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Abstract
Heisenberg’s uncertainty principle is one of the main tenets of quantum theory. Nevertheless, and despite its fundamental importance for our understanding of quantum foundations, there has been some confusion in its interpretation: Although Heisenberg’s first argument was that the measurement of one observable on a quantum state necessarily disturbs another incompatible observable, standard uncertainty relations typically bound the indeterminacy of the outcomes when either one or the other observable is measured. In this paper, we quantify precisely Heisenberg’s intuition. Even if two incompatible observables cannot be measured together, one can still approximate their joint measurement, at the price of introducing some errors with respect to the ideal measurement of each of them. We present a tight relation characterizing the optimal tradeoff between the error on one observable vs. the error on the other. As a particular case, our approach allows us to characterize the disturbance of an observable induced by the approximate measurement of another one; we also derive a stronger error-disturbance relation for this scenario.
The discovery and development of quantum theory have generated passionate debates among its founding fathers. The surprising features of the theory [e.g., its probabilistic nature, its uncertainty principle (1), its nonlocality (2, 3)] were indeed too counterintuitive to satisfy all physicists: Einstein, for instance, famously argued that “God does not play dice” (4) and could not accept the apparent “spooky action at a distance” (5) that seemed to be allowed by the theory. Interestingly, it has since been realized that what first seemed to be limitations of the theory—the impossibility of perfectly predicting measurement outcomes and of explaining them with local hidden variables—can turn out to allow for useful applications for information processing, such as quantum cryptography (6). With the advent of quantum information science, it becomes all the more essential to clarify what can or cannot be done quantum mechanically.
The well-known uncertainty principle is typically expressed in terms of “uncertainty relations.” To fix the notations, let us define the SDs 
of two observables A and B in the state 
as
with 
and 
, and the “value” of the commutator 
in the state 
, divided by 2i, as
Robertson’s well-known uncertainty relation (7) then imposes that
Such uncertainty relations are often wrongly interpreted—even, historically, by some of the most illustrious authors (8⇓⇓–11)—as saying that one cannot jointly measure the observables A and B on the state 
when 
, or that the measurement of one observable necessarily disturbs the other. Although this last observation indeed corresponds to Heisenberg’s intuition (1), this is actually not what standard uncertainty relations imply, let alone quantify (12). Rather than referring to joint (or successive) measurements of two observables on one state, they indeed bound the statistical deviations of the measurement results of A and B when each measurement is performed many times on several independent, identically prepared quantum states.
In this paper, we aim instead at precisely quantifying Heisenberg’s original formulation of the uncertainty principle. Even if two observables A and B are incompatible and can indeed not be jointly measured on a state 
, it is still possible to approximate their joint measurement. How good can such an approximation be? What is the optimal tradeoff between the error induced on the measurement of A and the error on B? What is the optimal tradeoff between the error in the approximation of one observable and the disturbance implied on the other? We answer these questions below by deriving tight error-tradeoff and error-disturbance relations.
Approximate Joint Measurements
Let us start by setting up our general framework for approximate joint measurements. Our presentation is inspired by those of Ozawa (13, 14) and Hall (15), and it is restricted here to the basics; more details are given in SI Text, section A.
To approximate the measurement of an observable A on a quantum system in the state 
(in some Hilbert space 
), a general strategy consists of measuring another “approximate” observable 
, possibly on an extended Hilbert space (i.e., on the joint system composed of the state 
and an ancillary system in the state 
of another Hilbert space 
). In this picture, the impossible joint measurement of two incompatible observables A and B on 
can thus be approximated by the perfect joint measurement of two compatible (i.e., commuting) observables 
and 
on 
. Note that in full generality, we do not assume a priori (for now at least) that 
and 
must have the same spectrums as A and B.
Following Ozawa (13, 14, 16⇓–18), we characterize the quality of the approximations 
and 
of A and B, respectively, by defining the rms errors
These rms errors, which generalize standard definitions in classical estimation theory (19), quantify the statistical deviations between the approximations 
and 
and the ideal measurements of A and B. We refer to work by Ozawa (17, 18), Hall (20), and Lund and Wiseman (21) for discussions on the motivations and appropriateness of such definitions. There has been a controversy (22, 23) on the question of whether these quantities were experimentally accessible; two different indirect methods have nevertheless been proposed (18, 21) and recently implemented (24, 25).
Error-Tradeoff Relations for Joint Measurements
The fact that quantum theory forbids perfect joint measurements of incompatible observables implies that the rms errors 
can generally not take arbitrary values. Some limitations on their possible values have been obtained previously (13, 14, 16, 26⇓–28), which we review below. For historical reasons, such limitations are often referred to as uncertainty relations (for joint measurements). We will keep this terminology when we refer to previously derived relations; however, because such relations are not, strictly speaking, about uncertainty but about errors in the approximation of joint measurements, we prefer the terminology “error-tradeoff relations (for joint measurements).”
Heisenberg–Arthurs–Kelly Relation.
In his seminal paper, Heisenberg (1) argued that the measurement of the position q of a particle necessary implies a disturbance 
on its momentum p, and that this disturbance is all the more important as the precision of the measurement of q is large (or as the “error” 
is small), such that 
, where h is the Planck constant.
The formalization of Heisenberg’s intuition rapidly led to the derivation of general uncertainty relations in terms of SDs (as in 4) rather than in terms of error and disturbance. Nevertheless, it is commonly believed that a relation similar to Robertson’s uncertainty relation (7) should also restrict the possible values of the errors 
and 
on A and B in an approximate joint measurement in such a way that
Although it is debatable whether this is really how Heisenberg’s claims (1) should be interpreted and generalized, this relation is commonly attributed to Heisenberg in the literature (13, 14, 17, 18, 21, 24, 25, 28). Because it was actually first explicitly derived by Arthurs and Kelly (26) [for position and momentum measurements; it was generalized to arbitrary observables by Arthurs and Goodman (27)], we will call it the Heisenberg–Arthurs–Kelly relation.
This relation was indeed proven to hold, under some restrictive assumptions on the approximate joint measurements (14, 16, 26⇓–28); namely, it holds when 
and 
are such that the mean errors 
and 
are independent of the state 
. Because we are only interested here in one particular state 
, for which we may want to adapt our approximation strategy, such an assumption is quite unsatisfactory for our purposes: We indeed aim at characterizing the tradeoff between 
and 
for all possible approximate measurements, in which case the Heisenberg–Arthurs–Kelly relation [7] does not generally hold (12).
Ozawa’s Uncertainty Relation.
Only recently did Ozawa (14) show how one could derive a universally valid uncertainty relation for joint measurements, by adding two additional terms to the left-hand side of 7. His relation writes
[We also note that a very similar but inequivalent relation was derived by Hall (15), which involves the SDs 
and 
rather than 
and 
; a discussion is provided in SI Text, section D].
The three terms in Ozawa’s relation come from three independent uses of Robertson’s relation [4] to different pairs of observables. Although this indeed leads to a valid relation and allows one to exclude a large set of impossible values 
, this is not optimal because the three Robertson’s relations (and therefore Ozawa’s relation) generally cannot be saturated simultaneously.
A Tight Error-Tradeoff Relation for Joint Measurements.
Using a general geometric inequality for vectors in a Euclidean space (Lemma 1 in Methods), one can improve on the suboptimality of Ozawa’s proof and derive the following error-tradeoff relation for approximate joint measurements:
or, in its dimensionless version, when 
, with 
, 
, and 
:
The proof is detailed in Methods. It can easily be checked (SI Text, section D) that Ozawa’s relation [8] can be directly derived from our relation [9]. Interestingly, one observes that Ozawa’s relation [8] remains valid even if one drops the term 
, which is precisely the term that appears in the Heisenberg–Arthurs–Kelly relation [7].
Not only is our relation stronger than Ozawa’s, but it is actually tight: for any A, B, and 
, any values 
saturating inequality [9 and 10] can be obtained. This can even be achieved by projective measurements on 
, without introducing any ancillary system (explicit examples are provided in SI Text, section C). Hence, contrary to previously derived relations, ours does not tell only what cannot be done quantum mechanically but what can be done.
Fig. 1 illustrates the constraints imposed by the three error-tradeoff relations [7–10] in the plane 
. Our relation [10] thus characterizes precisely the optimal tradeoff between 
and 
in the general context of approximate measurements. The values below the thick red curve (Fig. 1) cannot be reached, whereas all values on and above the curve can be obtained by tuning the actual measurements 
and 
depending on how well one wants to measure one observable, at the expense of increasing the error on the other.
Fig. 1.
Error-tradeoff and error-disturbance relations. The figure illustrates (in the case 
) how the different error-tradeoff and error-disturbance relations [7–10 and 12] restrict the possible values of the normalized rms errors 
. Contrary to the Heisenberg–Arthurs–Kelly relation [7] (dashed blue curve), Ozawa’s relation [8] (thin dashed red curve) is always valid; however, it does not fully characterize the whole set of forbidden values for 
(dark-shaded area), which is precisely delimited by our relation [9 and 10] (thick solid red curve). Imposing the same-spectrum assumption can imply strictly stronger constraints, such as 12 for the case where 
and 
(thick dashed red curve); however, more values of 
are forbidden (light-shaded area). The theoretical values expected from the experiment of Erhart et al. (24) are also shown (dotted blue curve and + symbols); they do not saturate 12, except for 
or 
. On the other hand, in an ideal implementation, the experiment of Rozema et al. (25) would saturate our inequality [12].
Error-Disturbance Scenario and the Same-Spectrum Assumption.
Let us now consider a special case of our general framework for approximate joint measurements: that of the error-disturbance scenario, as first discussed by Heisenberg (1).
In this context, one considers the disturbance 
in the statistics of one observable, B, due to the unsharp measurement of another observable, A. The latter is typically approximated by the measurement 
of a probe (or ancillary system in the state 
), which interacts with the state 
via a unitary transformation U (13). In such a case, the approximation of A corresponds to the measurement of 
on 
, whereas the perturbed measurement of B after the interaction with the probe corresponds to the measurement of 
(note that 
and 
commute). This error-disturbance scenario can be cast into the same formalism as our joint measurement framework; the rms error 
is now interpreted as the rms disturbance 
of B, with formally the same definition (13): 
as defined in 6.
Any error-tradeoff relation derived in the more general framework of joint measurements thus remains valid in this error-disturbance scenario. In particular, when interpreting 
as the rms disturbance 
, Ozawa’s relation [8] writes:
This error-disturbance relation was actually introduced by Ozawa before its previous version 8 for joint measurements (13). In a similar manner, our error-tradeoff relation [9 and 10] also implies an error-disturbance relation, by simply replacing 
with 
.
The difference from the previous, more general scenario of joint measurements is not merely in the interpretation of 
, however. A crucial point is that 
now has the same spectrum as B; furthermore, it is typically (but often implicitly) assumed in the error-disturbance scenario that 
, and hence 
, also has the same spectrum as A (13, 21, 24, 25). Because of these constraints, one may expect stronger restrictions on the possible values of 
to hold and that stronger “error-disturbance relations” can be derived. (For simplicity, and by abuse of language, we call error-disturbance relation any error-tradeoff relation derived under the same-spectrum assumption, because this is the crucial difference between the two scenarios.)
To illustrate this, let us now restrict our study to the case of dichotomic observables A, B with eigenvalues ±1 (such that 
) and to states 
for which 
(which implies 
), as considered, for instance, in the experiments of Erhart et al. (24) and Rozema et al. (25). We show in Methods that in this particular case, and with the same-spectrum assumption (hence, 
as well), an analogous relation to our error-tradeoff relation [9 and 10] holds, where 
and 
are replaced by 
and 
, respectively:
This error-disturbance relation is strictly stronger than our error-tradeoff relation [9 and 10] (and stronger than Ozawa’s relation [11]). Furthermore, we show in SI Text, section C that it is tight when 
: For any A, B, and 
satisfying the constraints above, one can reach any values 
that saturate the inequality, using approximate measurements such that 
. The constraint that inequality [12] imposes on the possible values of 
is also illustrated in Fig. 1; note that contrary to our error-tradeoff relation [9 and 10], inequality [12] also bounds the possible values of 
from above (Fig. S1, Inset).
Let us finally mention that if one imposes the same-spectrum assumption on 
only (e.g., if one does not impose that 
in the specific error-disturbance scenario considered above has the same spectrum as A), one can also derive a similar tight error-disturbance relation (under the assumptions now that 
and 
), where only 
in our error-tradeoff relation [9 and 10] is replaced by 
[S20].
Example: Qubits.
As an illustration of our error-tradeoff and error-disturbance relations [9, 10, and 12], let us consider the simplest case of qubits. We choose 
to define the north pole of the Bloch sphere, and let 
and 
(where 
denotes a vector composed of the three Pauli matrices) be two ±1-valued qubit observables characterized by unit vectors 
and 
on the Bloch sphere, of polar and azimuthal angles 
and 
, respectively. We take 
, and we assume, for convenience, that 
.
For such a choice of 
, A, and B, one finds 
, 
, and 
. For 
, 10 then writes:
One can check that this error-tradeoff relation can simply be saturated by defining 
and 
to be projective measurements in the same eigenbasis, specified by any unit vector 
on the Bloch sphere with polar and azimuthal angles 
and 
. More specifically, for 
and 
, one obtains
Interestingly, 
and 
are independent of the polar angle θ of 
. In particular, note that one can thus have, for instance, 
even when 
is quite different from A; also, when 
comes close to the north or south pole of the Bloch sphere, one can have arbitrarily close projection directions 
leading to quite different values for 
and 
: in our case, 
for 
and 
for 
. These somewhat unexpected properties might only be artifacts of the particular definitions of errors we use; it would be interesting to investigate possible alternative definitions that do not exhibit such behaviors.
Let us now impose that 
and 
have the same spectrum as A and B (i.e., because A and B are here ±1-valued observables, 
). Furthermore, assuming that 
, we have 
. Inequality [12] then applies; it can be saturated in the error-disturbance scenario [with 
and 
] in the following way: Let 
for 
(i.e., 
with 
), and let 
be its (normalized) eigenvectors, corresponding to its eigenvalues ±1; we then define 
, 
with 
a unitary such that 
and 
[e.g., with 
, a CNOT unitary (29) in the 
basis], and 
. One then gets
Two experiments by Erhart et al. (24) and Rozema et al. (25) were recently reported, showing a violation of the Heisenberg–Arthurs–Kelly relation [7] (more specifically, of its error-disturbance version, where 
is replaced by 
) and a verification of Ozawa’s error-disturbance relation [11] in qubit systems.
The first experiment (24) measured neutron spins, using the indirect method proposed by Ozawa (18) to estimate the rms errors and rms disturbances 
. 
was estimated from the measurement of 
on 
(the eigenstate of σZ, corresponding to its eigenvalue +1), and it was followed by the measurement of 
; note that 
, 
, and 
. The expected theoretical values for the rms errors and rms disturbances were 
and 
. These are plotted in Fig. 1; one can see that they are not optimal because they do not saturate our tight error-disturbance relation [12]. From the analysis above, it appears that adding a rotation 
before the measurement of B would be enough to allow the experimental setup used by Erhart et al. (24) to saturate inequality [12], however.
The second experiment (25) measured the polarization of single photons, using weak measurements as proposed by Lund and Wiseman (21) to estimate the rms errors and rms disturbances. A was approximated from a measurement of variable strength based on a CNOT unitary. Because the weak measurements used to estimate 
and 
are not infinitely weak, they slightly perturb the state of the photon, adding some noise. However, in an ideal implementation, the experiment of Rozema et al. (25) would saturate the bound of our error-disturbance relation [12].
To finish with, let us emphasize that no experiment will ever demonstrate the universal validity of an uncertainty relation (or error-tradeoff or error-disturbance relations), however, despite what the title of the article by Erhart et al. (24) suggests. First, note that in order for such experiments to be conclusive, one needs to trust the implementation perfectly; otherwise, systematic errors in the preparation of 
or in the estimation procedure for 
and 
could radically change the values of the different terms in the relation, leading to unjustified conclusions (and possibly even “showing” a violation of a valid relation). All one can do then is check that in that particular (perfectly trusted) implementation, for some particular A, B, and 
and for the particular approximations 
and 
implemented in that experiment, the error-tradeoff or error-disturbance relation of interest is satisfied. There is indeed no way experimentally to test all possible approximate joint measurement strategies, and the particular choice of 
and 
could be nonoptimal (e.g., as in ref. 24). It is, of course, trivial to obtain data satisfying an error-tradeoff relation if one does not try to optimize the values of 
: If the relation is universally valid, then any measurement strategy (e.g., outputting random results) will satisfy it. One can even similarly trivially violate the Heisenberg–Arthurs–Kelly relation [7] [e.g., by actually measuring A perfectly (so that 
) and outputting any values to approximate B (as long as 
)]. What is less trivial, and therefore more interesting, is to show experimentally that a tight error-tradeoff or error-disturbance relation can indeed be saturated.
Discussion
We have presented a state-dependent error-tradeoff relation [9 and 10] in the general framework of approximate joint measurements. Our relation is universally valid, whether the Hilbert spaces of interest are of finite, as in our qubit example, or infinite dimensions (provided 
is in the domains of 
and of all their products that are involved in the proof of 9 and 10), for example, for the measurement of position and momentum, as first considered by Heisenberg (1). Note also that although the framework for joint measurements was presented for pure states, it can easily be generalized to mixed states, and 9 and 10 still hold. Importantly, our new error-tradeoff relation was shown to be tight, and therefore to characterize fully the whole set of possible values of rms errors 
(in the case of pure states; our relation may in general not be tight for mixed states).
Error-tradeoff relations imply error-disturbance relations as a particular case. However, because of the same-spectrum assumption, strictly stronger relations can generally be derived in the error-disturbance scenario; we presented an example of such an error-disturbance relation, for ±1-valued observables with 
, allowing us to highlight a quantitative difference between the two scenarios. The derivation of a more general relation under the same-spectrum assumption is left for future work.
Our relations apply to the projective measurement of two observables A and B. It would be interesting to see if these could be generalized to some positive operator-valued measures (see refs. 15, 20, however, for the difficulties encountered) or to more observables (30). In the error-disturbance scenario, it may also be desirable to quantify the disturbance of the quantum state directly rather than that of the statistics of another observable; this is left as an open problem.
Our relations bound the rms errors of A and B, as defined in Eqs. 5 and 6. In the context of quantum information, one may prefer to use information-theoretic definitions for the quality of approximations, however. Developing such definitions, and deriving corresponding universally valid and tight error-tradeoff or error-disturbance relations would certainly be an interesting direction of research. This may indeed give a clearer operational meaning to such relations, and would be more adapted to their use in possible applications [in the same way, e.g., as entropic uncertainty relations are useful to prove the security of quantum cryptographic protocols (31, 32)]. This will involve radically different proof techniques, which may also allow one to consider error tradeoffs in general probabilistic theories not restricted to quantum theory and to its Hilbert space formalism. This will undoubtedly give more insight on the still puzzling, multifaceted uncertainty principle.
Methods
To prove our error-tradeoff and error-disturbance relations [9, 10, and 12], we start by introducing two general inequalities for real vectors.
Lemmas: Geometric Inequalities.
Let 
be two unit vectors of a Euclidean space 
, and let us define 
. We prove in SI Text, section B the following lemmas.
Lemma 1.
For any two orthogonal vectors 
and 
of 
, one has
Lemma 2.
For any two orthogonal unit vectors 
and 
of 
, defining 
and 
, one has
Proof of Our Error-Tradeoff Relation.
Let us now define, in the nontrivial case 
, the ket vectors
By writing these vectors in any orthonormal basis of 
(e.g., the common eigenbasis of 
and 
) and denoting by Re and Im their real and imaginary parts, respectively, one can define the following real vectors:
One then has
Hence, the (normalized) rms errors 
, 
can be interpreted as distances between vectors (17, 20), whereas the commutativity of 
and 
translates into an orthogonality condition for 
and 
.
The vectors 
, 
, 
, and 
thus satisfy the assumptions of Lemma 1, that 
and 
(Fig. 2). Together with Eqs. 24–26, inequality [16] implies our general error-tradeoff relation for joint measurements [10]. After multiplication by ΔA2 ΔB2, we obtain 9 (for which the case ΔA ΔB = 0 is trivial, because it implies CAB = 0).
Fig. 2.
Geometric construction used in the proof of our general error-tradeoff relation [9 and 10]. The real vectors 
, 
, 
, and 
satisfy the assumptions of Lemma 1; the particular choice of vectors illustrated here [for which 
, 
, and 
] saturates inequality [16], which quantifies the optimal tradeoff between the distance from the unit vector 
to an axis along a direction 
, and from the unit vector 
to an axis along a direction 
, orthogonal to 
.
Proof of Our Error-Disturbance Relation for the Case Where 
and 〈A〉 = 〈B〉 = 0.
With the assumptions that A2 = B2 = 1 and 〈A〉 = 〈B〉 = 0 (hence, ΔA = ΔB = 1) and that 
and 
have the same spectrum as A and B (hence, 
), the real vectors 
and 
defined as in 20 and 21 are now such that (with 
in the error-disturbance scenario)
The vectors 
, 
, 
, and 
thus satisfy the assumptions of Lemma 2 (Fig. 3). Together with Eqs. 29–31, inequality [17] gives our error-disturbance relation [12].
Fig. 3.
Geometric construction used in the proof of our error-disturbance relation [12] (for the case where 
and 
). The real vectors 
, 
, 
, and 
satisfy the assumptions of Lemma 2; the particular choice of vectors illustrated here (for which 
, 
, and 
) saturates inequality [17], which quantifies the optimal tradeoff between the distance from the unit vector 
to another unit vector 
, and from the unit vector 
to another unit vector 
, orthogonal to 
.
Acknowledgments
I thank M. J. W. Hall for fruitful discussions and comments on an earlier version of this manuscript. This work was supported by a University of Queensland postdoctoral research fellowship.
Footnotes
- ↵1E-mail: c.branciard{at}physics.uq.edu.au.
Author contributions: C.B. designed research, performed research, and wrote the paper.
The author declares no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1219331110/-/DCSupplemental.
Freely available online through the PNAS open access option.
References
Error-tradeoff relations for joint measurements
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