Probing Asymmetry in Spatial-temporal Correlations in Quantum Causal Inference Quantum correlations in space-time encapsu-late


 Quantum correlations in space-time encapsulate the most defining aspects of quantum physics. The dual of the spatial and temporal perspectives are bind with a one-to-one correspondence between bipartite quantum states and quantum channels. Consequently, causal relations between quantum events can sometimes be inferred solely from correlation statistics, apparently contradicting the classical \textit{credo}, `correlation does not imply causation'[1-6]. However, since the spatial-temporal duality does not imply a full symmetry of measurement statistics between the two domains[7], the extent to which correlation alone identifies quantum causality ponders inquiry vital for both fundamental and practical interests. Here, demonstrating a unified geometrical representation of spatial-temporal quantum correlation, we show that certain non-unital channels create temporal correlation without spatial analogue and break the spatial-temporal symmetry. By implementing such channels in a photonic architecture, we observe this asymmetry and classify quantum correlations using a distance criterion, thus bringing empirical insight into causal inference in quantum mechanics.

Quantum correlations in space-time encapsulate the most defining aspects of quantum physics. The dual of the spatial and temporal perspectives are bind with a one-to-one correspondence between bipartite quantum states and quantum channels. Consequently, causal relations between quantum events can sometimes be inferred solely from correlation statistics, apparently contradicting the classical credo, 'correlation does not imply causation' [1][2][3][4][5][6]. However, since the spatial-temporal duality does not imply a full symmetry of measurement statistics between the two domains [7], the extent to which correlation alone identifies quantum causality ponders inquiry vital for both fundamental and practical interests. Here, demonstrating a unified geometrical representation of spatial-temporal quantum correlation, we show that certain non-unital channels create temporal correlation without spatial analogue and break the spatial-temporal symmetry. By implementing such channels in a photonic architecture, we observe this asymmetry and classify quantum correlations using a distance criterion, thus bringing empirical insight into causal inference in quantum mechanics.
A fully unified treatment of space-time has been one of the cornerstones of modern physics ever since the epoch-making discovery of special relativity [8]; however, in non-relativistic quantum theory, the role of time, because of the particular structure of its underlying antiunitary, exhibits some pronounced differences from other physical observables [9]. Thus, both the similarities and discrepancies between temporal and spatial quantum correlations (TQC and SQC, respectively) [10,11] have been continuous subjects of experimental interest [12,13]. Particularly, as there is no strict analogue to the monogamy of entanglement [14] governing TQC [15,16], three-point TQC exhibits a richer behavioural scope than SQC [17][18][19], which has rendered TQC indispensable for the interpretation of specific observed statistics [20].
Compared with the three-point quantum correlation, the two-point correlation has a simpler algebraic structure and is intimately related to causal inference [5,6]; notably, Ried et al. [1] considered causal inference for quantum variables and illustrated the potentially advantageous rooting in quantum coherence and entanglement. More specifically, the parity of Pauli correlation functions was used as a proxy for identifying causal structures: an odd parity signifies that the underlying correlation originate from a maximally entangled bipartite state (quantum common cause), whereas an even parity guarantees that the correlation is induced by a quantum cause-effect relation which can be described by a quantum channel. This parity signature is consistent with intuition based on the Choi-Jamio lkowski isomorphism [21], which sets a one-to-one correspondence between an arbitrary quantum channel described by a completely positive tracepreserving (CPTP) map ε B|A [22,23] and a bipartite quantum state: where I A denotes the identity super-operator acting on the subsystem A. Since the evolution of a quantum system between two time instances can be fully described by the form of ε B|A , one may postulate that the logic of quantum causal inference can be generalized straightforwardly to any observed non-separable twopoint correlation statistics and that common cause and cause-effect can always be distinguished. In this letter, in contrast with the above intuition, we empirically disclose an asymmetry between the sets of two-point TQCs and SQCs, affirmatively validating a prior theoretical prediction [7]. This reveals a geometrical characteristic shared by TQC and SQC [7,[24][25][26], providing us with a novel viewpoint for investigating the relationship between them. We experimentally demonstrate that the concurrence of a non-maximally mixed initial state and a non-unital quantum channel facilitates the onset of asymmetry, which in turn augments the set of possible two-point TQCs. We use a metric based on Euclidean distance to classify quantum correlations and to explore their utility as a causal witness, thereby further elucidating non-trivial structures inferred to have quantum common causal and direct cause-effect relations in more general cases than in existing knowledge [1,5,6]. A single qubit ρ0 is measured at two instances in time, tA and tB, by two Pauli measurements,Mσ i andMσ j , with a CPTP map ε B|A between them. The measurements collapse the qubit according to Lüder's rule [27]. The results are stored in two reservoirs RA and RB and retrieved later to extract the correlation. b: The two-point SQC. The two-qubit state ρAB are successively measured at time instances tA and tB by two Pauli measurementsMσ i andMσ j , respectively. The evolutions of the two subsystems after preparation occur independently of one another. Identifying whether the statistical correlation from the quantum comb (shown in c) is TQC or SQC can be interpreted as a quantum-causal inference.
We aim to distinguish between TQC and SQC based only on the observed statistics under a scenario best illustrated by the quantum comb representation [28]. As shown in Figs. 1a and 1b, the two-point TQC and SQC are obtained by measuring the Pauli operators σ A i and σ B j with i, j ∈ {1, 2, 3} and by extracting the statistics restored at the reservoirs R A and R B . The former reservoir describes the temporal correlation of a single-qubit state ρ 0 before and after a CPTP channel ε B|A , whereas the latter describes the spatial correlation between two subsystems connected by a two-qubit state ρ AB . Although the measurement of σ A i temporally precedes that of σ B j , the nature of the correlation between the two events is not a priori known, and the task of determining, based on the two-point correlations σ A i σ B j , whether the correlation is inherently temporal or spatial can be reasonably construed as quantum-causal inference (as shown in Fig. 1c). We focus upon the subset of two-point , which geometrically form a set of coordinators in Euclidean space. As shown in the Supplementary Information [29] and Ref. [7], these correlations in the temporal domain can be evaluated as where σ 0 is the identity matrix. Equation (2) is to be compared with the two-point correlation function in the spatial domain, Note that in equation (2), if the channel is non-unital (i.e., if it alters the maximally mixed state ε B|A (1 1) = 1 1), then the choice of the initial state will have a non-trivial augmentation effect upon the set of possible TQCs. In the preparation stage, heralded, horizontally-polarized single photons are generated by pumping a periodically poled potassium titanyl phosphate (ppKTP) crystal. The basis of the first measurement is selected by quarter-wave and half-wave plates (QWP1 and HWP1), whereas the spatial light modulators (SLM1) encodes the measurement outcomes in the spatial profiles of the photons. After HWP2 and QWP2 revert the initial polarization of the photon, it enters the non-unital channel implemented by an unbalanced Mach-Zehnder interferometer. Then, the photon is measured again by QWP3, HWP3, and polarization beam splitter (PBS). SLM2 and SLM3 transform different spatial modes into different propagating directions of the signal photon. Finally, the photon is detected by singlephoton avalanche detectors (SPADs) at the four exit angles.
Geometrically, let Q t denote the set of possible twopoint TQCs: induced by all possible CPTP maps and initial states. The four extremal points of Q t are given by the evenparity coordinates, {(−1) m , (−1) n , (−1) m+n } , m, n ∈ {0, 1}, which are also the vertices of the tetrahedron T t enclosing the set of TQCs induced by only unital channels or by arbitrary channels but with an initial state restricted to being maximally mixed. Conversely, the geometrical set of all two-qubit SQCs is known to be inscribed by the tetrahedron T s with odd-parity vertices (−1) m , (−1) n , (−1) 1+m+n [30,31]. Note that the Choi-Jamio lkowski isomorphism (Equation (1)) implies a symmetry about the σ A 2 σ B 2 = 0 plane between T t and T s ; every TQC with a maximally mixed input induced by a given channel is bijectively mapped into SQC in the corresponding Choi state. This spatial-temporal symmetry is broken in the presence of certain nonunital channels acting upon the non-maximally mixed input states. The relatively uncomplicated form of these non-unital channels allows us to experimentally observe such spatial-temporal asymmetry with readily attainable photonic techniques.
To implement a photonic non-unital channel evolution and illustrate the bounding set of Q t , we employ the experimental setup shown in Fig. 2; the detailed description of the setup goes to the Methods section.
The asymmetry in the geometrical relationship between TQC and SQC that is demonstrated by our experiment is shown in Fig. 3a. The restricted set of TQCs with a vanishing second term of Equation (2), T t , is represented by the red tetrahedrons, while its mirror reflection with respect to the plane σ A 2 σ B 2 = 0, i.e., the set of two-qubit SQCs, T s , is represented by the blue tetrahedron. The set of all possible TQCs is depicted by the red translucent 'inflated tetrahedron', Q t . Compared with the T t inside it, the inflated Q t inscribes a volume that is approximately 85% larger, which manifests additional allowable correlation statistics. Another legible way to visualize this relationship is displayed in Fig. 3b, where the upper red square describes the possible set of two-qubit TQCs with different cases of the initial state and the CPTP map (small red squares 1, 2, 3, 4), which together span Q t in Fig. 3a. The lower blue square describes the possible sets of two-point SQCs with different cases of marginal states ρ A = Tr B [ρ AB ] and ρ B = Tr A [ρ AB ] (small blue square 1, 2, 3, 4), which are captured by the blue tetrahedron, T s , in Fig. 3a. The tetrahedron T t can specify two classes of TQC scenarios: I. an arbitrary single-qubit state ρ 0 subjected to a unital CPTP map (small red squares 2 + 3); II. a maximally mixed initial state ρ 0 = 1 1/2 subjected to an arbitrary choice of the quantum channel (small red squares 1 + 2); however, it does not describe the TQC at the concurrence of the non-maximally mixed input state and the non-unital CPTP map (small red square 4), i.e., T t Q t . Thus, the geometrical symmetry between T s and T t about the plane σ A 2 σ B 2 = 0 (depicted using the diagonal dashed lines) is broken because T s represents all SQCs. Moreover, the Choi-Jamio lkowski isomorphism (Equation (1)) maps the CPTP channel to a bipartite quantum state only when the input state is maximally mixed. This is represented as mapping the small red square 1 + 2 to the small blue squares 1 + 2 + 3 + 4; as can be seen in Fig. 3a, Q t has some partial overlap with T s in addition to the octahedral region O s = T s ∩ T t , which physically represents the SQC that is generated by the two-qubit separable states. To the best of our knowledge, the causal structure when the correlations fall into (Q t ∩ T s ) \ O s has been little investigated.
We track the trajectory of the correlation values by scanning through the parametric space in the geometric representation of Fig. 3. Here, we calculate two sets of parameters and obtain two typical curves, as represented by the orange and cyan theoretical curves on the surface of Q t in Fig. 3a; the details are presented in the Methods. After calculating the correlation function [1] of these two exemplary curves, i.e. C ≡ 3 i=1 σ A i σ B i , we find that the scalar C shows an unprecedented domain of [− 1 8 , 1]. This outdoes the lower bound of all the correlations induced by the quantum direct-cause effect via the statistics C and breaks the symmetry in statistics of C between direct-cause effects (C ∈ − 1 27 , 1 ) and common cause effects(C ∈ −1, 1 27 ) [5]. Our experimen-tal results, represented by the orange and cyan points, confirm the theoretical predictions: after the evolution in the non-unital channels, most of the points lie outside T t . The maximal distance of the data points from the nearest surface of T t is 0.250 ± 0.014, falling just slightly short of the theoretical maximum of √ 3/6 because of experimental imperfections. The extremal point that coordinates (−0.493, 0.515, 0.427) in our experimental results yields C = −0.108 which is very close to the theoretical prediction. The standard deviation of the distance measure is numerically estimated via Monte Carlo simulation.
In the overlapping region (Q t ∩T s ), it is known that one cannot distinguish between TQC and SQC based only on the correlation statistics of σ A i σ B i [5,6]. In this case, further tools are required to reveal the underlying causal structure and the two-point quantum correlations. We now define some useful quantities for further inquiry into causal inference. For an arbitrary point in T s , the a. The red tetrahedron Tt describes the two-point TQC conditioned on a maximally mixed initial state or unital channel evolution, whereas the blue tetrahedron Ts describes the twoqubit SQC. Ts and Tt are symmetric about the σ A 2 σ B 2 = 0 plane. The overlaps of Tt and Ts forming the octahedron Os are denoted by the dashed line, which bounds the region of quantum correlation induced by separable states [30,31]. The red translucent 'inflated tetrahedron' Qt encircling Tt represents the set of TQCs resulting in a non-unital channel between two single-qubit measurements. The orange-and cyan-coloured points represents the experimental results of the two typical parametric Kraus operators with v = 2u and u ∈ [0, π/2], and u = 2π/3 and v ∈ [0, 2π/3], respectively; the two curves correspond to the theoretical predictions. The 3σ error bars are too small to be displayed. b. Schematic illustration of the duality between TQC and SQC. Upper: dependence of TQC upon the mixedness of the initial state ρ0 (X-axis) and the unitality of the CPTP map (Y-axis). Lower: dependence of SQC upon the mixedness of the two marginal states, ρA = TrB[ρAB] and ρB = TrA [ρAB]. The regions with dashed diagonal line exhibit geometrical symmetry between Tt and Ts in Euclidean space.
Euclidean distance between it and the nearest point in octahedron O s constitutes an entanglement measurement [32]. Explicitly, we define the signed distance to the nearest facet of T s from the outside (denoted by SC) as the value of the SQC. Thus SC > 0 signifies that the correlation can be interpreted by SQC. We analogously quantify the entanglement witness (EW). Here, the signed distance is derived from the facets of the octahedron, with the points inside the octahedron giving positive values and EW < 0 signifying the existence of an entanglement in accordance with common practice. Finally, we quantify TQC as T C = R AB tr − 1, where R AB is the two-point pseudo-density matrix (PDM) [24] and T C > 0 indicates that the observed quantum correlation can be described by TQC. Fig. 4 shows the experimental results for computing T C, SC and EW. The green, blue and red curves correspond to the theoretical T C, SC and EW values of the states, respectively, using data points computed from experimental observations. Using the PDM formalism, we see that in both sets of states, T C > 0 is satisfied by all data points. In Fig. 4a, where u > 0.90 rad, SC > 0; this indicates that TQC and SQC are not distinguishable solely through statistical analysis of the σ A i σ B i correlations. Furthermore, EW can be as low as −0.237 ± 0.016, indicating that these TQCs generated by our experimental non-unital channel can be used to mimic the correlations of two-qubit entanglement states. The same phenomenon can be observed in Fig. 4b, where 0.75 rad < v < 1.33 rad. It is clear from the geometrical picture that the imitation of entangled-state statistics cannot be achieved using only the unital channel.
To conclude, we have revealed the geometrical asymmetry between the bounding sets of two-point TQCs under certain non-unital CPTP channel and the twoqubit SQCs and demonstrated that the former is approximately inscribes a volume 85% larger than the latter one. Comparing with the unital cases, when using parity C as the criterion, our results augmented the set of statistics that the temporal correlation can manifest, from C ∈ − 1 27 , 1 [1,5] to C ∈ − 1 8 , 1 . In particular, these findings hold operational significance in quantum causality. Our results elucidate the fine-print in the task of identifying causal structures from two-point correlation statistics, and our criterion based on the Euclidean distance consumes less resource of measurements comparing with the method of causal tomography [1,24]. Our work proves necessary for understanding (and hence benefiting from) the utility of causal inference in the quantum setting. A natural future research avenue is to generalize our inquiries on two-point correlations to the cases of multi-point correlations or higher-dimensional systems.
From the experimental perspective, the versatile photonic non-unital channel devised and implemented in this work can be extended to construct generic quantum channels with the arbitrary parameterization of Kraus operators, which have possible applications in the investigation of the non-Markovian dynamics in open quantum systems [33][34][35]. Moreover, our parametric non-unital quantum channel realizes on demand the photonic non-Hermitian evolution, and thus may facilitate the investigation of the PT -symmetric theory [36][37][38], quantum simulation based on adiabatic quantum computation [39], and the enhancement of quantum sensing with non-Hermitian dynamics [40].