Abstract
Defining and measuring the error of a measurement is one of the most fundamental activities in experimental science. However, quantum theory shows a peculiar difficulty in extending the classical notion of rootmeansquare (rms) error to quantum measurements. A straightforward generalization based on the noiseoperator was used to reformulate Heisenberg’s uncertainty relation on the accuracy of simultaneous measurements to be universally valid and made the conventional formulation testable to observe its violation. Recently, its reliability was examined based on an anomaly that the error vanishes for some inaccurate measurements, in which the meter does not commute with the measured observable. Here, we propose an improved definition for a quantum generalization of the classical rms error, which is statedependent, operationally definable, and perfectly characterizes accurate measurements. Moreover, it is shown that the new notion maintains the previously obtained universally valid uncertainty relations and their experimental confirmations without changing their forms and interpretations, in contrast to a prevailing view that a statedependent formulation for measurement uncertainty relation is not tenable.
Introduction
The notion of the mean error of a measurement of a classical physical quantity was first introduced by Laplace^{1} (p. 324) as the mean of the absolute value of the error. Subsequently, the rootmeansquare (rms) error was introduced by Gauss^{2} (p. 39) as a mathematically more tractable definition to derive the principle of the least square, and has been broadly accepted as the standard definition for the mean error of a measurement. In those approaches the error of a measurement of a quantity Θ is defined as N = Ω − Θ, where Ω is the quantity actually observed, here we call the meter quantity. Then Gauss’s rms error is defined as 〈N^{2}〉^{1/2}, where \(\langle \cdots \rangle\) stands for the mean value, while Laplace’s mean error as 〈N〉. From the above definition, Gauss’s rms error ε_{G} is determined by the joint probability distribution
of Θ and Ω as
so that ε_{G}(μ) = 〈N^{2}〉^{1/2}, and it perfectly characterizes accurate measurements: ε_{G}(μ) = 0 if and only if Ω = Θ holds with probability 1, i.e., \({\sum} \{ \mu (\theta ,\omega )\theta = \omega \} = 1\).
A straightforward generalization of Gauss’s definition to quantum measurements has been introduced as follows.^{3,4,5} Let A be an observable of a system S, described by a Hilbert space \({\cal H}\), to be measured by a measuring process M. Let M be an observable representing the meter of the observer in the environment E described by a Hilbert space \({\cal K}\). The Hilbert spaces \({\cal H}\) and \({\cal K}\) are supposed to be finite dimensional throughout the present paper for simplicity of the presentation, although the arguments supporting the main results are extended to the infinite dimensional case with wellknown mathematical methods. The time evolution of the total system S + E during the measuring interaction with the total Hamiltonian H determines the Heisenberg operators A(0), M(τ) with 0 < τ, where
To obtain the outcome x of this measurement the observer measures the observable M(τ) (i.e., measures the meter observable M just after the interaction), instead of measuring A(0) (i.e., measuring A just before the interaction). The error of this measurement is naturally identified with the observable, called the noise operator, defined by
(refs. ^{6,7}). Let ψ〉 and ξ〉 be the initial states of S and E, respectively. The noiseoperator based quantum rootmeansquare (qrms) error of this measurement is defined as
where ψ, ξ〉 = ψ〉ξ〉.^{3,4,5}
This notion was used to reformulate Heisenberg’s uncertainty relation for the accuracy of simultaneous measurements to be universally valid^{8,9,10,11,12,13,14,15,16,17} and made the conventional formulation testable to observe its violation.^{18,19,20,21,22,23,24}
Recently, Busch, Lahti, and Werner (BLW)^{25} raised a reliability problem for quantum generalizations of the classical rms error, comparing the noiseoperator based qrms error with the Wasserstein 2distance, another error measure based on the distance between probability measures, and pointed out several discrepancies between those two error measures in favor of the latter.
In order to resolve the conflict, here we introduce the following requirements for any sensible error measure generalizing the classical rootmeansquare error: (I) the operational definability, (II) the correspondence principle, (III) the soundness, and (IV) the completeness. The operational definability ensures that the error measure is definable by the operational description of the measuring process. The correspondence principle ensures that the error measure is consistent with the classical rms error in the case when the latter is also applicable. The soundness ensures that the error measure vanishes for any accurate measurements, while the completeness ensures that the error measure does not vanish for any inaccurate measurements. As shown later, the noiseoperator based qrms error ε_{NO} satisfies all the requirements (I)–(III) except (IV), whereas any error measures based on the distance of probability measures, such as the Wasserstein 2distance, satisfy (I) and (III) but do not satisfy (II) nor (IV). We propose an improved definition for a quantum generalization of the classical rms error, which is still based on the noise operator but satisfies all requirements (I)–(IV). Moreover, it is shown that the new error measure maintains the previously obtained universally valid uncertainty relations^{8,9,10,11,12,13,14,15,16,17} and their experimental confirmations^{18,19,20,21,22,23,24} without changing their forms and interpretations, in contrast to a prevailing view that a statedependent formulation for measurement uncertainty relation is not tenable.^{25,26,27}
Results
Operational definability
The probability distribution of the output x of the measurement is given by
where P^{M(τ)}(x) is the spectral projection of M(τ) for \(x \in {\Bbb R}\), i.e., P^{M(τ)}(x) is the projection with range {Ψ〉 ∈\({\cal H}\)⊗\({\cal K}\) M(τ)Ψ〉 = xΨ〉}. It is fairly wellknown that every measuring process has its probability operatorvalued measure (POVM) that operationally describes the statistics of the measurement outcome.^{28,29,30,31} The POVM Π of the measuring process M is a family \({\mathrm{{\Pi}}} = \{ {\mathrm{{\Pi}}}(x)\} _{x \in {\Bbb R}}\) of positive operators on \({\cal H}\) defined by
and satisfies the generalized Born formula
We consider the requirements for any quantum generalization ε of the classical rootmeansquare error ε_{G} to quantify the mean error ε (A, M, ψ〉) of the measurement of an observable A in a state ψ〉 described by a measuring process M; we shall also write ε (A, M, ρ) if the state is represented by a density operator ρ. The first requirement is formulated using the notion of POVM as follows.
(I) Operational definability. The error measure ε should be definable by the POVM Π of the measuring process M, the observable A to be measured, and the initial state ψ〉 of the measured system S.
The operational definability determines the mathematical domain of the error measure and requires that the mean error (i.e, the value of the error measure in the given state) should be determined by the operational description of the statistics of measurement outcomes.
The nth moment operator \(\hat {\Pi}^{(n)}\) of the POVM Π is defined by
We write \({\hat{\mathrm {\Pi}}} = {\hat{\mathrm {\Pi}}}^{(1)}\). Then the relation
holds (ref. ^{11}, Theorem 4.5).
Thus, ε_{NO} can be defined by the observable A, the POVM Π, and the state ψ〉, so that it satisfies the operational definability. In what follows, we shall write ε_{NO}(A, Π, ψ〉) = ε_{NO}(A, M, ψ〉) if Π is the POVM of M.
Correspondence principle
The second requirement is based on a common practice in generalizing a classical notion to quantum mechanics. Even in quantum mechanics, there are cases where the original classical notions are directly applicable, and in those cases the generalized notions should be consistent with the original ones.
In the problem of generalizing the classical rootmeansquare error to quantum mechanics, this principle is applied to the case where A(0) and M(τ) commute as two operators. In this case, the observables A(0) and \(M(\tau )\) are jointly measurable and their joint probability distribution μ(x, y) is given by
Then we can apply the classical definition of the rootmeansquare error to the joint probability distribution μ to obtain the classical rootmeansquare error ε_{G}(μ) of this measurement; in this case, the measuring process is classically described as a blackbox with the input–output joint probability distribution μ(x, y). Thus, the quantum generalization ε should satisfy
Thus, we should require that Eq. (14) holds if A(0) and M(τ) commute. However, we should proceed further to avoid possible inconsistencies, since there is a case where a pair of observables commute only on a subspace and they have the joint probability distribution only for states in that subspace as discussed by von Neumann^{32} (p. 230). To include such a general situation, we define the notions of commutativity and joint probability distribution in a satedependent manner. We say that observables X and Y commute in a state Ψ〉 if
for any x, y. A probability distribution μ(x, y) on \({\Bbb R}^2\), i.e., μ(x, y) ≥ 0 and \(\mathop {\sum}\nolimits_{x,y} \mu (x,y) = 1\), is called a joint probability distribution (JPD) of observables X, Y in Ψ〉 if
for any polynomial f(X, Y) of observables X, Y. Then, there exists a JPD of observables X, Y in ψ〉 if and only if X and Y commute in ψ〉 as shown in Theorem 1 in Methods. In this case, the JPD μ is uniquely determined by
To prevent the inconsistency between the original classical notion and its quantum generalization we pose the following requirement.
(II) Correspondence principle. In the case where A(0) and M(τ) commute in the initial state ψ, ξ〉, then the relation
should hold for the JPD μ of A(0) and M(τ) in ψ, ξ〉.
Suppose that A(0) and M(τ) commute in ψ, ξ〉. Let μ be their JPD in ψ, ξ〉. From Eqs. (2), (7) and (16) we have
Thus, the noiseoperator based qrms error ε_{NO} satisfies the correspondence principle.
Soundness
To discuss the soundness we need to clarify what measuring process M is considered to accurately measure an observable A in a given state ψ〉. This fundamental problem has, to the best of our knowledge, not been discussed in the literature except for our previous investigations,^{33,34,35,36} in which we introduced the following definition. We say that the measuring process M accurately measures an observable A in a state ψ〉 if A(0) and M(τ) commute in ψ, ξ〉 and their JPD μ satisfies μ(A(0) = M(τ)) = 1, where \(\mu (A(0) = M(\tau )) = \mathop {\sum}\nolimits_{x,y:x = y} \mu (x,y)\). We will provide a justification of this definition including its operational accessibility in Methods: statedependent definition for accurate measurements of quantum observables.
Under the above definition we pose the soundness requirement.
(III) Soundness. The error measure ε should vanish for any accurate measurements.
Now, we can see that any error measure ε satisfying the correspondence principle, (II), also satisfies the soundness, (III), since in this case we have the JPD μ of A(0) and M(τ) satisfying ε (A, M, ψ〉) = ε_{G}(μ) = 0.
Since the noiseoperator based qrms error ε_{NO} satisfies the correspondence principle, (II), it also satisfies the Soundness, (III).
Note that if A is accurately measured in ψ〉, then A and Π are identically distributed in ψ〉.
Completeness
Now, we introduce the following requirement.
(IV) Completeness. The measurement should be accurate if the error measure ε vanishes.
Busch, Heinonen, and Lahti^{37} (p. 263) pointed out that there is a measuring process M such that ε_{NO}(A, M, ψ〉) = 0 but M does not accurately measure A in ψ〉. For a simple example, let
with Π(y) = P^{M}(y). Then we have ε_{NO}(A, Π, ψ〉) = 0, but the measurement is not accurate, since A and Π are not identically distributed as 〈ψP^{A}(2)ψ〉 = 1/2 but 〈ψΠ(2)ψ〉 = 0.
Thus, the noiseoperator based qrms error ε_{NO} does not satisfy the completeness requirement. As shown above, the noiseoperator based qrms error ε_{NO} satisfies all the requirements (I)–(III) but does not satisfy (IV).
Locally uniform quantum rootmeansquare error
We call any error measure ε satisfying (I) and (II) a quantum rootmeansquare (qrms) error. A qrms error ε is said to be sound if it satisfies (III). It is said to be complete if it satisfies (IV). A sound and complete error measure correctly indicates the cases where the measurement is accurate and where not (Busch^{25}, p. 1263). A primary purpose of this paper is to find a sound and complete qrms error, and to establish universally valid uncertainty relations based on it.
We shall show that there is a simple method to strengthen the noiseoperator based qrms error to obtain a sound and complete qrms error. In addition to (I)–(IV), this error measure is shown to have the following two properties.
(V) Dominating property. The error measure ε dominates the noiseoperator based qrms error ε_{NO}, i.e., ε_{NO} (A, Π, ψ〉) ≤ ε (A, Π, ψ〉) for all A, Π, ψ〉.
(VI) Conservation property for dichotomic measurements. The error measure ε coincides with the noiseoperator based qrms error ε_{NO} for dichotomic measurements, i.e., ε_{NO} (A, Π, ψ〉) = ε(A, Π, ψ〉) if \(A^2 = {\hat{\mathrm {\Pi}}}^{(2)} = I\).
For any \(t \in {\Bbb R}\), define
We call \(\{ \varepsilon _t(A,{\Pi},\Psi \rangle )\} _{t \in {\Bbb R}}\) the qrms error profile for A and Π in ψ〉. If A(0) and M(τ) commute in the state ψ, ξ〉, then we have
for all \(t \in {\Bbb R}\). Thus, the qrms error profile is considered to provide additional information about the error of measurement M in the case where A(0) and M(τ) do not commute in the state ψ, ξ〉.
To obtain a numerical error measure from \(\{ \varepsilon _t(A,{\Pi},\psi \rangle )\} _{t \in {\Bbb R}}\), we define the locally uniform qrms error by
Then \(\bar \varepsilon\) is a sound and complete qrms error, satisfying both the dominating property, (V), and the conservation property for dichotomic measurements, (VI), as shown in Theorem 3 in Methods: sound and complete quantum rootmeansquare errors, where we introduce other two sorts of qrms errors to clarify the physical motivation behind the above definition.
For the example given in Eq. (20), we have
despite of the relation ε_{NO}(A, Π, ψ〉) = 0, the relation \(\bar \varepsilon (A,{\mathrm{{\Pi}}},\psi \rangle ) = 2\) correctly indicate that the measurement of A described by Eq. (20) is not an accurate measurement.
Discussion
Wasserstein 2distance
In what follows, we shall show that the Wasserstein 2distance satisfies the operational definability, (I), and the soundness, (III), but does not satisfy the correspondence principle, (II), nor the completeness, (IV).
Let \(\mu _{\psi \rangle }^A\,{\mathrm{and}}\,\mu _{\psi \rangle }^{\Pi}\) be the probability distributions of A and Π in state ψ〉, i.e.,
BLW^{25} advocated the Wasserstein 2distance \(W_2\left( {\mu _{\psi \rangle }^A,\mu _{\psi \rangle }^{\mathrm{{\Pi}}}} \right)\) between \(\mu _{\psi \rangle }^A\) and \(\mu _{\psi \rangle }^{\Pi}\) as an alternative quantum generalization of the classical rms error in comparison with the noiseoperator based qrms error ε_{NO}(A, Π, ψ〉). The Wasserstein 2distance is defined as
where the infimum is taken over all the probability distributions γ(x, y) on \(\Bbb R^{2}\) such that \(\gamma ({x},{\Bbb {R}}) = \mu _{{\psi} {\rangle}}^{A}(x)\) and \(\gamma ({\Bbb {R}},{y}) = \mu _{{\psi} {\rangle} }^{\mathrm{{\Pi}}}(y)\), where we write \(\gamma ({x},{\Bbb R}) = \mathop {\sum}\nolimits_{{y}{\in}{\Bbb R}}\gamma ({x},{y})\) etc. Thus, \(W_{2}\left( {\mu _{\psi \rangle }^{A},\mu _{\psi \rangle }^{\mathrm{{\Pi}}}} \right)\) satisfies the operational definability, (I). It should be pointed out that the Wasserstein 2distance \(W_{2}\left( {\mu _{\psi \rangle }^{A},\mu _{\psi \rangle }^{\mathrm{{\Pi}}}} \right)\) does not satisfy the correspondence principle, (II). To see this, suppose that A(0) and M(τ) commute in ψ, ξ〉. In this case, we have
where Cov = 〈(A(0) − a)(M(τ) − m)〉, Bias = a − m, a = 〈A(0)〉, and m = 〈M(τ)〉. The JPD μ(x, y) always satisfies the condition that \(\mu (x,{\Bbb R}) = \mu _{\psi \rangle }^A(x)\) and \(\mu ({\Bbb R},y) = \mu _{\psi \rangle }^{\mathrm{{\Pi}}}(y)\). Thus, we have
For the case where \(\mu _{\psi \rangle }^A = \mu _{\psi \rangle }^{\mathrm{{\Pi}}}\), we have \(W_2(\mu _{\psi \rangle }^A,\mu _{\psi \rangle }^{\mathrm{{\Pi}}}) = 0\), but ε_{G}(μ) = 0 only if μ(A(0) = M(τ)) = 1. To consider a typical case where ε_{G}(μ) > 0, suppose that A(0) and M(τ) are independent. Then we have σ(A(0)) = σ(M(τ)), Cov = 0, and Bias = 0, and hence \(\varepsilon _G(\mu ) = \sqrt {2\sigma } (A)\). Thus, ε_{G} (μ) > 0 whenever σ (A) > 0. For instance, let
Then, we have the joint probability distribution μ for A(0) and M(τ) in ψ, ξ〉 for arbitrary ξ〉 such that
We have \(\mu _{\psi \rangle }^A( + 1) = \mu _{\psi \rangle }^{\mathrm{{\Pi}}}( + 1) = 1/3\), \(\mu _{\psi \rangle }^A(  1) = \mu _\psi ^{\mathrm{{\Pi}}}(  1) = 2/3\), and A(0) and M(τ) are independent. Thus we have \(W_2(\mu _{\psi \rangle }^A,\mu _{\psi \rangle }^{\mathrm{{\Pi}}}) = 0\), but \(\sigma (A) = 2\sqrt 2 /3\) and ε_{G}(μ) = ε_{NO}(A, Π, ψ〉) = 4/3. Thus, the Wasserstein 2distance does not satisfy the correspondence principle, (II).
Note that the above example also shows that the Wasserstein 2distance \(W_2(\mu _{\psi \rangle }^A,\mu _{\psi \rangle }^{\mathrm{{\Pi}}})\) does not satisfy the completeness, (IV), whereas it satisfies the soundness, (III), since ε_{G}(μ) = 0 holds in Eq. (29) for any accurate measurement.
The logical relationships among requirements (I)–(IV) are summarized as follows. Under the major premise (I), we have shown that (i) (III) follows from (II), (ii) (II) does not follow from (III), since the Wasserstein 2distance satisfies (III) but does not satisfies (II), and that (iii) (II) and (IV) are independent, since ε_{NO} satisfies (II) but does not satisfy (IV) and since there exists an error measure ε satisfying (IV) but not satisfying (II), e.g. \(\varepsilon (A,{\Pi},\psi \rangle ) = \sup_{\phi \rangle }\varepsilon _{\mathrm{NO}}(A,{\Pi},\phi \rangle )\), where ϕ〉 varies over all the states. Note that there exists an error measure ε satisfying (I), (III), and (IV), but does not satisfy (II), e.g., \(\varepsilon (A,{\mathrm{{\Pi}}},\psi \rangle ) = \sup_{\phi \rangle }\varepsilon _{\mathrm{NO}}(A,{\mathrm{{\Pi}}},\phi \rangle )\), where ϕ〉 varies over the cyclic subspace \({\cal C}(A,\psi \rangle )\) generated by A and ψ〉.^{38}
Universally valid uncertainty relations
In what follows, we shall show that all the universally valid measurement uncertainty relations obtained so far^{8,10,11,13,14,15,16} for the noiseoperator based qrms error ε_{NO} are maintained by the locally uniform qrms error \(\bar \varepsilon\) with the same forms by property (V) and that their experimental confirmations reported so far^{18,19,20,21,22,23,24} for dichotomic measurements are also reinterpreted to confirm the relations for the new error measure \(\bar \varepsilon\) by property (VI). Moreover, the stateindependent formulation based on this notion maintains Heisenberg’s original form for the measurement uncertainty relation, whereas the statedependent formulation violates it. The new error measure \({\bar {\varepsilon}}\) thus clears a prevailing view that the statedependent formulation of measurement uncertainty relations is not tenable.^{25,26,27}
Let A, B be two observables of a quantum system S described by a Hilbert space \({\cal H}\). Any simultaneous measurement of A and B in a state ψ〉 defines a joint POVM Π(x, y) on \({\Bbb R}^2\) for the Hilbert space \({\cal H}\), for which the marginal POVM \({\mathrm{{\Pi}}}_A(x) = {\mathrm{{\Pi}}}(x,{\Bbb R})\) describes the Ameasurement and the marginal POVM \({\mathrm{{\Pi}}}_B(y) = {\mathrm{{\Pi}}}({\Bbb R},y)\) describes the Bmeasurement.^{12} Then the mean errors of the simultaneous measurement of A and B described by the joint POVM Π(x, y) in the state ψ〉 are defined as ε(A, Π_{A}, ψ〉) and ε(B, Π_{B}, ψ〉), respectively, for a given qrms error ε. In what follows we abbreviate ε(A) to ε(A, Π_{A}, ψ〉) and ε(B) to ε(B, Π_{B}, ψ〉) unless confusion may occur.
The above general formulation includes the errordisturbance relation for the Ameasurement error of a measuring process M and the thereby caused disturbance on B, since the Bdisturbance is generally defined by the error of the accurate Bmeasurement following the Ameasurement.^{8,11,39} This definition of the Bdisturbance is described in the Heisenberg picture as follows. Given a measuring process M, we can make an accurate simultaneous measurement of commuting observables M(τ) and B(τ). Then an approximate simultaneous measurement of A(0) and B(0) is obtained if the measurement of A(0) is replaced by the accurate measurement of M(τ) and the measurement of B(0) is replaced by the accurate measurement of B(τ). This simultaneous measurement is described by the joint POVM Π defined by
In this case, for a given qrms error measure ε, we define the mean error ε(A, M, ψ〉) of the A measurement carried out by M in ψ〉 as ε(A, M, ψ〉) = ε(A, Π_{A}, ψ〉) and the mean disturbance η(B, M, ψ〉) of B caused by M in ψ〉 as η(B, M, ψ〉) = ε(B, Π_{B}, ψ〉). In what follows we abbreviate ε(A) to \(\varepsilon (A,{\mathbf{M}},\psi \rangle )\) and η(B) to η(B, M, ψ〉) unless confusion may occur.
As above, any general relation for ε(A) and ε(B) implies a general relation for ε(A) and η(B), while any counter example for a general relation for ε(A) and η(B) is also a counter example for the corresponding relation for ε(A) and ε(B).
In this respect, it should be noted that the recent claim by Korzekwa, Jennings, and Rudolph (KJR)^{27} of the impossibility of statedependent errordisturbance relations is unfounded. In fact, KJR admitted that their basic assumption called the operational requirement (RO) should be applied to the notion of disturbance, but cannot be applied to the notion of error (KJR^{27}, p. 0521086); it can be easily seen that if (RO) were to be applied to the error, it would contradict the correspondence principle. However, such a discrimination between the disturbance and the error contradicts the above standard definition of the disturbance as the error of a successive measurement.
Heisenberg’s original formulation of the uncertainty principle states that canonically conjugate observables Q, P can be measured simultaneously only with a characteristic constraint (Heisenberg^{40}, p. 172)
where the unambiguous lower bound \(\hbar /2\) is due to a subsequent elaboration by Kennard^{41} (see also ref. ^{42}). Heisenberg justified this relation under the repeatability hypothesis or its approximate version, an obsolete assumption on the state change in measurement; see ref. ^{42} for a detailed discussion.
A counter example of Heisenberg’s relation (38) was given in ref. ^{43} in the errordisturbance scenario with ε = ε_{NO}, using a position measuring model originally constructed in ref. ^{44} to invalidate the standard quantum limit for gravitationalwave detectors with freemass probe.^{45,46} In ref. ^{47} continuously many linear position measuring processes including the above have been constructed that violate Heisenberg’s relation (38) in the errordisturbance scenario for an arbitrary choice of the qrms error ε. Thus, the violation of Heisenberg’s relation (38) is not due to a particular choice of the qrms error ε.
In contrast to the violation of Eq. (38) in the statedependent formulation, Appleby^{48} showed the relation
holds for ε = ε_{NO}, except for the case where \(\sup_{\psi \rangle }\varepsilon_{\mathrm{NO}} (Q,{\mathrm{{\Pi}}}_Q,\psi \rangle ) = 0\) or \(\sup_{\psi \rangle }\varepsilon_{\mathrm{NO}} (P,{\mathrm{{\Pi}}}_P,\psi \rangle ) = 0\), where the supremum is taken over all the possible states ψ〉. An apparent drawback of the above relation is that the stateindependent error measures \(\sup_{\psi \rangle }\varepsilon_{\mathrm{NO}} (Q,{\mathrm{{\Pi}}}_Q,\psi \rangle )\) and \(\sup_{\psi \rangle }\varepsilon_{\mathrm{NO}} (P,{\mathrm{{\Pi}}}_P,\psi \rangle )\) are defined by the qrms error ε_{NO} that is not complete. However, this drawback turns out to be immediately cleared if one uses the locally uniform qrms error \(\bar \varepsilon\) instead, since the relation
holds obviously for any observable X. Thus, Eq. (39) holds for \(\varepsilon = \bar \varepsilon\), one of the sound and complete qrms errors. It should be noted that in the stateindependent formulation as above the error measures \(\sup_{\psi \rangle }\varepsilon (Q,{\mathrm{{\Pi}}}_Q,\psi \rangle )\) and \(\sup_{\psi \rangle }\varepsilon (P,{\mathrm{{\Pi}}}_P,\psi \rangle )\) often diverges.^{47,48} Even in the original γray thought experiment, the error measure \(\sup_{\psi \rangle }\varepsilon (Q,{\mathrm{{\Pi}}}_Q,\psi \rangle )\) diverges as the wave packet goes beyond the scope of the microscope. Thus, Heisenberg’s original form holds in the stateindependent formulation but not due to the tradeoff between the resolution power and the Compton recoil. The notion of the resolution power of a microscope is welldefined only in the case where the object is welllocalized in the scope of the microscope, and it cannot be captured by the stateindependent formulation. The above remarks are also applied to the recent revival of the stateindependent formulation by Busch, Lahti, and Werner;^{49,50} in fact, the BuschLahtiWerner formulation in ref. ^{49} is equivalent to Appleby’s formulation^{48} for any linear measurements.^{47} For detailed discussions, we refer the reader to ref. ^{47}.
A generalization of Heisenberg’s relation (38) to arbitrary pair of observables A and B is obtained by using the noiseoperator based rms error ε = ε_{NO} as the relation
where \(C_{A,B} = \frac{1}{2}\langle \psi [A,B]\psi \rangle \), holding for any joint POVMs with unbiased or independent noise operators^{3,4,6,7,10,51,52} (see also refs. ^{53,54}). By the dominating property, (V), the above relation also holds for the locally uniform rms error \(\varepsilon = \bar \varepsilon\).
Using the noiseoperator based qrms error ε = ε_{NO}, the first universally valid relation
was given in 2003 (refs. ^{8,10}), which is universally valid for any observables A,B, any system state ψ〉, and any joint POVM Π, where the standard deviations σ(A), σ(B) are taken in the state ψ〉. By the dominating property, (V), the above relation also holds for the locally uniform qrms error \(\varepsilon = \bar \varepsilon\). Thus, we have a statedependent universally valid uncertainty relation for simultaneous measurements described by a sound and complete qrms error.
Using the noiseoperator based qrms error ε = ε_{NO}, Branciard^{15,16} considerably strengthened the above universally valid relation (42) as well as the relations proposed by Hall^{13} and by Weston, Hall, Palsson, and Wiseman^{14} in several ways. All those Branciard relations also hold for the locally uniform qrms error \(\varepsilon = \bar \varepsilon\) by the dominating property, (V); see Branciard^{16} [Section IV] for the alternative forms of the above mentioned relations to which the dominating property can directly apply.
Those universally valid relations for the noiseoperator based qrms error have already been experimentally confirmed in the errordisturbance scenario for dichotomic measurements (i.e., A(0)^{2} = B(0)^{2} = M(τ)^{2} = B(τ)^{2} = I) with observing the violation of Eq. (41).^{18,19,20,21,22,23,24} Interestingly, the above experiments were intended to confirm relations for the noiseoperator based qrms error ε = ε_{NO}, but they also can be reinterpreted as confirmations for the corresponding relations and the violation of Eq. (41) with the locally uniform qrms error \(\varepsilon = \bar \varepsilon\), one of sound and complete qrms errors, since in those experiments we have \(\varepsilon _{\mathrm{NO}} = \bar \varepsilon\) by the conservation property for dichotomic measurements, (VI). Thus, we already have a welldeveloped theory of statedependent measurement uncertainty relations based on a sound and complete qrms error, in contrast to a prevailing claim that the statedependent formulation of measurement uncertainty relations is not tenable.^{25,26,27}
Methods
Statedependent commutativity and joint probability distributions
The statedependent notion of commutativity was originally discussed by von Neumann^{32} (p. 230) as follows. Suppose that Ψ〉 is a superposition of common eigenstates of X and Y, namely, there exists an orthonormal family {x, y〉} of states such that Xx, y〉 = xx, y〉, Yx, y〉 = yx, y〉, and that \({\mathrm{\Psi }}\rangle = \mathop {\sum}\nolimits_{x,y} x,y\rangle \langle x,y\Psi \rangle\). In this case, a measurement of the observable
with a onetoone assignment of real values \((x,y) \mapsto z_{x,y}\) gives a joint measurement of X and Y in the state Ψ〉 and their joint probability distribution μ(x, y) = Pr{X = x, Y = y} of X and Y is given by
In this case, X and Y commute on the subspace \({\cal M}\) spanned by {x, y〉} but do not necessarily commute on \({\cal M}^ \bot\).
Then we have the following theorem.
Theorem 1. For any pair of observables X, Y and state Ψ〉, the following conditions are all equivalent.

(i)
The state Ψ〉 is a superposition of common eigenstates of X and Y.

(ii)
The observables X and Y commute in the state Ψ〉, i.e., Eq. (15) holds for any x,y.

(iii)
There exists a JPD μ of X and Y in Ψ〉, i.e., there exists a probability distribution μ(x, y) on \({\Bbb R}^2\) satisfying Eq. (16) for any polynomial f(X, Y) of observables X, Y.

(iv)
\(\mathop {\sum}\nolimits_{x,y} \langle {\mathrm{\Psi }}P^X(x) \wedge P^Y(y)\Psi \rangle = 1\), where ∧ stands for the infimum of two projections.
In this case, the JPD μ is uniquely determined by
Proof. The following proof is obtained by adapting the more general arguments previously given in refs. ^{34,35,36,55,56} to the case discussed here.
(i) ⇒ (iv): Suppose that Ψ〉 is a superposition of common eigenstates of X and Y, namely, there exists an orthonormal family of states {x, y〉} such that Xx, y〉 = xx, y〉, Yx, y〉 = yx, y〉, and that \(\Psi \rangle = \mathop {\sum}\nolimits_{x,y} x,y\rangle \langle x,y{\mathrm{\Psi }}\rangle\). Then we have
and hence (iv) holds.
(iv) ⇒ (ii): Let u,v∈\({\Bbb {R}}\). It is easy to see that
It follows from condition (iv) that
By symmetry we obtain
Thus, (ii) holds.
(ii) ⇒ (iii): Let
Then μ(x, y) ≥ 0 for all \(x,y \in {\Bbb R}\), since
by assumption, and \(\mathop {\sum}\nolimits_{x,y} \mu (x,y) = 1\). Let \(f(X,Y) = X^{n_1}Y^{m_1} \cdots X^{n_N}Y^{m_N}\) with 0 ≤ n_{1}, m_{1},…, n_{N}, m_{N}. Then by assumption we have
Thus, we have
By linearity, the relation
holds for every polynomial f(X, Y).
(iii) ⇒ (i): Suppose that there exists a JPD μ(x, y) of X and Y in Ψ〉. Let f(X), g(Y) be polynomials of X and Y. We have
and hence
Taking f(X), g(Y) as f(X) = P^{X}(x) and g(Y) = P^{Y}(y), we have
so that X and Y commute in Ψ〉. It follows that P^{X}(x)P^{Y}(y)Ψ〉 is a common eigenvector of X and Y if P^{X}(x)P^{Y}(y)Ψ〉 ≠ 0. It follows from \({\mathrm{\Psi }}\rangle = \mathop {\sum}\nolimits_{x,y} P^X(x)P^Y(y){\mathrm{\Psi }}\rangle\) that Ψ〉 is a superposition of common eigenstate of X and Y.
Suppose that (i)–(iv) hold and let μ be a JPD of X, Y in Ψ〉. Then
It follows that
From Eq. (16) we have
Since f(x, y) was arbitrary, we obtain
This completes the proof. QED
It should be noted that if
for all \(x,y \in {\Bbb R}\), then Eq. (45) defines a probability distribution μ(x, y) on \({\Bbb R}^2\) satisfying the marginal probability conditions:
However, Eq. (46) does not ensure that μ(x, y) satisfies Eq. (16), so that μ(x,y) is not necessarily a JPD of X and Y in Ψ〉. In fact, let X = σ_{x}, Y = σ_{y}, and Ψ〉 = σ_{x} = + 1〉, where σ_{x}, σ_{y} are Pauli operators on \({\Bbb C}^2\). Let f(X, Y) = YXY. Then we have
Thus, Eq. (16) does not hold.
Statedependent definition for accurate measurements of quantum observables
To characterize accurate measurements of a quantum observable in a given state, here, we take two approaches, one based on classical correlation and the other based on quantum correlation, which will be eventually shown to be equivalent.
As discussed before, if A(0) and M(τ) commute in ψ, ξ〉, there exists the JPD μ(x, y) of A(0) and M(τ) in ψ, ξ〉, which describes the classical inputoutput correlation. Then according to the consistency with the classical description, the observable A is considered to be accurately measured if A(0) and M(τ) are perfectly correlated in their JPD μ, i.e., μ(A(0) = M(τ)) = 1. Thus, we reach the following condition for the measuring process M to accurately measure A in the state ψ〉:
(S) A(0) and M(τ) commute in ψ, ξ〉 and their JPD μ satisfies μ(A(0) = M(τ)) = 1.
In the second approach, we consider the weak joint distribution (WJD) ν(x, y) of A(0) and M(τ) in ψ, ξ〉 defined by
From Theorem 1, if A(0) and M(τ) commute in ψ, ξ〉, the WJD ν(x, y) coincides with the JPD μ(x, y) of A(0) and M(τ) in ψ, ξ〉. The WJD always exists, and is operationally accessible by weak measurement and postselection,^{57} but possibly takes negative or complex values. Then it is natural to consider the following condition:
(W) The WJD of A(0) and M(τ) in ψ, ξ〉 satisfies ν(x, y) = 0 if x ≠ y.
Since the WJD is operationally accessible, condition (W) is also operationally accessible. Obviously, (W) is logically weaker than or equivalent to (S). If condition (S) holds, the measurement should be considered an accurate measurement for the consistency with the classical description. On the other hand, if the measurement is accurate, any operational test for the possible error should be passed. Observing the WJD is one of available tests for the accurate measurement, and it is natural to consider that the test is passed if ν (x, y) = 0 for all x, y with x ≠ y and that the test is failed, or the error is witnessed, if \(\nu (x,y)\not = 0\) for some x, y with x ≠ y; this type of test has been discussed in detail by Mir et al.^{58} and Garretson et al.^{59} in the context of witnessing momentum transfer in a whichway measurement. Thus, condition (W) should be satisfied by any accurate measurement, since a failure of (W), or a nonzero value of ν (x, y) for a pair (x, y) with x ≠ y, witnesses an error of the measurement.
Therefore, condition (S) is a sufficient condition for the measurement to be accurate, and condition (W) a necessary condition. The following theorem shows that both conditions are actually equivalent so that both of them are necessary and sufficient conditions for the measurement to be accurate.
Theorem 2. For any measuring process M, an observable A, and a state ψ〉, condition (S) and condition (W) are equivalent.
Proof: The assertion was generally proved in refs. ^{33,34} after a lengthy argument. Here, we give a direct proof. Since (S) implies (W), it suffices to show the implication (W) ⇒ (S). Suppose that the WJD ν(x, y) of A(0) and M(τ) in ψ, ξ〉 satisfies ν(x, y) = 0 if x ≠ y. Then
Consequently,
and
Thus,
It follows that A(0) and M(τ) commute in ψ, ξ〉 and the condition in (S) holds. Thus the implication (W) ⇒ (S) follows. QED
Sound and complete quantum rootmeansquare errors
In addition to the locally uniform qrms error, here, we introduce the following two sorts of qrms errors. For any invertible density function f, we define the fdistributed qrms error ε_{f} by
For any invariant mean m on \({\Bbb R}\),^{60} define the mdistributed qrms error ε_{m} by
Then we have the following theorem.
Theorem 3. The following statements hold.

(i)
The error measures \({\bar {\varepsilon}}\), ε_{f}, and ε_{m} are sound and complete qrms errors.

(ii)
The error measure \({\bar {\varepsilon}}\) has the dominating property, (V).

(iii)
The error measures \({\bar {\varepsilon}}\), ε_{f}, and ε_{m} have the conservation property for dichotomic measurements, (VI).

(iv)
The relations
$$\varepsilon _{m} \le {\bar {\varepsilon}} ,\quad \sup_{f}{\varepsilon} _{f} = {\bar {\varepsilon}},$$hold for any invariant mean m, where f varies over all the invertible density functions.

(v)
The error measure ε_{m} satisfies the relation
if \(A = \mathop {\sum}\nolimits_n a_nP^A(a_n)\).
Proof. It is obvious from definition that \(\bar \varepsilon\) satisfies the operational definability, (I). From Eq. (22), \(\bar \varepsilon\) satisfies the correspondence principle, (II), and hence satisfies the soundness, (III). To prove the completeness, (IV), suppose \(\bar \varepsilon (A,{\mathrm{{\Pi}}},\psi \rangle ) = 0\). Then we have
Since t was arbitrary, we have
for any {a_{j}} and {t_{j}}. By Fourier expansion, the set of operators \(\mathop {\sum}\nolimits_j a_je^{  it_jA}\) includes all functions of A, so that we have
for all x ∈ \({\Bbb R}\). Thus, P^{A}(x)ψ〉ξ〉 is a common eigenstate of M(τ) and A(0) for a common eigenvalue x, if P^{A}(x)ψ〉 ≠ 0, and \(\psi \rangle \xi \rangle = \mathop {\sum}\nolimits_x P^A(x)\psi \rangle \xi \rangle\) is a superposition of those common eigenstates of \(M(\tau )\) and A(0) with common eigenvalues. It follows from Theorem 3 in ref. ^{33} that condition (W) holds. Thus, \(\bar \varepsilon\) satisfies the completeness requirement (IV). Therefore, we conclude that \(\bar \varepsilon\) is a sound and complete qrms error. The proofs for ε_{f} and ε_{m} are similar, and assertion (i) follows.
Assertion (ii) follows immediately from the definition.
To prove assertion (iii), suppose \(A^2 = \hat {\Pi}^{(2)} = I\). Let ψ_{t}〉 = e^{itA}ψ〉. We have the commutation relation
and hence
Thus, we have
Thus, assertion (iii) follows.
Assertion (iv) follows easily from the properties of integral and invariant mean.
Assertion (v) follows from Theorem 5.2 in ref. ^{60}.
This completes the proof. QED
Consider a quantum system with single degree of freedom described by a pair of canonically conjugate observables Q, P prepared in a state \(\phi \rangle\) such that 〈pϕ〉^{2} = f(p). By the relation
the above definition of ε_{f} is equivalent to making the canonical approximate Ameasurement with the Qmeter prepared in the state ϕ〉 such that 〈pϕ〉^{2} = f(p) in the Pbasis before evaluating the noiseoperator based qrms error.^{61,62} The definition of ε_{m} is also equivalent to making the canonical approximate Ameasurement with the Qmeter prepared in the mDirac state before evaluating the noiseoperator based qrms error.^{61} It is wellknown that there is no canonical choice of f or m in general to achieve the ideal measurement of an arbitrary A.^{60,61} By Theorem 3 (iv), our definition for \(\bar \varepsilon\) is equivalent to
where f varies over all the invertible wave functions. Thus, although there is no canonical choice of f in general, the definition of \(\bar \varepsilon\) can be interpreted as choosing the most errorsensitive f among all the invertible wave functions f.
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Acknowledgements
The author acknowledges the support of the JSPS KAKENHI, No. 26247016 and No. 17K19970, and of the IRINU collaboration.
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Ozawa, M. Soundness and completeness of quantum rootmeansquare errors. npj Quantum Inf 5, 1 (2019). https://doi.org/10.1038/s415340180113z
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DOI: https://doi.org/10.1038/s415340180113z
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