The uncertainty relation, which displays an elementary property of quantum theory, was originally described by Heisenberg as the relation between error and disturbance. Ozawa presented a more rigorous expression of the uncertainty relation, which was later verified experimentally. Nevertheless, the operators corresponding to error and disturbance should be measurable in the identical state if we follow the presupposition of Heisenberg’s thought experiment. In this letter, we discuss simultaneous measurability of error and disturbance and present a new inequality using error and disturbance in the identical state. A testable example of this inequality is also suggested.
The uncertainty relation, which displays an elementary property of quantum theory, was originally described by Heisenberg [1] as the relation between the error and disturbance of a particle’s position and momentum as
where h is Planck’s constant.
Subsequently, a more generalized inequality was shown [2,3]:
where is the standard deviation of a self-conjugate operator X, which corresponds to some physical quantity, defined as
with
and as the commutator of A and B.
In some literature (for example, [4]), (2) is considered to be a more formal expression of (1).
Several decades later, Ozawa presented a more rigorous expression of the uncertainty relation [5-7]. The rootmean-square noise and root-mean-square disturbance are defined as
The Noise operator is defined using the meter-observable of as
with the disturbance operator as
where in and out mean just before and just after measurement, respectively. The new uncertainty relation is written by means of (5), (6) and also (3) as
Recently, it was reported [8] that (9) was verified experimentally by a neutron spin experiment. Nevertheless, it is not clear whether verification of (9) is possible for continuous quantities such as position and momentum. In other words, it is not clear whether (5) and (6) are measurable for such quantities [9,10]. Watanabe et al. [11-13] suggested another inequality suitable for practical measurement.
Moreover, error and disturbance were defined in the identical state in Heisenberg’s thought experiment [1] referring to the uncertainty principle. If we follow his presupposition, the operators corresponding to error and disturbance should be simultaneously measurable. In many textbooks on quantum theory, commutativity of observables is regarded as a necessary and sufficient condition of possibility of simultaneous measurement. Ozawa, however, insists in his paper [14] that, in some states, two noncommutative observables, A and B, are simultaneously measurable if they satisfy
and their meter observables are commutative. Simultaneous measurability has been discussed with respect to contextuality and weak measurement [14-17].
The purpose of this letter is to discuss the simultaneous measurability of error and disturbance. Firstly, we define simultaneous measurability from the quantum logical aspect. According to our definition, there exists no state where noncommutative observables are simultaneously measurable. Then, we define commutative operators which correspond to the error and disturbance of noncommutative observables. This definition leads to the uncertainty relation of error and disturbance in the identical state. A testable example of this relation is also suggested, where definition of error in [8] is shown to be insufficient for other settings.
2. Simultaneous Measurability
To prepare for discussion about simultaneous measurability, we define observables according to a common quantum logical approach [18,19]. The proposition that a measured value of a physical quantity u belongs to a subspace A of space of real number R is written as. When the truth value of can be determined experimentally, u is called measurable. Logic L, which is nothing but a -complete orthomodular lattice, consists of such propositions. Classical logic is a Boolean lattice, namely, an orthocomplemented distributive lattice, while quantum logic is not.
We suppose -field, which consists of all open sets belonging to space of real number R. A map u from to logic L is called an observable of L if
where is the orthocomplement of and constitute an orthogonal set of projection operators. It is proved that observables are - homomorphism from to L.
There exists a one-to-one correspondence between the whole set of bounded observables and the whole set of bounded self-conjugate linear operators. If, and only if, two such operators, which correspond to observables u and v, are commutative, they satisfy for any pair of
and the orthomodular lattice whose elements are’s and’s is Boolean. Here, we assume, as usual, that all the measurable quantities are observables.
We define the simultaneous measurability of observables u and v as follows.
Definition u and v are called simultaneously measurable if the truth value of can be determined experimentally.
We present the following theorem:
Theorem Let u and v be observables of logic L and , , for the fixed. Then, , are observables if, and only if, they satisfy (14).
Proof (sufficiency)
We assume (14) is satisfied. Firstly, we show the whole set whose elements are, is a -complete orthocomplemented distributive lattice. Since’s and satisfy the distribution law,
and also satisfy the distribution law. Moreover, if we define
for, is the orthocomplement of. Thus is a -complete orthocomplemented distributive lattice. It is clear that, satisfy (11)-(13) because is a distributive lattice. Therefore, are observables of if they satisfy (14). (necessity).
Let, be observables. From (13)
if. This equation leads to
If we put,
QED.
From the above, it is shown that is not an observable if (14) is not satisfied, that is, two observables which correspond to mutually-noncommutative linear operators are not simultaneously measurable.
For example, let
be projection operators corresponding to and, respectively, where
and are Pauli spin matrices. Then, if, the projection operator corresponding to is, which is not an observable.
3. Uncertainty Relation
From the previous section, we can say such quantities as
are not measurable because (7) and (8) are noncommutative when. Note that this fact does not deny (9) where (16) does not appear but (5), (6) and (3) do. These are measured separately by using states belonging to the same statistical ensemble. What we would like to emphasize is that the uncertainty relation should be written by means of commutative quantities if it is thought to be the relation between quantities which are measured in the identical state. Thus we define
as operators which express error and disturbance from the expectation values, respectively.
Using these operators, we examine the following quantity:
Since and are observables in different systems, (19) becomes
If we use
and assume
(19) is written by the use of (3), (5) and (6) as
It is clear that (22) is not invariably realized. One of the simplest counter examples is the case where always indicates. Nevertheless, we regard (22) as a rather reasonable assumption, which means that and are independent stochastic variables, and so are and.
We can calculate the lower bound of (23) by means of (2) and (9) to obtain
If we use
in place of (9), the minimal value becomes almost double:
4. A Testable Example
In this section, we suggest an experiment with a setting which is a little modified from the experiment in [8] as a testable example of the inequality (24). We define A, B and OA instead of their definition in [8] as
where
and. (22), which is necessary to conclude with (24), is satisfied in this setting. If the root-meansquare noise is completely calculable by using A, B and OA as insisted in [8],
Then,
and
It comes down to that Ozawa’s inequality (9) is not realized within. This fact seems to show that includes uncontrollable error.
Accordingly, we will estimate the range of, including uncontrollable error, on the assumption that (25) or (9) is realized. We redefine as
where is the operator which gives uncontrollable error and is assumed to satisfy
This assumption may demand that the angular momentum of the particle should be measured continuously. Then, inequalities corresponding to (26) and (24) will be derived from (23).
Firstly, if we assume (25), independently of. Then,
The minimum value of the coefficient of the righthand side is 1 when.
Next, if (9) is assumed,
Then
The minimum value of the coefficient of the righthand side is when.
If
at some angles and
at each angle are shown experimentally, we can conclude that Inequality (24) is realized. This is also an experimental proof that Ozawa’s inequality is correct.
5. Conclusion
To summarize, we have defined simultaneous measurability from the quantum logical aspect and conclude that operators corresponding to the error and disturbance should be commutative if they operate in the identical state. Moreover, a new inequality using such operators and a testable example are presented.
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doi:10.1007/BF01397280E. H. Kennard, “Zur Quantenmechanik Einfacher Bewegungstypen,” Zeitschrift für Physik, Vol. 44, No. 4-5, 1927, pp. 326-352. doi:10.1007/BF01391200H. P. Robertson, “The Uncertainty Principle,” Physical Review, Vol. 34, No. 1, 1929, pp. 163-164.
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and disturbance 
of a particle’s position and momentum as
is the standard deviation of a self-conjugate operator X, which corresponds to some physical quantity, defined as
as the commutator of A and B.
and root-mean-square disturbance 
are defined as
is defined using the meter-observable 
of 
as
as
in [
. When the truth value of 
can be determined experimentally, u is called measurable. Logic L, which is nothing but a 
-complete orthomodular lattice, consists of such propositions. Classical logic is a Boolean lattice, namely, an orthocomplemented distributive lattice, while quantum logic is not.
-field
, which consists of all open sets belonging to space of real number R. A map u from 
to logic L is called an observable of L if
is the orthocomplement of 
and 
constitute an orthogonal set of projection operators. It is proved that observables are 
- homomorphism from 
to L.
’s and
’s is Boolean. Here, we assume, as usual, that all the measurable quantities are observables.
can be determined experimentally.
, 
, 
for the fixed
. Then, 
, 
are observables if, and only if, they satisfy (14).
whose elements are
, 
is a 
-complete orthocomplemented distributive lattice. Since
’s and 
satisfy the distribution law,
also satisfy the distribution law. Moreover, if we define
, 
is the orthocomplement of
. Thus 
is a 
-complete orthocomplemented distributive lattice. It is clear that
, 
satisfy (11)-(13) because 
is a distributive lattice. Therefore
, 
are observables of 
if they satisfy (14). (necessity).
, 
be observables. From (13)
. This equation leads to
,
is not an observable if (14) is not satisfied, that is, two observables which correspond to mutually-noncommutative linear operators are not simultaneously measurable.
and
, respectively, where
and 
are Pauli spin matrices. Then, if
, the projection operator corresponding to 
is
, which is not an observable.
. Note that this fact does not deny (9) where (16) does not appear but (5), (6) and (3) do. These are measured separately by using states belonging to the same statistical ensemble. What we would like to emphasize is that the uncertainty relation should be written by means of commutative quantities if it is thought to be the relation between quantities which are measured in the identical state. Thus we define
and 
are observables in different systems, (19) becomes
always indicates
. Nevertheless, we regard (22) as a rather reasonable assumption, which means that 
and 
are independent stochastic variables, and so are 
and
.
. (22), which is necessary to conclude with (24), is satisfied in this setting. If the root-meansquare noise 
is completely calculable by using A, B and OA as insisted in [
. This fact seems to show that 
includes uncontrollable error.
, including uncontrollable error, on the assumption that (25) or (9) is realized. We redefine 
as
is the operator which gives uncontrollable error and is assumed to satisfy
independently of
. Then,
.
when
.