Quantifying Quantum Causal Inﬂuences Lucas Hutter1Rafael Chaves23Ranieri Vieira Nery2George Moreno24and Daniel Jost Brod1 1Institute of Physics Federal University Fluminense Niter oi Brazil

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Quantifying Quantum Causal Inﬂuences

Lucas Hutter,1Rafael Chaves,2,3Ranieri Vieira Nery,2George Moreno,2,4and Daniel Jost Brod1

1Institute of Physics, Federal University Fluminense, Niter´

oi, Brazil

2International Institute of Physics, Federal University of Rio Grande do Norte, 59070-405 Natal, Brazil

3School of Science and Technology, Federal University of Rio Grande do Norte, 59078-970 Natal, Brazil

4Departamento de Computac¸˜

ao, Universidade Federal Rural de Pernambuco, 52171-900, Recife, Pernambuco, Brazil

Causal inﬂuences are at the core of any empirical science, the reason why its quantiﬁcation is of

paramount relevance for the mathematical theory of causality and applications. Quantum correla-

tions, however, challenge our notion of cause and effect, implying that tools and concepts developed

over the years having in mind a classical world, have to be reevaluated in the presence of quantum

effects. Here, we propose the quantum version of the most common causality quantiﬁer, the average

causal effect (ACE), measuring how much a target quantum system is changed by interventions on

its presumed cause. Not only it offers an innate manner to quantify causation in two-qubit gates but

also in alternative quantum computation models such as the measurement-based version, suggest-

ing that causality can be used as a proxy for optimizing quantum algorithms. Considering quantum

teleportation, we show that any pure entangled state offers an advantage in terms of causal effects as

compared to separable states. This broadness of different uses showcases that, just as in the classical

case, the quantiﬁcation of causal inﬂuence has foundational and applied consequences and can lead

to a yet totally unexplored tool for quantum information science.

In spite of the mantra in statistics that ”correlation

does not imply causation”, a central goal of any quanti-

tative science is precisely that: to infer cause and effect

relations from observed data [1,2]. In fact, as stated by

Reichenbach’s principle [3], correlations between two

events do imply some causation. Either one has a direct

inﬂuence over the other or a third event acts as a com-

mon cause for both of them. Within this context, given

variables Aand B, the basic aim of causal inference is

to distinguish how much of their observed correlations

are due to direct causal inﬂuences, rather than due to

a potential common cause Λ. However, if we do not

have empirical access to Λ, which is then treated as a la-

tent/hidden variable, both models—common cause or

direct causal inﬂuences—generate the same set of pos-

sible correlations that cannot be set apart from obser-

vations alone. With that aim, one has to rely on inter-

ventions [1]. By intervening on A, we ﬁx it to a value

independent of Λsuch that any remaining correlations

between Aand Bcan unambiguously be traced back to

a direct inﬂuence A→B, a fundamental result with a

wide range of applications [4–9].

Notwithstanding all the successes of causality the-

ory, since Bell’s theorem [10] it is known that the clas-

sical notions of cause and effect break down at the

quantum level. Not only the notion of a causal struc-

ture has to be generalized in order to include quan-

tum states [11–18] or the possibility of superposition of

causal orders [19–21]; but also the meaning and appli-

cability of Bell inequalities as a causal compatibility tool

[22] have to be reevaluated [23], and tests employed to

bound the causal effects [24] have to be readjusted [25].

Given that, a fundamental question reemerges: how

can we quantify quantum causal effects? Complemen-

tary frameworks for reasoning about quantum causal

inﬂuences have been developed [13,15,17,20,21] and

explicit quantiﬁers of causality have already been pro-

posed [26,27]. Nevertheless, the quantum generaliza-

tion of the most widely used and intuitive quantiﬁer

of causality in the classical case, the so-called average

causal effect (ACE) [1,7,24,25,28,29], has not yet been

achieved. That is the main goal of this Letter.

Using the trace distance, we propose a quantum ver-

sion of the ACE. It quantiﬁes the causal inﬂuence that

an intervention on a system might have on a result-

ing quantum state. We show the applicability of our

framework in a number of paradigmatic quantum in-

formation scenarios. We start quantifying causal in-

ﬂuences in two-qubit gates and discussing the advan-

tages of our approach as compared to other recent pro-

posals [26]. Within the context of measurement-based

quantum computation [30,31] and quantum teleporta-

tion [32], we show that separable states imply a limited

amount of causal inﬂuence, a restraint that can be sur-

passed by any pure entangled state. Thus, our quantum

causality quantiﬁer not only provides a natural exten-

sion of a widely used and acknowledged classical tool

but also can be seen as a novel witness of non-classical

behavior.

Quantum Average Causal Effect – Suppose we ob-

serve correlations between variables Aand B, that

is, their probability distribution does not factorize as

p(a,b)6=p(a)p(b). From Reichenbach’s principle [3],

the most general causal model explaining such correla-

tions might involve direct inﬂuences as well as a com-

mon cause Λthat, for a variety of reasons, might not be

arXiv:2210.04306v1 [quant-ph] 9 Oct 2022

directly observed. Thus, at least in a classical descrip-

tion, the conditional observational distribution p(b|a)

can be decomposed as

p(b|a) = ∑

p(λ|a)p(b|a,λ). (1)

If, however, an intervention is performed on A, an op-

eration denoted as do(a), the interventional distribution

is now

p(b|do(a)) = ∑

p(λ)p(b|a,λ), (2)

where p(b|do(a)) denotes the probability of bwhen

variable Ais set by force to be a, that is, any potential

inﬂuence it might have had from the common cause Λ

is erased. Importantly, interventions bring in a natu-

ral way for quantifying causality. For instance, if aand

bare binary variables, a widely used quantiﬁer of the

causal inﬂuence from Ato B, the average causal effect

(ACE) [1,7,24,25,28,29], is deﬁned as

ACE =|P(b1|do(a1)) −P(b1|do(a0))|, (3)

measuring the change in the distribution p(b1) = p(b=

1)of the variable Bdepending whether the value of A

is set to a=1 or a=0. Notice that

P(b1|do(a1)) −P(b1|do(a0)) = P(b0|do(a0)) −P(b0|do(a1)),

therefore Eq. (3) accounts for the inﬂuence Aon the

full probability distribution of values of B. In contrast,

when Aand Bcan assume more than two values, gen-

eralizations of Eq. (3) are not unique.

For simplicity, ﬁrst assume that only Bcan have more

than two values. If we want to measure the largest

causal inﬂuence Ahas over B, a natural generalization

is to maximize the right-hand side of Eq. (3) over all

values of b[25], such that

ACEmax =max

b|P(b|do(a1)) −P(b|do(a0))|. (4)

This deﬁnition, however, might not capture the full

causal inﬂuence from Ato B, if that inﬂuence is very

spread through the event space of B. To illustrate, sup-

pose that Bcan assume integer values from 1 to 2N.

If a=0 (resp. a=1), the probability distribution

over Bis the uniform distribution over integers from

1to N(resp. N+1 to 2N). The ACE, as deﬁned by

Eq. (4), decreases as Nincreases. And yet, changing

the value of Aclearly has a large effect on the distribu-

tion of B. This example illustrates the extent to which

ACEmax is sensitive to a coarse-graining of the prob-

ability distribution. Since our intention is to quantify

causal inﬂuence in quantum protocols, it makes sense

to allow for arbitrary coarse-grainings on outcomes of

quantum measurements—after all, we can always en-

code a coarse-graining strategy as degeneracies in the

measured observable.

Building on that, we propose a generalization of

Eq. (3) based on the well-know total variation distance

(TVD), the largest possible difference that the two dis-

tributions can assign to the same event, given by

δ(P,Q) = 1

2∑

x∈X|P(x)−Q(x)|, (5)

where Pand Qare two probability distributions over

X. The ACE can then be deﬁned as

ACETVD =1

2∑

b|P(b|do(a1)) −P(b|do(a0))|. (6)

While it reduces to Eq. (3) when Bis binary, it actually

returns the largest value of ACEmax over all possible

coarse-grainings of the distribution of B.

To generalize Eq. (6) for a quantum system, there are

two choices. The ﬁrst is to suppose we have some set of

possible measurement bases and compute the ACETVD

at the level of the probability distribution over mea-

surement outcomes in these bases. Often, this is desir-

able, since it operates directly at the level of outcomes

[23,25,33]—the success probability of a quantum game

or protocol might be stated in terms of these quantities,

as typically done within device-independent quantum

information [34]. However, there are in principle in-

ﬁnitely many choices of measurement bases, and differ-

ent protocols or setups can differ on how much infor-

mation the experimenter or measuring agent has over

which bases they should measure. Therefore, it can also

be meaningful to measure directly the causal inﬂuence

of a parent variable on the resulting quantum state, ag-

nostic to which basis (if any) it will be measured in.

Following this reasoning, we propose a generaliza-

tion of the ACE for quantum states in terms of the trace

distance (TD), a well-known generalization of the TVD

measuring the distance between two density matrices ρ

and σ, deﬁned as

TD(ρ,σ):=1

2Tr q(ρ−σ)2=1

2∑

i|λi|, (7)

where λiare the eigenvalues of the matrix (ρ−σ). Just

like the TVD accounts for all classical “strategies” (i.e.

choices of coarse-graining), the TD accounts for all pos-

sible quantum strategies. More concretely, the trace

distance between two states corresponds to the max-

imum TVD between the two probability distributions

that would arise from measuring those states with the

same POVM.

If Ais a classical binary variable, then the quantum

ACE is naturally deﬁned as

ACEQ=TD(ρB(do(a1)),ρB(do(a0)), (8)

where ρB(do(a0)) is the density matrix that describes

the state at Bgiven the intervention do(a0). In many

cases, however, and particularly for the applications we

consider later on, Aactually corresponds to some pure

(qubit) quantum state. More concretely, Acould corre-

spond to any state in the Bloch sphere, and so Eq. (8) is

no longer well deﬁned. We thus generalize it as follows

ACEQ=E

a0,a1

TD(ρB(do(a1)),ρB(do(a0)), (9)

where we now average over the choice of a0and a1.

Following [13], do-interventions on quantum states are

obtained simply by tracing whatever state represents A

and replacing it with a pure state, and subsequently

averaging over all possible states of A. Clearly, which

average must be performed depends on the nature of

the variable A. For instance, if it is an arbitrary state

in the Bloch sphere, the natural choice is the uniform

(Haar) distribution [35,36].

Causal inﬂuences in two-qubit gates– We consider a

two-qubit gate, U, acting on a pair of qubits labelled A

and B. We wish to compute the ACEQfrom the input

state of qubit Aonto the output of qubit B. We consider

that this gate might be embedded into a larger quantum

circuit, but that we perform a do-intervention on qubit

A, replacing it by some pure state |Ai[13]. As there is

no reason for Bto be diagonal in any particular basis,

we perform a Haar-random average over the input of

B. As we are also not interested in the output of qubit

A, it is traced out after the application of U. The entire

procedure, shown in Figure 1, can be summarized by

ACEQ(U) = E

|ai

|biTD(ρ(b|do(a),ρ(b|do(a⊥)), (10)

where we used the shorthand

ρ(b|do(a)) = trAU|a,biha,b|U†. (11)

The average over choices of intervention is done as

follows. First, we choose some state |ai, and assume the

intervention consisted of choosing either of the antipo-

dal states in the Bloch sphere |aiand a⊥. We then av-

erage the result uniformly over |ai. We could have cho-

sen to average uniformly over two independent choices

of states |a0iand |a1i. However, this is computationally

more costly and seems to lead simply to a reduction

by a constant multiplicative factor. It is also intuitive

that, given any state |ai, the largest inﬂuence over Bis

obtained by choosing between either |aiand a⊥.

FIG. 1. The setup used to calculate the causal inﬂuence of

quantum gates. Here the inﬂuence is measured from the entry

|aito the output of |bi(red lines). We perform a Haar-random

average over |biand a partial trace over the ﬁrst output qubit.

Gate ACEQ

Local 0

cnot π/8

cz π/8

Bgate 0.5878

√swap 0.6427

swap 1

TABLE I. ACEQ(U)for some noteworthy two-qubit gates. As

expected, the causal inﬂuence on a cnot gate is the same in

both directions. The Bgate was deﬁned in [37], and is an

optimal two-qubit gate in the sense that any other gate can be

decomposed using only two copies of it (compared to three

cnot gates). This distinction manifests in the fact that ACEQ

is much larger for the Bgate than for the cnot

Our results, for a few paradigmatic quantum gates,

are summarized in Table Iand detailed in the Supple-

mental Material [38]. It is natural that the swap gate

displays the largest possible value of causal inﬂuence:

if the states of qubits Aand Bget swapped, then A

has maximal inﬂuence over Birrespective of anything

else. Another virtue of our deﬁnition is that it is basis

invariant. As a consequence, consider the cnot gate:

it ﬂips the target qubit if the control qubit is in the

|1istate, and does nothing otherwise. Thus, one can

imagine that the inﬂuence only exists from the (input)

control qubit onto the (output) target qubit, or at least

that it is stronger in that direction. Our measure, how-

ever, attributes the same causal inﬂuence from the con-

trol to the target in a cnot gate as vice-versa, which

is to be expected since these roles can be ﬂipped by a

local change of basis. Finally, our deﬁnition has a nat-

ural scale, ranging from 0(for local gates) to 1(for the

swap gate). Thus, not only our causality measure has

a fundamental motivation since it is a generalization of

the well-known ACE [1], it also displays a number of

advantages that can be showcased by comparison with

another recent proposal [26]. There, the cnot gate does

not have the same value of causal inﬂuence in both di-

rections, and neither does their deﬁnition has a natural

scale, which is inferred by averaging over Haar-random

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QuantifyingQuantumCausalInuencesLucasHutter,1RafaelChaves,2,3RanieriVieiraNery,2GeorgeMoreno,2,4andDanielJostBrod11InstituteofPhysics,FederalUniversityFluminense,Niter´oi,Brazil2InternationalInstituteofPhysics,FederalUniversityofRioGrandedoNorte,59070-405Natal,Brazil3SchoolofScienceandTechnology,Fe...

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Quantifying Quantum Causal Inﬂuences Lucas Hutter1Rafael Chaves23Ranieri Vieira Nery2George Moreno24and Daniel Jost Brod1 1Institute of Physics Federal University Fluminense Niter oi Brazil

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