x is an incoherent unitary inducing the transformation (r x , r y , r z ) → (r x , −r y , −r z ), and thus it must be that C(r x , r y , r z ) = C(r x , −r y , −r z ). Combining these arguments completes the proof of Eq. (S3).

B. Estimating coherence of formation for qubits
We will now apply the collective measurement scheme to estimate the coherence of formation, which for singlequbit states takes the form [64] where is the binary entropy. We perform numerical simulations and an optical experiment, following the same procedure as for estimating C 1 . In particular, we use N = 1200 copies of the state with θ ∈ [0, π/2], and average the data over 1000 repetitions. The results of the numerical simulation and experiment are shown in Fig. S1. Also in this case we compare the numerical simulation of CMS with three alternative schemes for coherence estimation, see main text for details.

C. Qutrit state tomography
The density matrix of an unknown qutrit state is reconstructed by performing projective measurements in 4 mutually unbiased bases. In the following, vectors |ξ i j form an orthonormal basis for a given i, i.e., ξ i j |ξ ik = δ jk , and are mutually unbiased for different i: 3 for i k. The vectors can be explicitly given as follows: |ξ 00 = |0 , |ξ 01 = |1 , |ξ 02 = |2 , For numerical simulation of coherence estimation via qutrit state tomography reported in the main text (see Fig. 3), we use the sample size N = 1200, and repeat the procedure 1000 times to infer the state from the results of the measurement. Here we use the method of maximumlikelihood reconstruction presented in Ref. [65]. In the state preparation module, H 1 , H 2 , H 3 , Q 1 , Q 2 are the rotation angles of the HWPs and QWPs shown in Fig. 4. In the preparation of states |ψ + , |ψ − , |ϕ + , |ϕ − , the QWP corresponding to Q 2 is removed. Details of the angles of wave plates used to prepare all states are given in Table S1.