PDM formalism for measurements at multiple times, systems
The pseudo-density matrix (PDM) formalism, developed to treat space and time equally12, provides a general framework for dealing with spatial and causal (temporal) correlations. Research on single-qubit PDMs has yielded fruitful results34,35,36,37,38,39,40,41,42. For example, recent studies have utilised quantum causal correlations to set limits on quantum communication42 and to understand how dynamics emerge from temporal entanglement37. Furthermore, the PDM approach has been used to resolve causality paradoxes associated with closed time-like curves39.
The PDM generalises the standard quantum n-qubit density matrix to the case of multiple times. The PDM is defined as
$${R}_{1...m}=\frac{1}{{2}^{mn}}\mathop{\sum }\limits_{{i}_{1}=0}^{{4}^{n}-1}...\mathop{\sum }\limits_{{i}_{m}=0}^{{4}^{n}-1}{\langle {\{{\tilde{\sigma }}_{{i}_{\alpha }}\}}_{\alpha = 1}^{m}\rangle} \mathop{\otimes }\limits_{\alpha = 1}^{m}{\tilde{\sigma }}_{{i}_{\alpha }},$$
(1)
where \({\tilde{\sigma }}_{{i}_{\alpha }}\in {\{{\sigma }_{0},{\sigma }_{1},{\sigma }_{2},{\sigma }_{3}\}}^{\otimes n}\) is an n-qubit Pauli matrix at time tα. \({\tilde{\sigma }}_{{i}_{\alpha }}\) is extended to an observable associated with up to m times, \({\otimes }_{\alpha = 1}^{m}{\tilde{\sigma }}_{{i}_{\alpha }}\) that has expectation value \(\langle {\{{\tilde{\sigma }}_{{i}_{\alpha }}\}}_{\alpha = 1}^{m}\rangle\). We shall return later to what measurement this expectation value corresponds to. The standard quantum density matrix is recovered if the Hilbert spaces for all but one time, say \({t}_{{\alpha }^{{\prime} }}\) are traced out, i.e. \({\rho }_{{\alpha }^{{\prime} }}={{\rm{Tr}}}_{\alpha \ne {\alpha }^{{\prime} }}{R}_{1...m}\). The PDM is Hermitian with unit trace but may have negative eigenvalues.
The negative eigenvalues of the PDM appear in a measure of temporal entanglement known as a causal monotone f(R)12. Analogously to the case of entanglement monotones43, in general, f(R) is required to satisfy the following criteria: (I) f(R) ≥ 0, (II) f(R) is invariant under local change of basis, (III) f(R) is non-increasing under local operations, and (IV) ∑ipif(Ri) ≥ f(∑piRi). Those criteria are satisfied by12
$$f(R):= \parallel R{\parallel }_{tr}-1={\rm{Tr}}\sqrt{R{R}^{\dagger }}-1.$$
(2)
If R has negativity, f(R) > 0. An intuition for why f(R) serves as a sign of causal influence is that negative eigenvalues tell you that the PDM is associated with measurements multiple times; in the case of a single time, there would be a standard density matrix with no negativity.
The PDM negativity f(R) can thus be used to distinguish, at least in some cases, whether the PDM corresponds to two qubits at one time or one qubit at two times. This can be viewed as a simple form of causal inference, raising the question of whether the inference involving two parties (of multiple qubits) at multiple times depicted in Fig. 1 can be undertaken in a similar manner. A key challenge in this direction is to find a closed-form expression for the PDM R, from which one can see whether f(R) > 0.
Closed form for m-time n-qubit PDMs
We derive a closed-form expression for the PDM for n qubits and two times, before generalising the expression to m times.
Consider the PDM of n qubits undergoing a channel \({{\mathcal{M}}}_{2| 1}\) between times t1 and t2. In order to fully define the PDM of Eq. (1) it is necessary to further define how the Pauli expectation values \(\langle {\{{\tilde{\sigma }}_{{i}_{\alpha }}\}}_{\alpha = 1}^{m}\rangle\) are measured, since that choice impacts the states in between the measurements. We, importantly, choose coarse-grained projectors
$$\left\{{P}_{+}^{\alpha }=\frac{{\mathbb{1}}+{\tilde{\sigma }}_{{i}_{\alpha }}}{2},{P}_{-}^{\alpha }=\frac{{\mathbb{1}}-{\tilde{\sigma }}_{{i}_{\alpha }}}{2}\right\},$$
(3)
where α in iα labels the time of the measurement. These are coarse-grained in the sense of being sums of rank-1 projectors, and by inspection generate lower measurement disturbance than fine-grained projectors in general. The coarse-grained projectors’ probabilities determine the expectation values \(\langle {\{{\tilde{\sigma }}_{{i}_{\alpha }}\}}_{\alpha = 1}^{m}\rangle\). (See Supplementary Information for a circuit to implement these measurements.)
The closed form of the PDM that we shall derive employs the Choi-Jamiołkowski (CJ) matrix of the completely positive and trace-preserving (CPTP) map \({{\mathcal{M}}}_{2| 1}\)44,45. An equivalent variant of the definition of the CJ matrix is as follows:
$${M}_{12}:= \mathop{\sum }\limits_{i,j=0}^{{2}^{n}-1}\left\vert i\right\rangle {\left\langle j\right\vert }^{T}\otimes {{\mathcal{M}}}_{2| 1}\left(\left\vert i\right\rangle \left\langle j\right\vert \right),$$
(4)
where the superscript T denotes the transpose. We show (see Supplementary Information) that the two-time n-qubit PDM, under coarse-grained measurements, can be written in a surprisingly neat form in terms of M12.
Theorem 1
Consider a system consisting of n qubits with the initial state ρ1. The coarse-grained measurements of Eq. (3) are applied at times t1 and t2. The channel \({{\mathcal{M}}}_{2| 1}\) with CJ matrix M12 is applied in-between the measurements. The n-qubit PDM can then be written as
$${R}_{12}=\frac{1}{2}({M}_{12}\,\rho +\rho \,{M}_{12}),$$
(5)
where \(\rho := {\rho }_{1}\otimes {{\mathbb{1}}}_{2}\).
Theorem 1 extends an earlier known form for the single qubit case to multiple qubits that may have entanglement34,38. The theorem provides an operational meaning for a mathematically motivated spatiotemporal formalism22. Moreover, the n-qubit PDM will enable us to investigate phenomena that cannot be explored in the single qubit case, such as quantum channels with associated extra qubits constituting a memory42.
We next, for completeness, stretch the argument to multiple times. Consider initially an n-qubit state ρ1 measured at time t1, undergoing the channel \({{\mathcal{M}}}_{2| 1}\), measured at time t2, undergoing \({{\mathcal{M}}}_{3| 2}\) and measured at time t3. The central objects to determine are the joint expectation values of the observables at three times. These can be written as
$$\langle {\tilde{\sigma }}_{{i}_{1}},{\tilde{\sigma }}_{{i}_{2}},{\tilde{\sigma }}_{{i}_{3}}\rangle ={{\rm{Tr}}}_{23}[{M}_{23}\left({P}_{+}^{2}{\rho }_{2}^{({\tilde{\sigma }}_{{i}_{1}})}{P}_{+}^{2}-{P}_{-}^{2}{\rho }_{2}^{({\tilde{\sigma }}_{{i}_{1}})}{P}_{-}^{2}\right)\otimes {\tilde{\sigma }}_{{i}_{3}}],$$
(6)
where we denote the CJ matrices for channels \({{\mathcal{M}}}_{2| 1},{{\mathcal{M}}}_{3| 2}\) by M12, M23 respectively, and (see Supplementary Information)
$${\rho }_{2}^{({\tilde{\sigma }}_{{i}_{1}})}={{\rm{Tr}}}_{1}[{R}_{12}\,{\tilde{\sigma }}_{{i}_{1}}\otimes {{\mathbb{1}}}_{2}].$$
(7)
Eqs. (6) and (7) then together imply that
$$\langle {\tilde{\sigma }}_{{i}_{1}},{\tilde{\sigma }}_{{i}_{2}},{\tilde{\sigma }}_{{i}_{3}}\rangle =\frac{1}{2}{\rm{Tr}}[({M}_{23}{R}_{12}+{R}_{12}{M}_{23}){\tilde{\sigma }}_{{i}_{1}}\otimes {\tilde{\sigma }}_{{i}_{2}}\otimes {\tilde{\sigma }}_{{i}_{3}}],$$
(8)
where implicit identity matrices are now omitted for notational convenience.
From Eq. (8), demanding that
$${R}_{123}=\frac{1}{2}({R}_{12}{M}_{23}+{M}_{23}{R}_{12}),$$
(9)
gives expectation values consistent with the PDM definition of Eq. (1). Since the expectation values uniquely determine the PDM, Eq. (9) must be the correct expression.
The above derivation can be directly generalised to more than three times:
Theorem 2
The n-qubit PDM across m times is given by the following iterative expression
$${R}_{12...m}=\frac{1}{2}({R}_{12...m-1}{M}_{m-1,m}+{M}_{m-1,m}{R}_{12...m-1})$$
(10)
with the initial condition \({R}_{12}=\frac{1}{2}(\rho \,{M}_{12}+{M}_{12}\,\rho )\) where Mm−1,m denotes the CJ matrix of the (m − 1)-th channel.
This iterative expression, proven in Supplementary Information, can be written in a (possibly long) closed-form sum in a natural manner. We have thus extended a key tool in the PDM formalism from the cases of single qubits, two times or two qubits single time to the case of n qubits at m times for any n and m.
Relation between PDM negativity and the possibility of common cause
PDM negativity (f > 0) was linked to cause-effect mechanisms for the case of one qubit at 2 times or 2 qubits at one time in ref. 12. We now consider the case of several qubits and several times, such that there may be combinations of temporal and spatial correlations. We use Eq. (5) to derive a relation between the negativity of parts of the PDM and the possibility of a common cause, meaning correlations in the initial state.
We model the possible directional dynamics of Fig. 1 as so-called semicausal channels46,47. Semicausal channels are those bipartite completely positive trace-preserving (CPTP) maps that do not allow one party to signal or influence the other. If the channel \({\mathcal{P}}\) does not allow B to influence A, it must admit the decomposition \({\mathcal{P}}={{\mathcal{M}}}_{BC}\,{\circ}\; {{\mathcal{N}}}_{AC}\)47. The circuit representation of \({\mathcal{P}}\) on A and B across two times t1, t2, is depicted in Fig. 2. The following theorem shows that when there is no signalling from B at time 1 to A at time 2, the PDM \({R}_{{B}_{1}{A}_{2}}\) has no negativity for any input state.
Semicausal channels are bipartite channels which can be decomposed into either \({{\mathcal{M}}}_{BC}\,{\circ}\; {{\mathcal{N}}}_{AC}\) or \({{\mathcal{N}}}_{AC}\,{\circ}\; {{\mathcal{M}}}_{BC}\) where C is an ancilla, as in the above circuit. The circles here indicate possible measurements. In this example, which is consistent (only) with cases 1, 3 and 5 in Fig. 1, A can causally influence B, while the inverse is not true.
Theorem 3
(null PDM negativity for semicausal channels) If a quantum channel \({\mathcal{P}}\) does not allow signalling from B to A, then, for any state \({\rho }_{{A}_{1}{B}_{1}}\) at time t1, the PDM \({R}_{{B}_{1}{A}_{2}}\) is positive semidefinite and the PDM negativity \(f({R}_{{B}_{1}{A}_{2}})=0\).
The theorem implies that only the existence of causal influence between B and A allows for \(f({R}_{{B}_{1}{A}_{2}}) > 0\). In particular, if there is no causation from B1 to A2, any initial correlations between A1 and B1 cannot make the PDM negativity \(f({R}_{{B}_{1}{A}_{2}}) > 0\). In contrast, several other observation-based measures such as the mutual information can be raised from initial correlations alone48.
Theorem 3 additionally has value for the more restricted task of characterising whether channels are signalling, as considered in refs. 46,47. If \(f({R}_{{B}_{1}{A}_{2}}) > 0\) the channel must be signalling from B to A. In this restricted task, one may vary over input states. There are reasons to believe pure product states may maximise \(f({R}_{{B}_{1}{A}_{2}})\) for a given channel. From property IV of f, with a given R = ∑ipiRi, the most negative pure state \({R}_{i* }:= {\rm{argmax}}\,f({R}_{i})\) respects f(Ri*) ≥ f(R). We moreover conjecture that if the channel is signalling from B to A, we can always find a pure product input state such that \(f({R}_{{B}_{1}{A}_{2}}) > 0\). We prove this conjecture for a quite general case of 2-qubit unitary evolutions49 in Supplementary Information.
Exploiting time asymmetry to distinguish cause and effect
Consider the case where there is negativity f(RAB) > 0, but it is not known which is the cause or effect, i.e. the time-label is unknown. We can then exploit the asymmetry of temporal quantum correlations50 to distinguish the cause and effect, and to determine whether there is a common cause.
The asymmetry of temporal quantum correlations can be defined by comparing forwards and time-reversed PDMs50. The time-reversed PDM,
$${\bar{R}}_{AB}:= S\,{R}_{AB}\,{S}^{\dagger },$$
(11)
where S denotes the n-qubit swap operator22,50. The methods given here to find a closed-form expression for RAB can be similarly applied to show that \({\bar{R}}_{AB}=\frac{1}{2}(\pi \,\bar{M}+\bar{M}\,\pi )\), where \(\pi := ({{\rm{Tr}}}_{A}{R}_{AB})\otimes {{\mathbb{1}}}_{A}\) and \(\bar{M}\) is the CJ matrix of the time reversed process. The CJ matrices M and \(\bar{M}\) can be extracted via a vectorisation of R and \(\bar{R}\), respectively50. Let T denote the transpose on the initial quantum system. The Choi matrices of the process and its time reversal are given by MT and \({\bar{M}}^{T}\), respectively. A process being CP is equivalent to its Choi matrix being positive44,45. When only one of the two Choi matrices is positive, we say there is an asymmetry of the temporal quantum correlations.
The asymmetry can be used to distinguish different causal structures. If there is no initial correlation (no common cause) the forwards process is CP but in general the reverse process may be not positive semidefinite (\({\bar{M}}^{T}\) ≱ 0). Furthermore, if both Choi matrices are not positive semidefinite (\({\bar{M}}^{T},{M}^{T}\) ≱ 0), then neither process is CP, and there must be a common cause (initial correlations).
Protocol for quantum causal inference
We will now make use of the results from previous sections to give a protocol that determines the compatibility of the experimental data with the causal structures shown in Fig. 1. In line with causal inference terminology2, we say that the data and a causal structure are compatible if experimental data could have been generated by that structure. As in causal inference in general, compatibility is not guaranteed to be unique.
The causal structures of Fig. 1 are as follows. Case 1 is the cause-effect mechanism in one direction, when there are two instances of quantum systems A and B located in space and actions on A influence the reduced state on B and the actions on B do not influence the reduced state on A. Case 2 is the same mechanism as Case 1 but in the opposite direction. Case 3 is the pure common cause mechanism, with no influence between A and B. There is a common cause, meaning correlations at the initial time t1, iff \({R}_{{A}_{1}{B}_{1}}\ne {R}_{{A}_{1}}\otimes {R}_{{B}_{1}}\). Cases 4 and 5 is when there is a common cause mechanism and also a cause-effect mechanism. Cases 4 and 5 are distinguished by the directionality of the cause-effect mechanism.
Recall that the setting involves two systems A and B and two times ti and tj. We are given the data that constructs the PDM \({R}_{{A}_{i}{B}_{j}}\) and assume that the data has correlations (\({R}_{{A}_{i}{B}_{j}}\ne {R}_{{A}_{i}}\otimes {R}_{{B}_{j}}\) for whatever i, j we are given data for) so that there is a non-trivial causal structure. We are not given the data that constructs the PDM \({R}_{{A}_{i}{B}_{i}{A}_{j}{B}_{j}}\) and do not have enough data to reconstruct the full channel on AB in general. We are, moreover, not told which time is measured first. The protocol is as follows:
- (1)
Evaluating compatibility with a common-cause mechanism. Consider the case of no negativity (\(f({R}_{{A}_{i}{B}_{j}})=0\)). Theorem 3 implies that only the existence of causal influence between Ai and Bj can allow for negativity. The purely common cause mechanism (case 3 in Fig. 1, \({R}_{{A}_{1}{B}_{1}}\,\ne\, {R}_{{A}_{1}}\otimes {R}_{{B}_{1}}\)) is, in contrast, compatible with no negativity. Thus for no negativity, the protocol is to conclude that the data \({R}_{{A}_{i}{B}_{j}}\) is compatible with the (purely) common cause mechanism.
- (2)
Evaluating compatibility with different cause-effect mechanisms. Consider the case of negativity (\(f({R}_{{A}_{i}{B}_{j}}) > 0\)). Theorem 3 rules out the common cause mechanism, and we are left to evaluate the compatibility of the data with cases 1, 2, 4, and 5 in Fig. 1. We make use of the time asymmetry results described around Eq. (11) for this evaluation. In particular, we extract the two Choi matrices MT, \({\bar{M}}^{T}\) associated with \({R}_{{A}_{i}{B}_{j}}\) and its time reversal \({\bar{R}}_{{A}_{i}{B}_{j}}\). The basic idea is that MT > 0 means there is a CP map on A that gives B, indicating that A could be the cause and B the effect. More specifically,
-
– If MT ≥ 0 and \({\bar{M}}^{T}\) ≱ 0, the data is compatible with A → B (case 1 in Fig. 1).
-
– If MT ≱ 0 and \({\bar{M}}^{T}\ge 0\), the data is compatible with A ← B (case 2 in Fig. 1).
-
– If MT ≥ 0 and \({\bar{M}}^{T}\ge 0\), the data is compatible with case 1 and/or case 2 in Fig. 1.
-
- (3)
If none of the above conditions are satisfied, i.e. f(RAB) > 0, MT ≱ 0 and \({\bar{M}}^{T}\) ≱ 0, the causal structure is compatible only with case 4 or 5 in Fig. 1.
Detailed justifications for the above protocol are given in the Supplementary Information. The Supplementary Information also contains a semidefinite programme motivated by a technical subtlety when extracting the CJ matrix from the PDM. When both ρ and π are of full rank, M and \(\bar{M}\) can be uniquely extracted using the vectorisation technique. However, when they are rank deficient, there are infinitely many solutions for M and \(\bar{M}\). Ref. 51 also showed how solving for the process in the case where the marginal is rank deficient is a semidefinite problem for the case of a single qubit. Therefore, we design a semidefinite programming problem to find all possible CJ matrices where \({M}^{{T}_{1}}\) and \({\bar{M}}^{{T}_{1}}\) are the least negative.
The protocol identifies compatibility, and it is natural to wonder whether it uniquely identifies the structure used to generate the data. For at least part of the protocol this appears to be the case. Numerical simulations of 2-qubit cases show a near unit probability that if \(f({R}_{{A}_{i}{B}_{j}}) > 0\) the data is indeed not generated by the common cause mechanism (see Supplementary Information).
Example: cause-effect mechanism
We now consider an example that shows how our light-touch protocol can resolve the causal structure even for channels that do not preserve quantum coherence. Let systems A and B be uncorrelated single qubit systems, and the end effect of the compound channel \({{\mathcal{M}}}_{BC}\,{\circ}\; {{\mathcal{N}}}_{AC}\) on the compound system AB be the channel that measures the system A, recording the outcome in C and then preparing a state on system B that depends on C, as in Fig. 2. Denote the effective channel on AB by \({{\mathcal{L}}}_{A\to B}={{\rm{Tr}}}_{CA}\,{\circ}\; {{\mathcal{M}}}_{BC}\,{\circ}\; {{\mathcal{N}}}_{AC}\). For concreteness, we choose \({{\mathcal{N}}}_{AC}({\rho }_{A}\otimes {\left\vert 0\right\rangle }_{C}\left\langle 0\right\vert )=\left\langle 0\right\vert {\rho }_{A}\left\vert 0\right\rangle {\left\vert 00\right\rangle }_{AC}\left\langle 00\right\vert +\left\langle 1\right\vert {\rho }_{A}\left\vert 1\right\rangle {\left\vert 11\right\rangle }_{AC}\left\langle 11\right\vert\) and \({{\mathcal{M}}}_{BC}({\rho }_{B}\otimes {\rho }_{C})=S({\rho }_{B}\otimes {\rho }_{C}){S}^{\dagger }\) where S is the unitary swap. Thus the action of \({{\mathcal{L}}}_{A\to B}\) on the state is \({{\mathcal{L}}}_{A\to B}({\rho }_{A})=\left\langle 0\right\vert {\rho }_{A}\left\vert 0\right\rangle {\left\vert 0\right\rangle }_{B}\left\langle 0\right\vert +\left\langle 1\right\vert {\rho }_{A}\left\vert 1\right\rangle {\left\vert 1\right\rangle }_{B}\left\langle 1\right\vert\). Therefore, the CJ matrix of \({\mathcal{L}}\) in the Pauli basis is
$$L=\frac{1}{2}\mathop{\sum }\limits_{i=0}^{3}{\sigma }_{i}\otimes {\mathcal{L}}({\sigma }_{i})=\frac{1}{2}({\sigma }_{0}\otimes {\sigma }_{0}+{\sigma }_{3}\otimes {\sigma }_{3}).$$
(12)
Substituting Eq. (12) into Eq. (5), the PDM
$$\begin{array}{lll}{R}_{{A}_{1}{B}_{2}}\,=\,\left(\frac{1}{2}{\rho }_{{A}_{1}}+\frac{1}{4}{\sigma }_{3}+\frac{z}{4}{\sigma }_{0}\right)\otimes \left\vert 0\right\rangle \left\langle 0\right\vert\\\qquad\qquad+\,\left(\frac{1}{2}{\rho }_{{A}_{1}}-\frac{1}{4}{\sigma }_{3}-\frac{z}{4}{\sigma }_{0}\right)\otimes \left\vert 1\right\rangle \left\langle 1\right\vert ,\end{array}$$
(13)
where \(z:= {\rm{Tr}}({\rho }_{{A}_{1}}{\sigma }_{3})\). The eigenvalues of \({\rho }_{{A}_{1}}+\frac{1}{2}{\sigma }_{3}+\frac{z}{2}{\mathbb{1}}\) are \(\frac{1}{2}(1+z\pm \sqrt{{(1+z)}^{2}+{x}^{2}+{y}^{2}})\) with \(x:= {\rm{Tr}}({\rho }_{{A}_{1}}{\sigma }_{1}),y:= {\rm{Tr}}({\rho }_{{A}_{1}}{\sigma }_{2})\). When x2 + y2 = 0, the PDM is positive (\(f({R}_{{A}_{1}{B}_{2}})=0\)) without coherence in the Pauli-z basis. However, the PDM is negative (\(f({R}_{{A}_{1}{B}_{2}}) > 0\)) exactly when x2 + y2 > 0, i.e. when the initial state \({\rho }_{{A}_{1}}\) is coherent in the Pauli-z basis.
For concreteness, we now assume the initial state is given by \({\rho }_{{A}_{1}{B}_{1}}=\left[(1-\lambda )\frac{{\mathbb{1}}}{2}+\lambda \left\vert +\right\rangle \left\langle +\right\vert \right]\otimes \left\vert 0\right\rangle \left\langle 0\right\vert ,\lambda \in (0,1).\) The Choi matrix of the time reversal process (Eq. (11)) can be calculated to be
$${\bar{L}}^{T}=\frac{1}{2}\left(\begin{array}{cc}2&\lambda \\ \lambda &0\end{array}\right)\otimes \left\vert 0\right\rangle \left\langle 0\right\vert +\frac{1}{2}\left(\begin{array}{cc}0&\lambda \\ \lambda &2\end{array}\right)\otimes \left\vert 1\right\rangle \left\langle 1\right\vert .$$
(14)
Clearly, LT ≥ 0 and \({\bar{L}}^{T}\) ≱ 0 for any λ ∈ (0, 1).
Applying the causal inference to the above case we would firstly note \(f({R}_{{A}_{1}{B}_{2}}) > 0\) so case 3 is ruled out. Since LT ≥ 0 and \({\bar{L}}^{T}\) ≱ 0, the data is compatible with A → B (case 1 in Fig. 1).
The example has implications for when the apparent quantum advantage of not requiring interventions for causal inference exists. An earlier observational protocol25 showed this advantage existing for a case of coherence- preserving channels. The above example using our observational protocol indicates that coherence-preserving channels is not required for this apparent quantum advantage. In the above example, there is coherence in the initial state but a decoherent channel. A further example of applying the protocol to a cause-effect mechanism with a common cause is given in the Supplementary Information.
