Quantum Disturbance without State Change:
Soundness and Locality of Disturbance Measures

Heisenberg’s error-disturbance relation (EDR)

$$\varepsilon(A)\eta(B)\geq\frac{1}{2}|{\langle{[A,B]}\rangle}|$$

(1)

for the mean error $\varepsilon(A)$ of a measurement of an observable $A$ in any state and the mean disturbance $\eta(B)$ caused on an observable $B$, originally introduced by the $\gamma$-ray microscope thought experiment [1], has been commonly believed as a dynamical aspect of Heisenberg’s uncertainty principle, which is formally represented by a rigorously proven relation

$$\sigma(A)\sigma(B)\geq\frac{1}{2}|{\langle{[A,B]}\rangle}|$$

(2)

for the indeterminacies, defined as the standard deviations $\sigma(A),\sigma(B)$, of arbitrary observables $A,B$ in any state [1, 2, 3]. There have been longstanding research efforts to prove Heisenberg’s EDR [4, 5, 6, 7, 8], while the universal validity has not been reached. Instead, a recent study [9, 10] revealed a universally valid form of EDR

$$\varepsilon(A)\eta(B)+\varepsilon(A)\sigma(B)+\sigma(A)\eta(B)\geq\frac{1}{2}|{\langle{[A,B]}\rangle}|,$$

(3)

where $\sigma(A)$ and $\sigma(B)$ are the standard deviations just before the measurement, and made Heisenberg’s EDR testable [10, 11] to observe its violations, confirming the new relation as well [12, 13, 14]. Subsequently, stronger EDRs were derived [15, 16, 17, 18], and confirmed experimentally [19, 20, 21, 22, 23, 24].

In order to define the error $\varepsilon(A)$ and disturbance $\eta(B)$ in Eq. (3), we suppose that the measurement $\mathbf{M}$ of $A$ is described by an interaction from time $t=0$ to $t=\tau$ between the system $\mathbf{S}$ in a state $|{\psi}\rangle$ and the probe $\mathbf{P}$ prepared in a fixed state $|{\xi}\rangle$, and that the outcome of the measurement is obtained by the measurement of the meter observable $M$ in the probe $\mathbf{P}$ at time $t=\tau$. ¹¹1 Note that this general description of a measuring process, also called an indirect measurement model [10], is introduced and proved in Ref. [25] to be equivalent to the most general description using a completely positive instrument, or a so-called quantum instrument, which is a reformulation of the Davies-Lewis instrument [26] with the additional requirement of complete positivity.

In the Heisenberg picture, we shall write $X(0)=X\otimes I$, $X(\tau)=U^{\dagger}X(0)U$, $Y(0)=I\otimes Y$, $Y(\tau)=U^{\dagger}Y(0)U$ for observables $X$ in $\mathbf{S}$ and $Y$ in $\mathbf{P}$, where $U$ is the unitary evolution operator for $\mathbf{S}+\mathbf{P}$ from $t=0$ to $t=\tau$. The error $\varepsilon(A)=\varepsilon_{O}(A,\mathbf{M},|{\psi}\rangle)$ and disturbance $\eta(B)=\eta_{O}(B,\mathbf{M},|{\psi}\rangle)$ in Eq. (3) are defined by

			(4)

See Ref. [10] for details. We call $\varepsilon_{O}$ and $\eta_{O}$ as the operator-based error measure and the operator-based disturbance measure. We shall write $\varepsilon_{O}(A)=\varepsilon_{O}(A,\mathbf{M},|{\psi}\rangle)$ and $\eta_{O}(B)=\eta_{O}(B,\mathbf{M},|{\psi}\rangle)$ when no confusion may occur.

In the previous work [18], we have investigated the properties of the operator-based error measure (called therein as noise-operator-based quantum root-mean-square error) $\varepsilon_{O}$, and we have introduced its completion $\overline{\varepsilon}$, the locally uniform quantum root-mean-square error, and subsequently we have experimentally tested [27] the completeness of $\varepsilon_{O}$ and $\overline{\varepsilon}$ to show how hidden error in $\varepsilon_{O}$ manifests in the defining procedure of $\overline{\varepsilon}$.

In the present work, we focus on the properties, soundness and locality, of the operator-based disturbance measure $\eta_{O}$, where soundness generally requires a disturbance measure to assign the value 0 to “non-disturbing” measurements, and locality generally requires a disturbance measure to assign the value 0 to “non-disturbing” local measurements.

We say that a measurement is distributionally non-disturbing to an observable $B$ in the system state $|{\psi}\rangle$ if $B(0)$ and $B(\tau)$ have identical probability distributions in the initial state $|{\psi,\xi}\rangle$. Korzekwa, Jennings, and Rudolph [28] criticized the use of the operator-based disturbance measure, based on the following requirement for disturbance measures.

Distributional requirement (DR) for disturbance measures. Any disturbance measure should assign the value 0 to distributionally non-disturbing measurements.²²2 Note that KJR [28] called distributionally non-disturbing measurements as “operationally non-disturbing measurements”; see Eq. (2) in KJR [28].

KJR [28] called the DR “the commonly accepted and operationally motivated requirement that all physically meaningful notions of disturbance should satisfy”. They claimed that the operator-based disturbance measure does not satisfy the DR and has even an ‘unphysical’ property, since it takes a positive value for a measurement that does not change the state at all. Further, they concluded that state-dependent formulations of EDRs are not tenable.

In this paper, we examine the validity of the DR. For this purpose, we consider a more fundamental principle in quantum mechanics, the correspondence principle, stating that if the classical description is available, quantized concepts should be consistent with the classical description. We argue that the DR violates the correspondence principle. We generally show that even if the measurement does not change the state, the disturbance is operationally detectable as long as the operator-based disturbance measure takes a positive value. Thus, the claims made by KJR are groundless. The DR requires that disturbance measures only count the change of the probability distribution in time, but according to the correspondence principle, valid disturbance measures should also count the change of the observable that manifest in the time-like correlation, as the operator-based disturbance measure does.

Moreover, we show that the DR violates another fundamental requirement for no-signaling under local operations, called the locality requirement. Subsequently, we show that disturbance measures satisfying the DR cannot be used to demonstrate the security of quantum cryptography, because they do not properly describe the disturbance caused by the eavesdropper. In contrast, we show that the operator-based disturbance measure satisfies the correspondence principle and the locality requirement. Based on those arguments, we shall conclude that state-dependent formulations of EDRs based on the operator-based disturbance measure reliably represent the originally motivated dynamical aspect of Heisenberg’s uncertainty principle.

The correspondence principle generally states that quantum theory should be consistent with classical theories in the case where the classical descriptions are also available.³³3 “The term [the correspondence principle] codifies the idea that a new theory should reproduce under some conditions the results of older well-established theories in those domains where the old theories work [Wikipedia https://en.wikipedia.org/wiki/Correspondence˙principle (August 1, 2022)]”. In fact, it is a common practice to apply classical descriptions to commuting observables through their joint probability distributions. In Ref. [18] we consider the correspondence principle as a requirement for error measures. Here, we extend the consideration to disturbance measures.

It is well-known that any commuting observables $X,Y$ have their joint probability distribution in any state. Here, for a given state $|{\Psi}\rangle$, the joint probability distribution (JPD) of any two observables $X,Y$ is defined as a 2-dimensional probability distribution $\mu(u,v)$ satisfying

$\displaystyle\langle{\Psi|f(X,Y)|\Psi}\rangle=\sum_{u,v}\,f(u,v)\mu(u,v)$

(5)

for every (non-commutative) polynomial $f(X,Y)$ of $X$ and $Y$. In general, two observables $X,Y$ have their JPD in a state $|{\Psi}\rangle$ if and only if they commute in $|{\Psi}\rangle$ in the sense that

$\displaystyle P^{X}(u)P^{Y}(v)|\Psi\rangle=P^{Y}(v)P^{X}(u)|\Psi\rangle,$

(6)

where $P^{X}(u)$ and $P^{Y}(v)$ are the spectral projections of $X$ and $Y$ (Ref. [18], Theorem 1). In this case, the JPD $\mu$ is uniquely determined by

$\displaystyle{\mu(u,v)=\langle{\Psi|P^{X}(u)P^{Y}(v)|\Psi}\rangle.}$

(7)

The JPD $\mu$ determines the (classical) root-mean-square deviation $\delta_{G}(\mu)$ between the classical random variables ${\bf u}=u$ and ${\bf v}=v$, the notion originally introduced by Gauss [29], by

$\displaystyle{\delta_{G}(\mu)=\left(\sum_{u,v}\,(u-v)^{2}\mu(u,v)\right)^{1/2}.}$

(8)

We say that a disturbance measure $\eta$ satisfies the correspondence principle (CP) if $\eta(B)=\delta_{G}(\mu)$ provided that $B(\tau)$ and $B(0)$ have their JPD $\mu$ in the initial state $|{\psi,\xi}\rangle$. An important property of the operator-based disturbance measure $\eta_{O}$ is that it satisfies the CP, as easily follows from Eq. (5). Similarly, the operator-based error measure $\varepsilon_{O}$ also satisfies CP in the sense that $\varepsilon_{O}(A)=\delta_{G}(\mu)$ provided that $M(\tau)$ and $A(0)$ have their JPD $\mu$ in the initial state $|{\psi,\xi}\rangle$ as shown in Ref. [18].

If $B(0)$ and $B(\tau)$ have their JPD $\mu$, the correlation between $B(0)$ and $B(\tau)$ has the classical picture described by $\mu$, and the classical notion $\delta_{G}(\mu)$ of the root-mean-square deviation is applicable to quantifying the disturbance of $B$. In this case, according to the correspondence principle, any quantum definition of a disturbance measure $\eta$ should be consistent with the classical measure $\delta_{G}$. Thus, we say that a quantum disturbance measure $\eta$ satisfies the correspondence principle if two measures, the quantum $\eta$ and the classical $\delta_{G}$, are consistent, whenever the classical picture is available, as a desirable property of a quantum disturbance measure. In this sense, the correspondence principle determines the value of the disturbance on $B$, when the joint probability distribution of $B(0)$ and $B(\tau)$ exists, in an analogous way as the probability distribution of $B(0)$ determines its standard deviation $\sigma(B)=\sigma(B(0))$ appearing in Eq. (3).

KJR [28]  identified as ‘unphysical’ the property of the operator-based disturbance measure $\eta_{O}$ that it does not assign the value 0 in a case where the state has not changed at all. In such a case, the probability distribution of every observable has not changed, so that this is a stronger violation of the DR. However, we shall show here that this is not a peculiarity of the operator-based disturbance measure, but a straightforward consequence of the CP.

Consider a qubit measurement. The projective measurement of $A=\sigma_{z}$ in the state $|{0}\rangle:=|{\sigma_{z}=+1}\rangle$ does not change the initial state $|{\psi}\rangle=|{0}\rangle$. In this case, it was shown [30] that the operator-based disturbance measure indicates that $B=\sigma_{x}$ is disturbed by the amount $\eta_{O}(\sigma_{x})=\sqrt{2}$, and this value was actually obtained by a neutron optical experiment [19]. However, according to the DR, every disturbance measure $\eta$ should assign the value 0, and KJR [28] identified the above property of $\eta_{O}$ as a very unphysical property. In contrast, we shall show that every disturbance measure $\eta$ satisfying the CP assigns the value $\sqrt{2}$.

It is well-known that the projective measurement of $\sigma_{z}$ is carried out by the controlled-NOT operation

$\displaystyle{U=|{0}\rangle\langle{0}|\otimes I+|{1}\rangle\langle{1}|\otimes\sigma_{x}}$

(9)

for the measured qubit $\mathbf{S}$ and the probe qubit $\mathbf{P}$ prepared in the fixed state $|{\xi}\rangle=|{0}\rangle$ from $t=0$ to $t=\tau$ and by the subsequent meter measurement for $M=\sigma_{z}$ in $\mathbf{P}$ (Figure 1) . The Schrödinger time evolution satisfies

	$\displaystyle U(|{0}\rangle\otimes|{0}\rangle)$	$\displaystyle=|{0}\rangle\otimes|{0}\rangle,$		(10)
	$\displaystyle U(|{1}\rangle\otimes|{0}\rangle)$	$\displaystyle=|{1}\rangle\otimes|{1}\rangle.$		(11)

For $B=\sigma_{x}$ the Heisenberg time evolution is given by

	$\displaystyle\sigma_{x}(0)$	$\displaystyle=\sigma_{x}\otimes I,.$		(12)
	$\displaystyle\sigma_{x}(\tau)$	$\displaystyle=\sigma_{x}\otimes\sigma_{x}.$		(13)

Here, Eq. (13) follows from

$\displaystyle U^{\dagger}(\sigma_{x}\otimes I)U=|1\rangle\langle 1|\sigma_{x}|0\rangle\langle 0|\otimes\sigma_{x}+|0\rangle\langle 0|\sigma_{x}|1\rangle\langle 1|\otimes\sigma_{x}=\sigma_{x}\otimes\sigma_{x}.$

It follows that $\sigma_{x}(\tau)$ and $\sigma_{x}(0)$ commute and they have the JPD $\mu(u,v)$ in the state $|{\psi,\xi}\rangle=|{0,0}\rangle$ as

	$\displaystyle\mu(u,v)$	$\displaystyle=\langle{0,0|P^{\sigma_{x}(\tau)}(u)P^{\sigma_{x}(0)}(v)|0,0}\rangle$
		$\displaystyle=\langle{0,0|P^{\sigma_{x}\otimes\sigma_{x}}(u)P^{\sigma_{x}\otimes I}(v)|0,0}\rangle.$		(14)

Then we obtain

$$\mu(u,v)=\frac{1}{4}$$

(15)

(cf. Section 9). Thus, if the disturbance measure $\eta$ satisfies the CP, we have

$\displaystyle{\eta(\sigma_{x})^{2}=\delta_{G}(\mu)^{2}=\sum_{u,v=\pm 1}(u-v)^{2}\mu(u,v)=2.}$

(16)

Therefore we conclude $\eta(\sigma_{x})=\sqrt{2}$. Thus, the non-zero value $\eta_{O}(\sigma_{x})=\sqrt{2}$ is not a peculiar property of the operator-based disturbance measure.

It will be instructive to compare the above scenario (1) that the system is prepared in the sate $|0\rangle$ and then a projective measurement of $\sigma_{z}$ is performed and another scenario (2) that the system is prepared in the sate $|0\rangle$ but no measurement is performed. In both scenarios, the system state $|0\rangle$ is unchanged and the probability distribution of any observable does not change. How can we operationally distinguish the two scenarios. In scenario 1 we have shown that the observable $B=\sigma_{x}$ is disturbed. For the time $t=0$ just before the measurement and the time $t=\tau$ just after the measurement, we obtain $B(0)=\sigma_{x}\otimes I$ and $B(\tau)=\sigma_{x}\otimes\sigma_{x}$ (cf. Eqs. (12) and (13)). Their joint probability distribution $p_{2}(u,v)$ satisfies $p_{2}(u,v)=1/4$ for any $u,v$ (cf. Eq. (15)) that leads to $\eta(B)=\sqrt{2}$. On the other hand, scenario 2 is easily analyzed, so that we obtain $B(0)=B(\tau)=\sigma_{x}\otimes I$, and their joint probability distribution $p_{1}(u,v)$ satisfies $p_{1}(u,v)=\delta_{u.v}/2$ that leads to $\eta(B)=0$. The joint probability distributions can be experimentally obtained by weak measurements and post-selections as proposed by Lund and Wiseman [11]. Thus, we can operationally distinguish between the above two scenarios.

This conclusion might sound counter-intuitive, as the pure sate has the “maximal information” about the system. However, unchanging the pure state does not imply unchanging the observable, because the “maximal information” about the system does not include the “maximal information” about an observable, analogously with the fact that the “maximal information” about the whole system does not include the “maximal information” about subsystems.

In fact, according to the available classical description, the conditional probability

$\displaystyle{\Pr\{\sigma_{x}(\tau)=u|\sigma_{x}(0)=v\}=\mu(u|v)=\frac{1}{2}}$

(17)

shows that the value of $\sigma_{x}(0)$ has been completely randomized, although their marginals have not changed at all as

$\displaystyle{\Pr\{\sigma_{x}(\tau)=u\}=\Pr\{\sigma_{x}(0)=u\}=\frac{1}{2}.}$

(18)

Thus, the DR neglects the disturbance caused by the randomization by measurement without changing the probability distribution.

We have shown that the DR with the notion of probability non-disturbing measurements contradicts the CP. To reconcile the conflict, we shall characterize non-disturbing measurements from the two fundamental requirements: the CP and the operational accessibility.

Consider the following condition.

(S) $B(\tau)$ and $B(0)$ have their JPD $\mu$ in $|{\psi,\xi}\rangle$ satisfying that $\mu(u,v)=0$ if $u\neq v$.

From the point of view of the CP, if condition (S) holds, we should conclude that the measurement $\mathbf{M}$ does not disturb $B$ in $|{\psi}\rangle$. Thus, condition (S) is considered as a sufficient condition for a proper definition of non-disturbing measurements.

On the other hand, from the point of view of operational accessibility, it is convenient to consider the weak joint distribution (WJD) $\nu(u,v)$ of $B(\tau)$ and $B(0)$ in $|{\psi,\xi}\rangle$ defined by

$\displaystyle{\nu(u,v)=\langle{\psi,\xi|P^{B(\tau)}(u)P^{B(0)}(v)|\psi,\xi}\rangle.}$

(19)

The WJD always exists, though possibly takes negative or complex values, and is operationally accessible by weak measurement and post-selection [31, 32, 33]; see also Ref. [34] for a short survey. Then it is natural to consider the following condition.

(W) The WJD of $B(\tau)$ and $B(0)$ in $|{\psi,\xi}\rangle$ satisfies that $\nu(u,v)=0$ if $u\neq v$.

If the measurement $\mathbf{M}$ does not disturb the observable $B$ in $|{\psi}\rangle$, any operational tests for witnessing the disturbance should fail. Since measuring WJD is one of such operational tests for which the disturbance is detected if $\nu(u,v)\neq 0$ for some $u\neq v$ [35, 36], condition (W) is considered as a necessary condition for a proper definition of non-disturbing measurements.

Obviously, (W) is logically weaker than or equivalent to (S). However, Theorem 1 (Section 10) shows that both conditions are actually equivalent. In fact, according to the theory of quantum perfect correlations [37, 38], both conditions (S) and (W) equivalently require that $B(\tau)$ and $B(0)$ are perfectly correlated in the state $|{\psi,\xi}\rangle$ [34]. Thus, the above argument justifies the following definition of non-disturbing measurements. We say that the measurement $\mathbf{M}$ is properly non-disturbing to an observable $B$ in $|{\psi}\rangle$ if one of the conditions (S) or (W) is satisfied. Since the WJD is operationally accessible, this definition is also operationally accessible.

To consider the reliability of the operator-based disturbance measure, we examine the following requirements: (i) the CP, (ii) soundness, (iii) operational accessibility, and (iv) completeness.

We have already shown that the operator-based disturbance measure $\eta_{O}$ satisfies the CP, i.e., $\eta_{O}(B)=\delta_{G}(\mu)$ if $B(\tau)$ and $B(0)$ have the JPD $\mu$. We introduce the soundness requirement: Any disturbance measure $\eta$ should assign the value 0 to any properly non-disturbing measurements. It is interesting to see that the CP implies soundness. To show this, suppose that the measurement is properly non-disturbing to $B$ in $|{\psi}\rangle$. Then $B(\tau)$ and $B(0)$ have the JPD $\mu$ satisfying that $\mu(u,v)=0$ if $u\neq v$. It follows that $\varepsilon_{G}(\mu)=0$ and by the CP we have $\eta(B)=\varepsilon_{G}(\mu)=0$. Accordingly, the operator-based disturbance measure $\eta_{O}$ satisfies the soundness requirement. We conclude, therefore, that even if the measurement does not change the state, the disturbance can be operationally detected as long as the operator-based disturbance measure takes a positive value.

It has been known that the operator-based disturbance measure $\eta_{O}$ is operationally accessible in the two ways: (i) the tomographic three state method, proposed by Ozawa [10] and experimentally realized by Erhalt et al. [12] and others [14, 19, 23] and (ii) the weak measurement method, proposed by Lund and Wiseman [11] and experimentally realized by Rozema et al. [13] and others [20, 22, 21].

As the converse of soundness, a disturbance measure $\eta$ is said to be complete if $\eta$ assigns the value 0 only to properly non-disturbing measurements. There is an example in which $\eta_{O}$ does not satisfy completeness (Ref. [38], p. 750). However, it is known that $\eta_{O}$ satisfies completeness if (i) (commutative case) $B(\tau)$ and $B(0)$ commute in $|{\psi,\xi}\rangle$ or if (ii) (dichotomic case) $B^{2}=I$ (Ref. [18], Theorem 3).

We have seen that the operator-based disturbance measure satisfies all requirements (i)–(iii), and partially satisfies requirement (iv) above.

Analogously from an argument for the operator-based error measure $\varepsilon_{O}$ in Ref. [18], it follows that $\eta_{O}$ can be modified to satisfy completeness by defining the operator-based locally uniform disturbance measure $\overline{\eta}$ as

$\displaystyle{\overline{\eta}(B,\mathbf{M},|{\psi}\rangle)=\sup_{t\in\mathbb{R}}\eta_{O}(B,\mathbf{M},e^{-itB}|{\psi}\rangle).}$

(20)

Then the error measure $\overline{\eta}$ satisfies requirements (i) – (iv) and also (v) (Dominating property) $\eta_{O}(B,\mathbf{M},|{\psi}\rangle)\leq\overline{\eta}(B,\mathbf{M},|{\psi}\rangle)$ for any $|{\psi}\rangle$, and (vi) (Conservation property for dichotomic measurements) $\overline{\eta}(B)=\eta_{O}(B)$ if $B^{2}=I$. Thus, all the EDRs for $\eta_{O}$ also holds for $\overline{\eta}$; see analogous discussions for the operator-based error measure in Ref. [18].

In the following we shall discuss another requirement on locality, which the operator-based disturbance measure satisfies, but contradicts the DR.

We have argued that state-dependent formulations of error-disturbance relations are well-founded by the operator-based disturbance measure, which is a sound disturbance measure according to the notion of properly non-disturbing measurement that is supported by the CP and operational accessibility, in contrast to KJR’s claim that the operator-based disturbance measure is not sound under the notion of distributionally non-disturbing measurements, which we have shown to contradict the CP.

Yet, there is a prevailing view that only probability distributions of outcomes of measurements can be operationally compared [39], despite the fact that the new experimental techniques enable us to operationally detect the change of an observable in time: (i) the tomographic three state method [10, 12, 14, 19, 23] and (ii) the weak measurement method [11, 13, 20, 22, 21].

In what follows, we shall show below another drawback of the DR that the notion of probability non-disturbing measurements violates a locality requirement to be posed below.

Consider a composite system $\mathbf{S}_{1}+\mathbf{S}_{2}$ in a state $|{\Psi}\rangle$. Since any local measurement of $\mathbf{S}_{2}$ does not interact with the system $\mathbf{S}_{1}$, we naturally take it for granted that any local measurement of $\mathbf{S}_{2}$ non-disturbing to an observable $B_{2}$ in $\mathbf{S}_{2}$ should be non-disturbing to the observable $B_{1}\otimes B_{2}$ for any observable $B_{1}$ in $\mathbf{S}_{1}$. We call this requirement the locality requirement for a definition of disturbing measurements. We shall show that the definition of distributionally non-disturbing measurements does not satisfy this requirement, whereas the definition of properly non-disturbing measurements does satisfy the requirement as shown in Theorem 2 (Section 11),

For this purpose, we consider a maximally entangled two-qubit system $\mathbf{S}_{1}+\mathbf{S}_{2}$ in the Bell state $|{\Phi^{+}}\rangle=(|{00}\rangle+|{11}\rangle)/\sqrt{2}$. Since $|{\Phi^{+}}\rangle=(|{0_{x}0_{x}}\rangle+|{1_{x}1_{x}}\rangle)/\sqrt{2}$, the outcomes of the joint local measurements of the observables $\sigma_{x}^{(1)}=\sigma_{x}\otimes I$ and $\sigma_{x}^{(2)}=I\otimes\sigma_{x}$ show a perfect correlation. From Theorem 4 (Section 11), measurements properly non-disturbing to $\sigma_{x}^{(2)}$ does not change the JPD of $\sigma_{x}^{(1)}$ and $\sigma_{x}^{(2)}$, so that the perfect correlation between $\sigma_{x}^{(1)}$ and $\sigma_{x}^{(2)}$ is not disturbed. However, we shall show that a probability non-disturbing measurement breaks the perfect correlation, and this concludes that the definition of probability non-disturbing measurements does not satisfy the locality requirement, according to Theorem 3 (Section 11).

In order to examine the reliability of the operator-based disturbance measure, KJR [28]  introduced the following definition. A state $|{\psi}\rangle$ is called a zero-noise zero-disturbance (ZNZD) state with respect to observables $A$ and $B$ if the projective measurement of $A$ in the state $|{\psi}\rangle$, which always satisfies $\varepsilon(A)=0$, is distributionally non-disturbing to $B$. Then they proved that for every pair of non-commuting observables $A$ and $B$, there exists a ZNZD state $|{\psi}\rangle$ such that $|\langle{\psi|[A,B]|\psi}\rangle|\not=0$. Thus, if the disturbance measure $\eta$ satisfies the DR, any relation of the form

$\displaystyle\sum_{m,n=0}^{\infty}f_{mn}(A,B)\varepsilon(A,\rho)^{m}\eta(B,\eta)^{n}\geq|{\langle{[A,B]}\rangle}|,$

(34)

where $f_{00}(A,B)=0$, must be violated. From this, KJR [28]  concluded that any state-dependent EDR, based on the expectation value of the commutator as a lower bound, is not tenable, and that state-independent formulations are inevitable.

We have two objections to their claims. First of all, the universally valid relation (3) with $\varepsilon_{O}(A)=0$ leads to the relation

$\displaystyle\eta_{O}(B)\geq\frac{|{\langle{[A,B]}\rangle}|}{2\sigma(A)}>0$

(35)

for any projective measurement of $A$ in any ZNZD state such that $|\langle{\psi|[A,B]|\psi}\rangle|\not=0$. Thus, the measurement is properly disturbing to $B$ by the soundness of $\eta_{O}$, and consequently the disturbance is operationally detectable by the operational accessibility of the definition of properly non-disturbing measurements, so that the assumption by KJR [28]  that $\eta(B)=0$ in any ZNZD state is unfounded.

Secondly, they concluded that state-independent formulations are inevitable for alternative formulations. However, currently proposed state-independent formulations of EDRs [41, 39, 42] do not appear to capture the essence of Heisenberg’s original idea. Recall that Heisenberg derived his EDR by the $\gamma$-ray microscope thought experiment, in which the EDR is derived from the relation between the resolution power and the Compton recoil, reciprocally relating to the wave length of the incident light. Since the wave length is independent of the state of the object, the above formulation might be considered as state-independent. However, the analysis is valid only state-dependently, since the resolution power of the microscope can be defined by the wave length only in the limited situation in which the object is properly placed in the scope of the microscope. Thus, we can adequately define the error of the $\gamma$-ray microscope only state-dependently. In the state-independent formulations, currently one defines the state-independent error as the worst case of the state-dependent error, which must diverge to infinity as the object wave function spreads out of, or moves far apart from, the scope of the microscope. Such state-independent definitions would facilitate to reproduce the form of Heisenberg’s original formulation, but do not keep the physics underlying it. Thus, state-dependent formulations are inevitable to represent Heisenberg’s original idea underlying the uncertainty principle.

In this paper, we have given a definition of non-disturbing measurement from the point of view of the correspondence principle and operational accessibility. Subsequently, we have established the reliability of the operator-based disturbance measure. We have already discussed the reliability of the operator-based error measure in our previous work [18]. Both accounts ensure that universally valid EDRs [9, 15, 16, 17] reliably represent a dynamical aspect of Heisenberg’s uncertainty principle besides the well-established relation for the indeterminacy in quantum states representing a kinetic aspect of the principle. Thus, the objections to state-dependent formulations of EDRs shown in [39, 28] are unfounded, although those views appear to still prevail in the literature [43, 44]. We conclude that the theory [9, 10, 11, 15, 16, 17, 18] and experiments [12, 13, 19, 14, 21] for state-dependent formulations of EDRs are reliable and that state-dependent formulations are inevitable to represent Heisenberg’s original idea underlying the uncertainty principle.

The new quantitative methods developed in this paper for universally valid EDRs with the well-defined operator-based disturbance measure incorporating with the methods of weak values and weak measurements will provide new quantitative methods to understand the change, transfer, or disturbance of observables in time, which does not manifest in the change of the probability distribution, but which does manifest in the time-like correlation. This quantity will be useful and even inevitable for exploring foundational problems in quantum physics including the long-lasting controversy over the roles of uncertainty principle in which-way measurements for interferometers (Refs. [45, 46, 35, 36, 47] and the references therein). In addition to the foundational problems, it will be expected that universally valid EDRs call for new research interests in exploring various frontiers in physics including fault-tolerant quantum computing [48, 49, 50], quantum metrology [51, 52, 53], and multi-messenger astronomy [54], in which technological limits would be overcome by the fundamental principle independent of particular models. We hope that the methods of operator-based disturbance measures as well as operator-based error measures will be accepted for broad areas of quantum physics.

Here, we shall give a derivation of Eq. (15). The JPD $\mu$ of $\sigma_{x}(\tau)=\sigma_{x}\otimes\sigma_{x}$ and $\sigma_{x}(0)=\sigma_{x}\otimes I$ in the state $|{0,0}\rangle$ is given by

$\displaystyle\mu(u,v)$

$\displaystyle=\langle{0,0|P^{\sigma_{x}\otimes\sigma_{x}}(u)P^{\sigma_{x}\otimes I}(v)|0,0}\rangle.$

We have

	$\displaystyle P^{\sigma_{x}\otimes\sigma_{x}}(+1)P^{\sigma_{x}\otimes I}(\pm 1)$	$\displaystyle=[P^{\sigma_{x}}(+1)\otimes P^{\sigma_{x}}(+1)+P^{\sigma_{x}}(-1)\otimes P^{\sigma_{x}}(-1)](P^{\sigma_{x}}(\pm 1)\otimes I)$
		$\displaystyle=P^{\sigma_{x}}(\pm 1)\otimes P^{\sigma_{x}}(\pm 1),$
	$\displaystyle P^{\sigma_{x}\otimes\sigma_{x}}(-1)P^{\sigma_{x}\otimes I}(\pm 1)$	$\displaystyle=[P^{\sigma_{x}}(+1)\otimes P^{\sigma_{x}}(-1)+P^{\sigma_{x}}(-1)\otimes P^{\sigma_{x}}(+1)](P^{\sigma_{x}}(\pm 1)\otimes I)$
		$\displaystyle=P^{\sigma_{x}}(\pm 1)\otimes P^{\sigma_{x}}(\mp 1).$

Consequently,

	$\displaystyle\mu(+1,\pm 1)$	$\displaystyle=\langle{0,0|P^{\sigma_{x}}(\pm 1)\otimes P^{\sigma_{x}}(\pm 1)|0,0}\rangle=\langle{0|P^{\sigma_{x}}(\pm 1)|0}\langle{0|P^{\sigma_{x}}(\pm 1)|0}\rangle$
		$\displaystyle=\frac{1}{4},$
	$\displaystyle\mu(-1,\pm 1)$	$\displaystyle=\langle{0,0|P^{\sigma_{x}}(\pm 1)\otimes P^{\sigma_{x}}(\mp 1)|0,0}\rangle=\langle{0|P^{\sigma_{x}}(\pm 1)|0}\langle{0|P^{\sigma_{x}}(\mp 1)|0}\rangle$
		$\displaystyle=\frac{1}{4}.$

Therefore, we obtain Eq. (15), i.e.,

$$\mu(u,v)=\frac{1}{4}.$$