Experimental hierarchy of two-qubit quantum correlations without state tomography

A Werner state, which is the singlet Bell state affected by white noise, is a prototype example of states, which can reveal a hierarchy of quantum entanglement, steering, and Bell nonlocality by controlling the amount of noise. However, experimental demonstrations of this hierarchy in a sufficient and necessary way (i.e., by applying measures or universal witnesses of these quantum correlations) have been mainly based on full quantum state tomography, corresponding to measuring at least 15 real parameters of two-qubit states. Here we report an experimental demonstration of this hierarchy by measuring only six elements of a correlation matrix depending on linear combinations of two-qubit Stokes parameters. We show that our experimental setup can also reveal the hierarchy of these quantum correlations of generalized Werner states, which are any two-qubit pure states affected by white noise.


INTRODUCTION
Quantum correlations reveal not only the strangeness of quantum mechanics, but are the main resources for quantum technologies, including quantum sensing and quantum information processing [1].Thus, the detection, control, and quantification of these resources is of paramount importance.
Among different types of correlations, a special interest has been paid to quantum entanglement [2], Einstein-Podolsky-Rosen (EPR) steering (also called quantum steering) [3,4], and Bell nonlocality that can be revealed by testing the violation of a Bell inequality [5].These types of correlations coincide for two-qubit pure states, but can be different for mixed states.Probably, the most intuitive distinction between these three types of quantum correlations for two systems (parties) can be given from a cryptographic perspective with the use of trusted and untrusted detectors.Specifically, according to Refs.[6,7]: (i) quantum entanglement can be revealed if both parties use only trusted detectors, (ii) EPR steering can be tested if one party uses trusted detectors and the other untrusted ones, and (iii) Bell nonlocality can be demonstrated if both parties use untrusted detectors.We experimentally determined and compared measures of these correlations for Werner states.
It is theoretically well known that by gradually adding noise to a pure state, one can reveal a hierarchy of different types of quantum correlations, including quantum entanglement, EPR steering, and Bell nonlocality.These effects are equivalent for two-qubit pure states, however they are in general different for mixed states.Werner [8] reported in 1989 that the singlet Bell state affected by white noise can be entangled without exhibiting Bell nonlocality, i.e., without violating any Bell inequality.It was further found that the Werner state with a proper amount of white noise can be entangled but unsteerable, or steerable without exhibiting Bell nonlocality, in addition to the trivial cases when a given state is Bell nonlocal (so also steerable and entangled) or separable (so also unsteerable and Bell local).Even a more refined hierarchy can be revealed by considering generalized Werner states, defined as mixtures of an arbitrary two-qubit pure states and white noise [9].Thus, the Werner and Werner-like states can be considered prototype examples of states indicating such a hierarchy.Their generation and the detection of their quantum correlations are a central topic of this paper.
Here we study a hierarchy of quantum correlations via their measures.We note that various other hierarchies of non-universal witnesses of quantum correlations have been investigated in detail.These include studies of sufficient conditions (i.e., nonuniversal witnesses) for observing specific types of correlations via matrices of moments of, e.g., the annihilation, creation, position, momentum, or Pauli operators.For example: (i) hierarchies of various witnesses of spatial [10] and spatiotemporal [11,12] correlations of bosonic systems revealing their nonclassicality via a nonpositive Glauber-Sudarshan P function; (ii) a hierarchy of entanglement witnesses [13,14] based on the Peres-Horodecki partial transposition criterion or their generalizations [15] using contraction maps (e.g., realignment) and positive maps (e.g., those of Kossakowski, Choi, and Breuer); (iii) a hierarchy of necessary conditions for the correlations that arise when performing local measurements on separate quantum systems, arXiv:2302.10159v2[quant-ph] 30 May 2023 which enabled finding a hierarchy of upper bounds on Bell nonlocality [16,17] (iv) a hierarchy of EPR steering witnesses [18] based on entanglement criteria with the constraint that measurement devices of one party cannot be trusted.Especially powerful methods for finding infinite hierarchies of quantum-correlation criteria are those formulated as semidefinite programs [16][17][18][19][20].Note that semidefinite programming has been found very effective in calculating not only nonuniversal witnesses but also measures (or universal witnesses) of quantum steering [3,21,22], Bell nonlocality [5], and entanglement [2].It could also be noted that a hierarchy of quantum nonbreaking channels, which is closely related to a hierarchy of temporal [23] and spatial quantum correlations, has been studied very recently both theoretically and experimentally in Ref. [24], where the effects of white noise (or, equivalently, a qubit-depolarizing channel) on quantum memory, temporal steerability [25,26], and nonmacrorealism were revealed by applying a full quantum process tomography.
The use of measures or universal witnesses of these quantum correlations, however, is required to demonstrate experimentally such hierarchies in a sufficient and necessary manner.For example, to our knowledge, no experiment has been performed to determine standard entanglement measures of a general two-qubit mixed state without full quantum state tomography (QST).These measures include the concurrence [27], which is a measure of the entanglement of formation, the negativity [28] quantifying the Peres-Horodecki entanglement criterion, and the relative entropy of entanglement [29].Thus, a QST-based approach to study a hierarchy of quantum correlations was applied in our former related study [9], as based on measuring 16 real parameters for two-qubit Werner states.
A hierarchy of quantum-correlation measures enables efficient estimations of one measure for a given value of another.More specifically, estimating the range of a measure of a given type of quantum correlation for a certain value of a measure (or bounds) of another type of quantum correlations were reported for arbitrary or specific classes of two-qubit states.These estimations include comparisons of: (i) entanglement and Bell nonlocality [30][31][32][33], (ii) steering and Bell nonlocality [34], as well as (iii) entanglement and steering [35].Note that such estimations can also be applied to compare nonequivalent measures describing the same type of correlations, including two-qubit entanglement [36] or singlequbit nonclassicality [37].Explorations of the relationships between measures of entanglement, steering, and Bell nonlocality for specific types of two-qubit states have been also attracting a considerable interest.Recent studies include, e.g., theoretical analyses of two-qubit Xstates [38] and two-mode Gaussian states [39].Experimental QST-based hierarchies of quantum entanglement, steering, and Bell nonlocality for specific classes of twoqubit states in relation to the above-mentioned estimations were also reported.These include experiments with mixtures of partially entangled two-qubit pure states [40] and GWSs based on full QST [9] or full quantum process tomography [24].A hierarchy of the latter states is also experimentally studied here but without applying a full QST.
We note that an experimental method for testing polarization entanglement without QST of general two qubit states was proposed in Ref. [41] based on measuring a collective universal witness of Ref. [42].However, the method, to our knowledge, has not been implemented experimentally yet.Another experimental approach to determine entanglement of a given state without QST can be based on measuring a bipartite Schmidt number, which satisfies various conditions of a good entanglement measure [43,44] and can be determined experimentally via a witnessing approach [45].However, it is not clear how the same method can also be used to experimentally determine also steering and nonlocality measures.Note that we want to apply a versatile experimental setup, which can be used to determine various measures of all the three types of quantum correlations.
Multiple indicators of quantum steering have been demonstrated experimentally (for a review see Ref. [4]).We note a very recent Ref. [46], where it was shown experimentally that a critical steering radius is the most powerful among practical steering indicators.Its scaling property allows classifying as steerable or non-steerable various families of quantum states.This approach is useful in testing theoretical concepts of the critical radius in real experiments prone to unavoidable noise.The authors used a setup introducing losses and measured elements of a correlation matrix to determine the steering indicators.Similar quantifiers, but describing nonlocality and entanglement, were measured in Ref. [47] using the parameters M and F , which are also applied in this paper.
Here, we report the first (to our knowledge) experimental demonstration of the hierarchy of measures of entanglement, steering, and Bell nonlocality without applying full QST, i.e., by measuring only six elements of a correlation matrix R (corresponding to linear combinations of two-qubit Stokes parameters) for the Werner states.Moreover, we show that the generalized Werner states (GWSs), which are mixtures of an arbitrary two-qubit pure state and white noise, can reveal a more refined hierarchy of the quantum-correlation measures using our experimental setup.
The setup also bears some similarities with a previously proposed and implemented scheme by Bovino et al. [55].Our experimental method of measuring R for general two qubit states is conceptually similar to that reported in Ref. [55] for measuring a nonlinear entropic witness.We find that the witness, defined in the next section, can actually be interpreted as the threemeasurement steering measure S.However, the advantage of our method is that it is more versatile.As shown in Ref. [47], one can perform a full tomography of all the elements of the R matrix rather than only determining its trace.Thus, from the set of six numbers (determining a correlation matrix R) we can learn much more about quantum correlations compared to the original method of Ref. [55].In addition to that, our design provides several practical benefits with respect to Ref. [55].Namely from the experimental point of view, it only requires a single Hong-Ou-Mandel interferometer instead of two.Moreover, our design shares the same geometry with the entanglement-swapping protocol [47].As a result, it can be deployed in future teleportation-based quantum networks to acquire various entanglement measures of distributed quantum states.
This experimental method of measuring the R matrix enables us a complete determination of not only steering measures, but also a fully entangled fraction (FEF) [56] and Bell nonlocality measures [57].We note that for the GWSs, the FEF is exactly equal to the two most popular measures of entanglement, i.e., the negativity and concurrence [2].Thus, the hierarchy of the three measures can be experimentally determined from the R matrix for the Werner states, which is the main goal of this paper.

CORRELATION MATRIX R FOR WERNER AND WERNER-LIKE STATES
We study quantum effects in two qubits by means of the 3 × 3 correlation matrix R = T T T , which is defined by the matrix T composed of the two-qubit Stokes parameters T ij = Tr[ρ(σ i ⊗ σ j )], which are the mean values of the Pauli matrices σ i (i = 1, 2, 3).Superscript T denotes transposition.The standard Bloch representation of a general two-qubit state ρ can be given by the elements T ij together with the single-qubit Stokes parameters where denote the Bloch vectors of the first and second qubits, respectively.
, and I n is the nqubit identity operator.We analyze in detail a special type of the general states given in Eq. (1).Specifically, we have experimentally generated the polarization Werner states by mixing the singlet Bell state, |ψ − ⟩ = (|HV ⟩−|V H⟩)/ √ 2, with white noise (i.e., the maximally mixed state) [8]: assuming various values of the mixing (noise) parameter p ∈ [0, 1].Here, |H⟩ and |V ⟩ denote horizontal and vertical polarization states, respectively.The correlation matrix R for the Werner states simplifies to R(ρ W ) = p 2 I 3 .
We also theoretically analyze GWSs, which can be defined by replacing the singlet state |ψ − ⟩ in Eq. ( 2) by a pure state The state can also be obtained by transmitting a photon in the state |ψ q ⟩ through a depolarizing channel.Note that GWSs also often defined slightly differently, i.e., via 3), as experimentally studied in, e.g., Ref. [9].A special case of such states, i.e., a modified Werner state, when |ψ − ⟩ is replaced by |ϕ q=1/2 ⟩, is referred to as an isotropic state.Such modifications of the Werner states or the GWSs do not affect their quantum correlation measures.
The correlation matrix R for the GWSs, given in Eq. ( 3), is diagonal and reads We note that the correlation matrices T and R are in general nondiagonal (including the non-perfect Werner state measured by us experimentally), although they are diagonal for the perfect GWSs states given in Eq. ( 3).Anyway, as shown in Ref. [58], an arbitrary state ρ described by a nondiagonal T , can be transformed (via a singular-value decomposition) into a state with a diagonal T by local unitary operations, thus, without changing its quantum correlations, including those studied below.

MEASURES OF QUANTUM CORRELATIONS FOR WERNER AND WERNER-LIKE STATES Fully entangled fraction and entanglement measures
The FEF [56] for an arbitrary two-qubit state ρ in Eq. ( 1) can be defined as [47]: given in terms the function θ(x) = max(x, 0).In general, the FEF is only a witness of entanglement; however, for some special classes of two-qubit states, including the GWSs, the FEF becomes a good entanglement measure, and it reduces to the concurrence and negativity: For completeness, we recall that the concurrence C(ρ) of an arbitrary two-qubit state ρ is defined as [27]: , the superscript * denotes complex conjugation, and σ 2 is the second Pauli matrix.Moreover, we recall the definition of the negativity N of a two-qubit state ρ, which reads [28]: N (ρ) = θ(−2µ min ) with µ min denoting the smallest eigenvalue of ρ Γ , i.e., min[eig(ρ Γ )], where the superscript Γ indicates partial transposition.It is seen that the negativity, concurrence, and FEF reduce to the same function for the GWSs.
Let p E (q) denote the largest value of the mixing parameter p as a function of the superposition parameter q for which ρ GW (p, q) is separable.This can be obtained by solving FEF(ρ GW ) = 0 resulting in: which means that ρ GW (p, q) is entangled iff p ∈ (p E (q), 1].In the special case of the standard Werner states, Eq. ( 6) simplifies to which implies the well known fact [8] that the Werner state is separable iff the mixing parameter p ∈ [0, 1/3].
It should be noted that entanglement measures for general states, given in Eq. ( 1), depend not only on the correlation matric R, but also on the single-qubit Stokes parameters ⟨σ i n ⟩ for n = 1, 2, 3 and i = 1, 2. It is seen that the FEF is not a universal witness of two-qubit entanglement, because it solely depends on the R matrix.Nevertheless, the FEF is a good measure of the entanglement of the GWSs.

Quantum steering measures
The effect of quantum steering of a two-qubit state refers to the possibility to affect at a distance one qubit (say of Bob) via local measurements performed on the other qubit (say by Alice).The quantum steerability of a given two-qubit state ρ can be experimentally tested, assuming that each party is allowed to measure n observables in their sites (qubit), by the inequality derived by Cavalcanti, Jones, Wiseman, and Reid (CJWR), which reads [59]: where r = {r A 1 , ..., rA n , rB 1 , ..., rB n } is the set of measurement directions with rA i , rB i ∈ R 3 (for i = 1, ..., n) denoting unit and orthonormal vectors, respectively.Moreover, A measure of steering can be obtained by maximizing F n (ρ, r) over the set of measurement directions, i.e., F n (ρ) = max r F n (ρ, r).More specifically, Costa and Angelo [60] suggested the following steering measures depending on the number n of measurements per qubit: where is the normalization constant such that S n (ρ) ∈ [0, 1] for any two-qubit ρ.Hereafter, we focus on analyzing the steering measures S 2 and S 3 (and related quantifiers) in the two-and threemeasurement scenarios, denoted as 2MS and 3MS, which correspond respectfully to measuring two and three Pauli operators on qubits of both parties.Costa and Angelo found that these two-qubit steering measures can be compactly written as [60]: respectively, given in terms of c = c 2 1 + c 2 2 + c 2 3 and c min = min |c i |, where {c i } = svd(T ) are singular values of T .Note that the original formulas for S 2 and S 3 in Ref. [60] were given assuming the diagonal form of the matrix T , so c i were simply given by T ii .The steering measures given in (11) can be rewritten in terms of the correlation matrix R as follows: The Costa-Angelo measure S 3 of steering in the 3MS is sometimes modified as (see, e.g., Refs.[35,40]): and we also apply this measure in the following, because of a useful property that S reduces to the negativity and concurrence for any two-qubit pure states.Note that S, S 3 ∈ [0, 1] and they are monotonically related to each other for any two-qubit states: For the GWSs, described by the correlation matrix R given in Eq. ( 4), we find Let p S (q) denote the largest value of the mixing parameter p for a given value of the superposition parameter q for which ρ GW (p, q) is unsteerable.Thus, by solving S(ρ GW ) = 0, we have: which means that a given GWS, ρ GW (p), is steerable assuming three measurements per qubit iff the mixing parameter p ∈ (p S (q), 1].In the special case of the Werner states, Eq. ( 16) simplifies to the formulas: . Quantum steerability in the 2MS, as based on S 2 or related measures, corresponds to Bell nonlocality and it is discussed in detail in the next section.
We note that to quantify steering, assuming three measurements on both Alice' and Bob's qubits, we can interchangeably use S 3 , defined in Eq. ( 12), S in Eq. ( 14), as well the steerable weight [21] (as applied in our closely related paper [9]), or the steering robustness [22] in the 3MS.Indeed, if one of the steering measures vanishes, then all the other measures vanish too.However, the steering measure S 2 , as defined in Eq. ( 13) in the 2MS, although it is equivalent to the Bell nonlocality measure B, but it is fundamentally different from another steering measure S 2 (for clarity denoted here as S ′ 2 ) studied by us in Ref. [9], because it corresponds to the case when Alice (Bob) performs two (three) measurements on her (his) qubit.Thus, S 2 (ρ) = 0 (corresponding to vanishing Bell nonlocality of a given state ρ) does not imply that also S ′ 2 (ρ) = 0, which was shown experimentally in [9]).This is possible because an extra measurement performed by Bob on his qubit, as allowed in the S ′ 2 scenario, can reveal the steerability of ρ.

Bell nonlocality measures
The Bell nonlocality of a given two-qubit state ρ can be tested by the violation of the Bell inequality in the Clauser-Horne-Shimony-Holt (CHSH) form [61] (20) where a, a ′ , b, b ′ ∈ R 3 are unit vectors describing measurement settings, and B is referred to as the Bell-CHSH operator.Bell nonlocality can be quantified by the maximum possible violation of the CHSH inequality in (20) over all measurement settings, which lead Horodecki et al. to the following analytical formula [57] max where the nonnegative quantity M (ρ) is the sum of the two largest eigenvalues of R(ρ).The CHSH inequality in (20) is satisfied iff M (σ) ≤ 1.For a better comparison with other measures of quantum correlations defined in the range [0,1], the Bell nonlocality measure of Horodecki et al. [57] can be given by (see, e.g., [40,47,62]) or, equivalently, as [60] which guarantee that B, B ′ ∈ [0, 1].It is seen that B ′ is exactly equal to the steering measure S 2 , given in Eq. ( 13), in the 2MS.Hereafter, we apply both nonlocality measures because their specific advantages.In particular, as shown explicitly below, B ′ depends linearly on the mixing parameter p of the Werner states and GWSs, thus its experimental estimation results in smaller error bars compared to those of B. On the other hand, B is equal to the negativity and concurrence [36,62], but also to the steering measure S and the FEF: for an arbitrary two-qubit pure state |ψ⟩ = a|HH⟩ + b|HV ⟩ + c|V H⟩ + d|V V ⟩, where a, b, c, d are the normalized complex amplitudes.This useful property of B is not satisfied for B ′ .We also study B to enable a more explicit comparison of our present experimental results with those in our former closely related papers [9,47].Anyway, B and B ′ are monotonically related to each other: The Bell nonlocality measures B and B ′ for the Werner states read which explicitly shows that the states are nonlocal iff p > 1/ √ 2. By comparing Eq. ( 26) with Eq. ( 8), it is clearly seen that the Werner states for the mixing parameter p ∈ (1/3, 1/ √ 2) are entangled, but without violating the CHSH inequality, as first predicted in Ref. [8].For the GWSs, formulas in Eq. ( 26) generalize to: Let p B (q) denote the largest value of the mixing parameter p for a given value the superposition parameter q for which ρ GW (p, q) is Bell local.Thus, by solving B(ρ GW ) = 0 one finds: which means that ρ GW (p, q) is Bell nonlocal if p ∈ (p B (q), 1].This function implies for q = 1/2 the wellknown result that the  The hierarchy parameter H[ρGW(p, q)], defined in Eq. ( 33), versus the superposition (q) and mixing (p) parameters.A given GWS, ρGW(p, q), is separable if

Hierarchy of quantum correlations
The following hierarchy of the discussed quantum correlation measures hold for a general two-qubit state ρ: or, equivalently, We also note that S 2 (ρ) ≤ B(ρ) and S 3 (ρ) ≤ S(ρ).The inequalities in (31) for the GWSs reduce to To visualize this hierarchy, we define the following hierarchy parameter of quantum correlations for the GWSs, which is given in terms of the Heaviside function χ(x) equal to 1 for x > 0 and zero for x ≤ 0. This parameter is plotted in Fig. 1 as a function of the parameters p and q uniquely specifying ρ GW (p, q).

EXPERIMENTAL SETUP
The experiment is implemented on the platform of linear optics with qubits encoded into polarization states of discrete photons.These photons are generated in the process of spontaneous parametric down-conversion occurring in a cascade of two Type-I BBO crystals in the Kwiat et al. configuration [63].A femtosecond fundamental laser pulse is frequency doubled to 413 nm and pumps the crystal cascade on its way there and back (as depicted in Fig. 2).Each time the pulse impinges on the crystals, a pair of photons can be generated in the polarization singlet Bell state.To achieve a high degree of entanglement, the pumping pulse is diagonally polarized (by the half-wave plate HWP A ) and subject to a polarization dispersion line [64].In our case, this dispersion line is implemented by two beam displacers (BDs) enveloping HWP B .The photons generated, while the pulse propagates forward are labelled 1 and 2 while the photons generated in the pulses second-time travel through the crystals are denoted 3 and 4.
The investigated state is encoded both into photons 1 and 2 (the first copy) and into photons 3 and 4 (the second copy).A collective measurement on both copies is then performed by projecting photons 2 and 4 onto the singlet Bell state using a fiber beam splitter (FBS) followed by post-selection onto coincidence detection on its output ports.The remaining photons 1 and 3 are projected locally by means of the sets of quarter and halfwave plates (QWPs and HWPs) and polarizing beam splitters (PBSs).We recorded the number of four-fold coincidence detections for various settings of the wave plates; namely, for all combinations of the projections onto the horizontal, vertical, diagonal, anti-diagonal, and both circular polarization states.We have subsequently calculated the expectation values of the Pauli matrices, that is Note that this formula is almost identical to the one in our previous paper [47], except that in the paper instead of the Π projection, the 1 − Π projection was applied there.
When adjusting the setup to generate the requested Bell state (or the maximally mixed state), we have tuned the polarization of the pump beam, so that locally the generated photons have equal probabilities to be horizontally and vertically polarized.(The probability for a single photon being horizontally polarized is p H = 0.50 ± 0.03.)Balancing these probabilities for the horizontal and vertical polarizations implies also balancing in any single-photon polarization basis.Note that the single-photon state is fully incoherent, because the other photon from a pair is ignored and, hence, mathematically one traces over its state.As a consequence, we can consider With respect to that we conclude that the prepared two copies of the Bell state are balanced enough to warrant the replacement of 1 − Π by Π.This is also supported by the fact that the numerically closest Bell state producing the observed values for the three measures has its parameter q = 0.474 -see Eq. (38) and comments in the surrounding paragraph.
Despite narrow frequency filtering on all photons (see the parameters of the bandpass filters in Fig. 2) and a relatively thin crystal cascade of twice 1 mm, there is a generation-time jitter, which causes the visibility of twophoton interference on the FBS to decrease.We have performed a calibration measurement that reveals that 56.7% of the photons do not interfere on FBS.Moreover, the laser power fluctuates over time yielding variable rates of photon-pairs generation.In order to compensate for these two effects, we have performed all the measurements in the two regimes with a temporal delay between photons 2 and 4: (a) tuned for interference and (b) detuned (controlled by the motorized translation M).These two measurements together with the calibration measurement allow us to estimate the net probability of the two copies of the investigated state to pass simultaneously the Bell-state projection on the FBS, as well as the local polarization projections resulting in a four-fold detection event.With the repetition rate of the laser pulse of 80 MHz, we achieve about 1 such an event per 5 minutes.
While the crystals generate two copies of the singlet Bell state, we can readily modify the detection electronics to effectively perform the measurement on the two copies of a maximally mixed state.So far the coincidence window, i.e., the time within all photons must be detected to be considered a coincidence event, had to be very narrow (5 ns) to assure detection of photon pairs originating from a single laser pulse.By considerably widening that window by several orders of magnitude, we effectively aggregate also detections that are completely unrelated and mutually random.This way, the observed state becomes effectively white noise.
Having all the measurements performed on a pure entangled state (two copies of the singlet Bell states) as well as on the maximally mixed state (i.e., the two copies of the maximally mixed state), we can easily interpolate the results for any Werner state with mixing parameter p.In order to do so, we make use of the fact that when two polarization states of single photons interact on a beam splitter and one of them is being a maximally mixed state, the resulting probability of coincidence detection is independent of the state of the other photon.As a result, we interpolate the measurement for any Werner state by combining with probability p 2 the outcomes observed on two copies of maximally entangled states and with probability of 1−p 2 the results observed on a maximally mixed state.
Note that, contrary to reconstructing the R matrix, there is no experimental advantage of reconstructing the 3 × 3 matrix T ≡ T 3 compared to a full QST of a two qubit state ρ, which corresponds to reconstructing the 4 × 4 matrix T 4 = [⟨σ n ⊗ σ m ⟩] for n, m = 0, ..., 3, where σ 0 = I 2 is the qubit identity operator.It might look that reconstructing all the 9 elements of T 3 is much simpler than reconstructing 16 (or 15) elements of T 4 .But this is not the case, because the required types of measurements are the same in both reconstructions.Note that the optical reconstruction T 3 for a given two-qubit polarization state ρ is usually based on projecting ρ on all the eigenstates of the three Pauli operators for each qubit, i.e., projections onto the six polarization singlequbit states (so 36 two-qubit states): diagonal (|D⟩), antidiagonal (|A⟩), right-(|R⟩) and left-circular (|L⟩), horizontal (|H⟩), and vertical (|V ⟩).Analogously, a standard QST of ρ also corresponds to reconstructing T 4 via the same 36 projections as those for T 3 , and the single-qubit identity operator is given by I 2 = |H⟩⟨H| + |V ⟩⟨V |.So, the required measurements for reconstructing T 3 and T 4 are the same, but only their numerical reconstructions are different, although can be based on exactly the same measured data.

EXPERIMENTAL HIERARCHY OF QUANTUM CORRELATIONS
In this section we test the experimental Werner states generated in the setup described in the former section and compare experimental results with theoretical predictions for ideal Werner states.One can calculate the correlation matrix elements R ij following the derivations [47]: (34) noting that A ij and B ij can be experimentally determined.As a result the physical correlation matrices R ij of the Bell singlet state and the maximally mixed state were obtained using a maximum likelihood method.First we derive the correlation matrix R |ψ − ⟩ for the singlet Bell state.
Then we evaluated also the correlation matrix for the maximally mixed state corresponding to white noise, 0.017 0.006 −0.007 0.006 0.013 0.016 −0.007 0.016 0.006 Using definition (2) we can derive the correlation matrix R W (p) of the Werner states for selected values of the mixing parameter p as follows, Now we apply the above-described definitions of the quantifiers of quantum correlations including the defined measures of Bell nonlocality (B and B ′ = S 2 ), steering in the 3MS (S and S 3 ), and entanglement (FEF) based on this correlation matrix.
Our experimental results are summarized in Tables I and II and plotted as symbols in Figs. 3 and 4. The error bars were derived using a Monte Carlo method following the normal distribution of the correlation matrix components with variance corresponding to the number of detected photocounts.Asymmetry of estimated error bars results from presence of the θ function in the formulas for estimated quantities as well as from the requirement on physicality of the R matrices.Figure 3 shows also the theoretically predicted correlation measures plotted with solid curves, which were calculated for the ideal Werner states.
In the theoretical section we considered the Costa-Angelo steering measures S 2 and S 3 that can be calculated also from the R matrix.We evaluated these steering measures using Eqs.( 12) and ( 13).The results are plotted in Fig. 3(b).It is clear that these measures linearly depend on the mixing parameter p.The nonzero regions of the correlation measures, shown in both panels of Fig. 3, are the same.Experimental results shown in Fig. 3(b) are also summarized in Table II.
The original correlation matrices R were derived from measured coincidences using two methods of maximum likelihood estimation of Ref. [65].Both the methods lead to the R matrices that are essentially the same.Our experimental results shown in Fig. 3 demonstrate a very good agreement with theoretical predictions.It is clear that S 2 , S 3 , and FEF are the most stable measures at least for the Werner states and GWSs by exhibiting the smallest errors because of their linear dependence on the mixing parameter p.By contrast to those quantifiers, the measures of steering S in the 3MS and of Bell nonlocal- In the experiment all imperfections of individual components decrease the resulting correlation measures.Together with the instability and a natural Poisson randomness of the measured coincidences, these effects result in measurement uncertainties.Also our experimentally generated singlet-like state is not perfect.We tried to simulate all these mentioned imperfections degrading the input Bell-like state assuming the rest of the measurement to be nearly perfect.These expected imperfections result in a class of generalized states in the form of where |ψ ± ⟩ = √ q|HV ⟩ ± √ 1 − q|V H⟩.The correlation measures for our most entangled experimental Bell-like state read: B = 0.9933, S = 0.9691, and FEF = 0.9685.We found that these results are most consistent with a state of the type specified in Eq. (38) for the parameters q ≈ 0.474 and p ≈ 0.994.This implies the purity of this Bell-like state of about 98.9%.

CONCLUSIONS
We reported the detection of quantum correlation measures of two optical polarization qubits without QST.Specifically, we have measured all the elements of the correlation matrix R (which is symmetric by definition) for the Werner states with different amount of white noise.These elements correspond to linear combinations of twoqubit Stokes parameters.With the matrix R, we were able to determine various measures of quantum entanglement, steerability, and Bell nonlocality of the Werner and Werner-like states.
Most notably, our experiment allows us to show the hierarchy of the tested quantum correlation measures.This means that iff the mixing parameter p ≤ 1/3, the Werner state is separable.For p ∈ (1/3, 1/ √ 3], the Werner state exhibits entanglement (as revealed by the FEF), but it is unsteerable and Bell local.Subsequently, for p ∈ (1/ √ 3, 1/ √ 2] the state is entangled and steerable, but without exhibiting Bell nonlocality.Finally, for p > 1/ √ 2 the state is also Bell nonlocal.These regions, separated by the three values of p = {1/3, 1/ √ 3, 1/ √ 2} ≈ {0.333, 0.577, 0.707}, are depicted with different background colors in Fig. 3.We have also analyzed theoretically a hierarchy (shown in Fig. 1) of some measures of quantum correlations for generalized Werner states, which are defined as arbitrary superpositions of a twoqubit partially-entangled pure state and white noise.
The problem of detecting measures of quantum correlations is essential to assess their suitability for quantuminformation protocols especially for quantum commu-nication and cryptography when considering not only trusted but also untrusted devices.We believe that experimental determination of various measures of entanglement, steering, and Bell nonlocality without full QST, as reported in this work, clearly shows its advantage compared to standard methods based on a complete QST.Specifically, our method relies on measuring only 6 real elements instead of 15 (or even 16) elements in a complete two-qubit QST.
Moreover, experimental studies of a hierarchy of quantum-correlation measures might be useful for effective estimations of one measure for a specific value of another without full QST for experimental quantum channels.

FIG. 3 .
FIG. 3. Experimental demonstration of the hierarchy of quantum correlations of the Werner states without full QST: the Bell nonlocality measures (a) B and (b) B ′ = S2 (solid blue lines and curves), the 3MS steering measures (a) S and (b) S3 (dashed red), and (a,b) the FEF (dot-dashed black lines) shown versus the mixing parameter p. Symbols depict experimental results and curves represent theoretical predictions.

FIG. 4 .
FIG. 4. Experimental and theoretical predictions of different measures: (a) S3 vs S quantifying steering in the 3MS and (b) S2 = B ′ vs B describing Bell nonlocality and, equivalently, steering in the 2MS.Symbols depict the measures calculated for the experimental Werner states for the indicated values of the mixing parameter p.The error bars are marked by solid red curves that follow the dotted curves.Arbitrary twoqubit states lie on the dotted curves.The dashed diagonal lines are added just to show the curvature of the solid curves more clearly and stress the mutual inequality of the plotted quantities.