Quantification of Entanglement and Coherence with Purity Detection

Entanglement and coherence are fundamental properties of quantum systems, promising to power near future quantum technologies, such as quantum computation, quantum communication and quantum metrology. Yet, their quantification, rather than mere detection, generally requires reconstructing the spectrum of quantum states, i.e., experimentally challenging measurement sets that increase exponentially with the system size. Here, we demonstrate quantitative bounds to operationally useful entanglement and coherence that are universally valid, analytically computable, and experimentally friendly. Specifically, our main theoretical results are lower and upper bounds to the coherent information and the relative entropy of coherence in terms of local and global purities of quantum states. To validate our proposal, we experimentally implement two purity detection methods in an optical system: shadow estimation with random measurements and collective measurements on pairs of state copies. The experiment shows that both the coherent information and the relative entropy of coherence of pure and mixed unknown quantum states can be bounded by purity functions. Our research offers an efficient means of verifying large-scale quantum information processing.


INTRODUCTION
Entanglement is a fundamental trait of many-body quantum systems and a key resource for quantum information processing [1][2][3][4][5].Recently, theoretical methods to characterize quantum superpositions have been generalized to evaluate quantum coherence in single systems [6] and explore its uses for quantum technologies [7][8][9][10][11][12][13][14][15].Quantification of such resources provides insights on the true computational power of quantum devices [16][17][18][19][20], and many important measures are defined in terms of the von Neumann entropy S(ρ) = −Tr(ρ log ρ).Besides, the von Neumann entropy has been found widespread applications in quantum date compression [21], quantum thermodynamics [22], capacity bounds for quantum channels [23] and many-body physics, from the characterization of topological matter [24][25][26], to dynamics out of equilibrium [27], to the understanding of tensor network methods [28] (see Ref. [29] for a review).However, the quantification of von Neumann entropy is hard both theoretically and experimentally, as it necessitates knowledge of the full spectrum of the system state ρ.Clever methods that enable to witness entanglement and coherence employ randomized measurements [30][31][32][33] and collective detections on many copies of quantum states to extract spectrum polynomials, e.g., the state purity Tr(ρ 2 ) [34][35][36][37][38][39].Yet, these protocols cannot be easily applied to quantify entanglement and coherence: there are not measures of quantum resources which can be expressed in terms of directly observable (polynomial) quantities.
In this letter, we address this challenge by proposing an efficient approach to identify quantitative bounds to entanglement and coherence of unknown quantum states in terms of purity functions, in contrast to other protocols based on local measurements [40,41].We focus on the coherent information and relative entropy of coherence, which are both defined in terms of the von Neumann entropy and are information measures with compelling operational interpretations.The coherent information is related to the distillable entanglement and the capacity of quantum channels with applications in quantum communication, one-way entanglement distillation, quantum state merging and quantum many-body physics [30,[42][43][44][45][46][47][48][49].The relative entropy of coherence lower bounds the distillable coherence and plays an important role in quantum thermodynamics, quantum metrology, quantum computing, quantum random number generation and quantum phase transitions [50][51][52][53].We prove analytical upper and lower bounds on the coherent information and relative entropy of coherence of arbitrary finite dimensional quantum states in terms of their local and global purities, which are measurable without spectrum reconstruction [54].Then, we experimentally demonstrate our proposal in an optical system by implementing the randomized measurements scheme on four-qubit states, and collective measurements on two copies of two-qubit states.The experiment results confirm that operationally useful en-arXiv:2308.07068v2[quant-ph] 25 Jun 2024 tanglement and coherence of unknown quantum states can be quantified without spectrum reconstruction.

RESULTS
Our study has two main merits.First, it discovers simple analytical functions that quantify, rather than only witness, key quantum resources in arbitrary systems of finite dimension.Second, it shows an experimental comparison between the well-established interference-based method for non-tomographic exploration of quantum properties [55][56][57][58], and the recently introduced "shadow estimation" techniques [59][60][61][62].Together, our study provides a theoretically universal and practically efficient means to benchmark features of unknown quantum systems.
A. Quantification of coherent information.
For quantum states ρ AB ∈ H d A ⊗ H d B , the coherent information is defined by where A and B are subsystems and ρ B = Tr A (ρ AB ) is the reduced density matrix on subsystem B. A positive value of I(A⟩B) signals operationally useful entanglement between subsystems A and B [48].
Measuring I(A⟩B) requires knowledge of the eigenvalues of the density matrices.We propose a method to obtain upper and lower bounds on the von Neumann entropy in terms of the global and marginal purity of the state.Given the spectral decomposition of a d-dimensional quantum state, i.e., ρ = i,ρ , where the logarithm is written in base 2 [63].The spectrum We can immediately use these results to bound the coherent information as follows (see Supplementary Note 1 for details).
Result 1-Given a quantum state ρ AB , its coherent information I(A⟩B) is bounded as follows: where The lower and upper bounds is tight for pure states (P(ρ AB ) = 1) with P(ρ B ) = 1 d B and the difference ϵ e = P(ρ B ) − 1/d B certifies the tightness of u e (ρ) and l e (ρ).In a way similar to how non-factorizable superpositions of multipartite states, e.g.i c i |ii . . .i⟩, yield entanglement, the quantumness of a system can be identified with the degree of coherence of its state |ψ⟩ = i c i |i⟩ , i |c i | 2 = 1, in a reference basis {|i⟩}.One natural way to quantify the coherence of a state in a reference basis {|1⟩ , |2⟩ , . . ., |d⟩} of a d-dimensional Hilbert space H d is by measuring how far it is to the set of incoherent states I [64,65].The choice of distance function is in principle arbitrary.Yet, an important operational interpretation is enjoyed by the relative entropy of coherence [64] where ρ d = i |i⟩⟨i| ρ |i⟩⟨i| is the state after dephasing in the reference basis.In the asymptotic limit of infinite system preparations, C RE (ρ) represents the maximal rate of extraction of maximally coherent qubit states 1/2 i,j=0,1 |i⟩ ⟨j| from ρ by incoherent operations.Like the coherent information, this quantity is bounded by purity function (see Supplementary Note 1 for details).
Result 2-The relative entropy of coherence C RE (ρ) is bounded as follows: where This inequality chain, like the one in Eq. 2, is tight for pure states (P(ρ) = 1) with diagonal matrix of . ϵ e (ϵ c ) → 0 indicates the maximally entangled state (maximally coherent state), which is of particular interest in quantum information science.
C. Detecting purity with shadow estimation.
We first use shadow tomography [59,66,67] to detect the purity of the four-qubit biased Greenberger-Horne-Zeilinger (GHZ) states in the form of (7) which are encoded on the polarization and path degrees of freedom (DOF) of photons.As shown in Fig. 1a, the polarization-entangled photons are generated from a periodically poled potassium titanyl phosphate (PPKTP) crystal set at Sagnac interferometer.Then, we then sent two photons into two beam displacers (BDs) as shown in Fig. 1b, which transmits the vertical polarization and deviates the horizontal polarization.Consequently, the biased GHZ state |GHZ θ ⟩ is obtained, where h (v) denotes the deviated (transmitted) spatial mode.
We prepare eleven ρ GHZ θ by setting θ ∈ [0, π 2 ] with interval of π 20 , and then use M = 2 × 10 4 measurements in shadow estimation on each ρ GHZ θ to bound the coherent information I(A⟩B) of ρ GHZ θ .We consider the bipartition of ρ GHZ θ with two subsystems A and B, where A ∪ B = {1, 1 ′ , 2, 2 ′ } and A ∩ B = ∅.Each subsystem contains |A| and |B| qubits, respectively.We consider three cases of where The results of P(ρ GHZ θ ) and P(ρ B ) are shown in Fig. 2a.To indicate the accuracy of estimated purities, we perform standard quantum tomography (SQT) [68][69][70] on the prepared ρ GHZ θ with 1.4 × 10 6 measurements, and treat the reconstructed state as target state.With the reconstructed ρ GHZ θ , we calculate the corresponding purities that are shown with black frames in Fig. 2a.The maximal error between purities (Eq.8 and Eq. 9) estimated from classical shadows and SQT is ϵ = 0.0132 ± 0.0109.The high accuracy (ϵ ≪ 1) agrees well with the theoretical prediction that the measurement cost of shadow tomography is in the order of 2 |AB| /ϵ 2 [33], while the SQT requires (at least) an order of 2 |AB| rank(ρ AB )/ϵ 2 measurements to reach the same accuracy [71,72].According to Eq. 3, the lower bound l e and upper bound u e of I(A⟩B) can be calculated with the estimated purities, and the results are shown with orange and blue dots in Fig. 2b  is sensitive to noise (See Supplementary Note 2 for analyzations).The results of calculated I(A⟩B)) are shown with green dots in Fig. 2b-Fig.2d, in which we observe that I(A⟩B)) is well bounded by l e and u e expect θ = 6π/20 in Fig. 2b.Similar phenomena are also observed in Fig. 2a, where the estimation of P(ρ GHZ θ,d ) (green bars) are larger than the results from SQT.There are two main reasons attributed to these discrepancies.The first one is that the randomized measurement and SQT are performed separately, i.e., they are not obtained from the same copies of prepared ρ GHZ θ .
There are unavoidable noises such as the slight drifts of the mounts holding BDs, which would accordingly introduces errors in state preparation and detection.The second one is that we use maximal likelihood estimation (MLE) in SQT to return a physical state from collected data .MLE is a biased estimation which underestimates properties of unknown quantum state [73], while the shadow tomography we implemented is an unbiased estimation of purity [59].
To bound C RE (ρ GHZ θ ), we calculate the purity of the diagonal matrix of ρ GHZ by P(ρ GHZ θ,d ) =  2a.Thus, u c and l c are deduced with estimated P(ρ GHZ θ,d ) and P(ρ GHZ θ ) according to Eq. 6.As C RE ≥ 0, we set l c = 0 whenever it takes negative values.The results of the calculated u c and l c are shown with red and yellow triangles in Fig. 2e, in which one observes they tightly bound C RE (ρ GHZ θ ) from SQT (cyan squares).

D. Detecting purity with collective measurements.
The purity of a quantum state ρ can be indicated from two copies of ρ by P(ρ) = Tr(ρ 2 ) = Tr(Vρ ⊗ ρ) with V being the swap operation on ρ ⊗ ρ [74][75][76][77].The purity from collective measurement has been demonstrated to extract Renyi entropy for violation of entropic inequalities to witness entanglement [74].This Renyi quantity, while able to certify entanglement as it is an entanglement witness [78], does not quantify it.We consider the case of two-qubit state in the form of The swap operation on V on ρ ⊗ ρ can be implemented by performing Bell-state measurement (BSM) between each qubit and its corresponding copy [36, 79,80].In our case, the BSM is performed between the polarization-encoded qubit 1( 2) and the path-encoded qubit 1 ′ (2 ′ ) [81], respectively.The outcome probability of the two BSMs on ρ 12 ⊗ρ 1 ′ 2 ′ is denoted by The purity of ρ ψ 2,θ and the subsystem purity of ρ ψ 2,θ,B with B = {2} are then obtained by Similarly, the purity of the diagonal matrix of ρ ψ 2,θ can be obtained by The results of P(ρ ψ 2,θ ), P(ρ ψ 2,θ,B ) and P(ρ ψ 2,θ,d ) are shown in Fig. 3a, with θ ∈ [0, π 2 ] with interval of π 20 .The lower bound l e and upper bound u e of I(A⟩B) are calculated according to Eq. 3 and shown in Fig. 3b.We observe u e < 0 for all ρ ψ 2,θ , which indicates the prepared ρ ψ 2,θ is less useful for entanglement distillation.Similarly, the lower bound l c and upper bound u c of C RE (|ψ 2,θ ⟩) can be calculated according to Eq. 6.The results are shown in Fig. 3c.Note that l c is much closer to u c compared to the case in Fig. 2e.This is because the bounds l c and u c are functions of the leading order term (purity) in Taylor expansion of the von Neumann entropy about pure states, so that l c and u c are tight for pure states.Experimentally, the prepared ρ 12 and ρ 1 ′ 2 ′ are quite close to the ideal form of |ψ 2,θ ⟩, while ρ GHZ θ is much more noisy.The high accuracy of l c and u c is also confirmed by C RE (ρ 12 ) with reconstructed ρ 12 from SQT, which is shown with cyan dots in Fig. 3c.

DISCUSSION
We demonstrated universal and computable theoretical bounds to operationally meaningful measures of entanglement and coherence in terms of purity functionals.Then, we experimentally extracted these bounds by implementing two purity detection methods: shadow estimation and collective measurements.The experiment showed that quantum resources can be estimated, rather than just witnessed, with a precision that does not scale with the rank of the state (guaranteed by theory [33,59,71,72]), conversely to state tomography.The scalability of the measurement network makes purity detection employable in testing the successful preparation of quantum superpositions in large computational registers, certifying that a complex device has run a truly quantum computation.The proposed bounds are sufficiently tight for practically useful quantum states, i.e., the high-fidelity GHZlike states or maximally coherent states, which are important entanglement and coherence resources that are widely used in quantum information protocols.The bounds Eqs. ( 2) and ( 5) represent the leading order term in Taylor's expansion of the von Neumann entropy.Thus, tightened bounds for noisy states can be extracted by evaluating the higher-order terms Tr(ρ 3 ), Tr(ρ 4 ), . . ., Tr(ρ d ), which can be efficiently detected with hybrid shadow estimation [82,83].In particular, the bounds become strict when we include moments of the system dimension.It would be interesting for future work to study the tightness of the bounds for the intermediate cases.Another unexplored direction is that one can extend the method proposed here to determine directly measurable bounds to the total correlations in multipartite systems {A i }.For instance, consider the quantum analogue of the multi-information between random variables [84,85] It is easy to verify that the product of the state marginals i ρ Ai solves the minimization, I(ρ A1,...,An ) = i S(ρ Ai ) − S(ρ A1,...,An ).Quantitative bounds to the total system correlations in terms of purities are given by a straightforward generalization of Eq. 2.
Our work has important and wide practical applications in various fields in quantum computation, communication, quantum thermodynamics, quantum many-body physics, etc.The proposed method has an immediate application in benchmarking current and near-term quantum technologies and serves as a basic and useful tool for analyzing and optimizing practical implementations of quantum information protocols.We use a continuous-wave laser operating at a central wavelength of 405 nm with a full width at half maximum (FWHM) of 0.012 nm as our pump light source.The pump light passes through a PBS followed by an HWP set at θ/2, which transforms the polarization of the pump light into cos θ |H⟩ p + sin θ |V ⟩ p .The pump light passes PBS that transmits the component of |H⟩ and reflects component of |V ⟩.Then, the PPKTP crystal is coherently pumped from anticlockwise and clockwise directions respectively, and the generated photons are superposed on the PBS leading to the outcome state of cos θ |HV ⟩ 12 + sin θ |V H⟩ 12 .An HWP set at 45 • is applied on photon 2, which leads to a biased polarization-entangled state in form of cos θ |HH⟩ 12 + sin θ |V V ⟩ 12 .To enhance collective efficiency, we employ lens L1 with a focal length of 200 mm and lens L2 with a focal length of 250 mm.The two photons pass through narrowband filters (NBFs) with an FWHM of 3 nm and then are coupled into single-mode fibres.  m) } which is further exploited for the estimation of various properties of the underlying state ρ [30,33].The random unitary operations U n ∈ Cl 2 on the polarization and path DOF are implemented with a combination of electrical-controlled half waveplate (E-HWP) and quarter waveplate (E-QWP) [61], and the projective measurements on the Pauli-Z basis are sequentially performed on the polarization and path DOF (See Supplementary Note 2 for more details).
Intuitively, the vector λ that maximizes S is the one that spread as uniformly as possible; while the vector λ that minimizes S is the one that has the minimal number of nonzero large values.In the following, we will analytically solve this problem and confirm this intuition.

Maximization
First, we focus on the maximization problem with d = 3.Note that when d = 2, the solution to the constraints of Eq. ( 14) is unique and the optimization problem will be trivial.Without loss of generality, we assume λ 1 ≥ λ 2 ≥ λ 3 .Then the problem can be stated as We prove that the maximum is reached with the following Lemma.
Lemma 1.The solution to the maximization problem in Eq. ( 15) is given by Proof.The differential of the entropy function S(ρ) and the constraints are given by and respectively.We rewrite Eq. ( 18) to Thus, the differential of the entropy function becomes Since the function log λ is concave for λ Thus, dS(ρ)/dλ 3 ≥ 0. To reach the maximum of S(ρ), we thus only need to set λ 3 to be its maximum, which happens when λ 2 = λ 3 .Together with the constraints, then we can solve the equations and show that the solution to the maximization problem is given in Eq. ( 16).Now, we can solve the maximization problem of Eq. ( 14) for a general case of d.
Theorem 1. Suppose λ 1 ≥ λ 2 ≥ . . .λ d .The solution to the maximization problem in Eq. ( 14) is Proof.The solution in Eq. ( 20) is exactly determined when setting λ 2 = λ 3 = • • • = λ d .Suppose the maximization problem solution is not this one, then we must have that λ 2 > λ d .In the following, we prove the contradiction by showing that changing the values of λ 1 , λ 2 , λ d would make the entropy S(ρ) larger while fixing all other values (λ 3 , λ 4 , . . ., λ d−1 ) and the constraints.Now the constraints for λ 1 , λ 2 , and λ d becomes Proof.Suppose we always have the solution in the form as Otherwise, there must exist three λ i , λ j , λ k such that λ i > λ j ≥ λ k and λ k ̸ = 0. Following a similar argument in the proof of Theorem 1, we can show that this contradicts Lemma 2. According to Eq. ( 29), we have

Upper and lower bounds to coherence and entanglement
We now call {λ M i,ρ }, {λ m i,ρ } the vectors solving the maximization and the minimization, respectively.Given a bipartite state ρ AB , by minimizing (maximizing) the marginal purity on B subsystem and maximizing (minimizing) the global purity, one has Result 1 (Eq.(2) of the main text)-Given a quantum state ρ AB ∈ H d A ⊗ H d B , and defining ρ B = Tr A ρ AB , its coherent information I(A⟩B) is bounded as follows: l e (ρ AB ) ≤ I(A⟩B) ≤ u e (ρ AB ), (30) le(ρ AB ) = By minimizing (maximizing) the coherence of the dephased state ρ d = i |i⟩⟨i| ρ |i⟩⟨i|, and maximizing (minimizing) the coherence of the state under study, we obtain lower (upper) bounds to the relative entropy of coherence: Result 2 (Eq.( 5) of the main text) -The relative entropy of coherence C RE (ρ) is bounded as follows: ρ , we determine the extreme values of the state entropy S(ρ) = − d i=1 λ i,ρ log λ i,ρ at fixed purity P(ρ) := d i=1 λ 2

B.
Quantification of quantum coherence.
-Fig2d respectively.We observe that l e > 0 with θ = 3π 20 , 4π 20 , 5π 20 and 6π 20 , which indicates the corresponding ρ GHZ θ admits distillable entanglement.To investigate the tightness of lower and upper bounds of I(A⟩B), we calculate the I(A⟩B) with reconstructed ρ GHZ θ instead of theoretical predictions as I(A⟩B)

) CREFIG. 2 .
FIG. 2. Experimental results of quantification of I(A⟩B) and CRE on the prepared ρ GHZ θ by shadow estimation.a The estimation of global purity P(ρGHZ θ ), marginal purity P(ρB) and the purity of diagonal matrix P(ρGHZ θ,d ).The colored bars represent the results from shadow estimation, while the black frames represent the results from SQT for comparison.b − d The upper bound ue and lower bound le of I(A⟩B) with B = {1}, B = {1, 1 ′ } and B = {1, 1 ′ , 2} respectively.e The upper bound uc and lower bound lc of CRE(ρGHZ θ ).The error bars represent the statistical error by repeating shadow estimation for 10 times.

16 i=1 d 2 i
with d i being the diagonal elements of ρGHZ θ = M m=1 ρ(m) GHZ θ .The results of P(ρ GHZ θ,d ) are shown with green bars in Fig.

F.
Shadow tomography.In shadow tomography, local random unitary operations U n ∈ Cl 2 are individually applied on each qubit of an Nqubit state ρ, where Cl 2 is the single-qubit Clifford group.Then the rotated state is measured on the Pauli-Z basis, producing a bit string |b⟩ = |b 1 b 2 • • • b N ⟩ , b n ∈ {0, 1}.The classical shadow of a single experimental run is constructed by ρ = N n=1 3U † n |b n ⟩ ⟨b n | U n − I 2 with I 2 being identity matrix.By repeating the measurement M times, one has a collection of classical shadows {ρ

SUPPLEMENTARY NOTE 1 :
DERIVATION OF THE BOUNDS Given a quantum state ρ in a d-dimensional Hilbert space, our task is to bound the Von Neumann entropy of ρ with a function of the state purity P(ρ) := Tr(ρ 2 ).The spectral decomposition of the quantum state is ρ = d i=1 λ i |ψ i ⟩ ⟨ψ i |, where {|ψ i ⟩} forms an orthonormal basis of the d-dimensional Hilbert space.The variational problem is then formulated as max / min S(ρ) = − d i=1 λ i log(λ i )