1. Introduction
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
(1)
where h is Planck’s constant.
Subsequently, a more generalized inequality was shown [2,3]:
(2)
where 
is the standard deviation of a self-conjugate operator X, which corresponds to some physical quantity, defined as
, (3)
with
(4)
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
(5)
(6)
The Noise operator 
is defined using the meter-observable 
of 
as
(7)
with the disturbance operator 
as
(8)
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
(9)
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
(10)
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
(11)
(12)
(13)
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 
(14)
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
(15)
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
(16)
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
(17)
(18)
as operators which express error and disturbance from the expectation values, respectively.
Using these operators, we examine the following quantity:
(19)
Since 
and 
are observables in different systems, (19) becomes
If we use
(20)
(21)
and assume
(22)
(19) is written by the use of (3), (5) and (6) as
(23)
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
(24)
If we use
(25)
in place of (9), the minimal value becomes almost double:
(26)
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
(27)
(28)
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],
(29)
(30)
(31)
(32)
Then,
(33)
and
(34)
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
(35)
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,
(36)
The minimum value of the coefficient of the righthand side is 1 when
.
Next, if (9) is assumed,
(37)
Then
(38)
The minimum value of the coefficient of the righthand side is 
when
.
If
(39)
at some angles and
(40)
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.