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1 THEORETICAL SETTING OF INNER REVERSIBLE QUANTUM MEASUREMENTS

1 THEORETICAL SETTING OF INNER REVERSIBLE QUANTUM MEASUREMENTS
1 THEORETICAL SETTING OF INNER REVERSIBLE QUANTUM MEASUREMENTS

THEORETICAL SETTING OF INNER REVERSIBLE

QUANTUM MEASUREMENTS

Paola A. Zizzi

Dipartimento di Matematica Pura ed Applicata

Università di Padova

Via Belzoni, 7

35131 Padova, Italy

zizzi@math.unipd.it

Abstract

We show that, at least formally, any unitary transformation performed on the quantum state of a closed quantum system, is a special case of generalized quantum measurement. More specifically, any unitary transformation describes an inner, reversible, generalized quantum measurement.

We also show that under some specific conditions it is possible to perform a unitary transformation on the state of the closed quantum system by means of a collection of generalized measurement operators. In particular, given a complete set of orthogonal projectors, it is possible to implement a reversible quantum measurement that preserves the probabilities (mirror measurement).

In this context, we introduce the concept of “Truth-Observable”, which is the physical counterpart of an inner logical truth. The Truth-Observable is related to a modification of the usual cut rule of sequent calculus in inner quantum logic, which reflects the identification of the quantum system with a quantum space background.

1. Introduction

The original measurement postulate of quantum mechanics, introduced by von Neumann [1], considered only projective measurements, where a “standard” quantum observable A has a spectral resolution in terms of orthogonal projection operators. The postulate states that during a measurement of A, the state vector of the quantum system reduces to the eigenvector of A corresponding to the measurement result. More recently, the notion of quantum measurement has been extended to generalized measurements [2] for which the concept of “generalized” quantum observable has been introduced, where the positive resolution of the identity replaces the spectral decomposition. The concept of a generalized quantum measurement is necessary to describe the quantum interaction between a quantum system and a measuring device. Moreover, the generalized measurement formalism can deal with phenomena not captured by projective measurements; in particular, it is quite useful for describing joint measurements of incompatible “standard” observables.

Finally, a generalized formalism appears to be more suitable for quantum computing (for a review of quantum computing see ref. [2]).

However, although generalized measurements can be implemented by the use of projectors augmented with unitary evolution, and ancillary system, one does not really overcome the measurement problem.

In the case of quantum computers, the above problem becomes even worse, as the irreversibility of a quantum measurement is dramatically incompatible with the reversibility of the whole computational process.

Thus, one should find a way to bind quantum measurement and quantum evolution/computation in a single mathematical formulation.

We believe that this unification is possible only when the "observer" is conceived internal to the original quantum system to be measured. To do so, the "observer" should enter a quantum space background whose quantum states are in one-to-one correspondence with the computational states of the quantum computer.

In some sense, this idea is the analogous of Feynman's idea [3] that the most efficient way to simulate a quantum system is to use another quantum system (a quantum computer).

In our case, the most adequate way to simulate a reversible measurement is to use another quantum system, which is not the apparatus, but the quantum space background of the quantum system itself.

In a previous paper [4] we looked at the quantum space background of the quantum computer as the space associated (by the non-commutative version of the Gelfand-Naimark's theorem [5]) with the algebra of quantum logic gates, that is the algebra of unitary matrices U(n). More specifically, in [4] we considered a quantum computer with N qubits, where the quantum logic gates are n

n unitary matrices where: N

. We found that the quantum space background of the quantum computer is the n2

fuzzy sphere [6] with N

elementary cells.

n2

The inner quantum logic of the quantum computer, based on this geometrical setting, was investigated in [7]. The main features of that logic are paraconsistency and symmetry, which are also present in basic logic [8].

Also, some consequences of [4] were exploited in quantum gravity [9] and in quantum computability [10].

The present paper is conceived manly to support the ideas and results of [4] and [7] by formalizing the notion of internal reversible measurement in the framework of the generalized measurement formalism.

In fact we show that any unitary transformation is formally a generalized quantum measurement.

Moreover, we prove that, once it is given a collection of generalized measurement operators which satisfy some particular requirements, it is possible to implement a reversible measurement. Such a result might be quite useful in quantum computing,meaning that, at least in principle, one might be able to concatenate quantum

measurements in such a way to preserve the reversibility of the whole computational process.

Finally, we implement the internal logical structure of a quantum isolated system discussed in [7] by introducing the new concept of Truth-Observable, which exhibits a dual nature between physics and logic.This paper is organized as follows:

In Sect. 2, we briefly review generalized measurements, projective measurements and POVM (Positive Operator-Valued Measure).

In Sect.3, we formally define inner reversible measurements as a special case of generalized measurements.

In Sect.4, we review the "mirror measurement", introduced in [4], which is a special case of reversible generalized measurement.

In Sect.5, we introduce the new concept of Truth-Observable, and its logical and philosophical implications.

Sect. 6 is devoted to the conclusions.

2. Generalized Quantum Measurements

2.1 Postulate of generalized quantum measurement

A generalized quantum measurement (of a closed quantum system S in the finite dimensional space of states) is described by a collection m M of measurement operators on the Hilbert space H of S, where the index m refers to the possible

outcomes of the measurement. The measurement operators satisfy the completeness relation:

m

m m I M M (1)

Where: M is the adjoint of M and I is the identity operator on H .

If the state of the system S is just before the measurement, the probability of getting the outcome m is:

m m M M m p )(. (2)After the measurement, the system S is left in the normalized state:

)

('m p M m

. (3)The completeness relation (1) ensures that the probabilities sum up to 1:

1)( I M M M M m p m

m m m m m m (4)

A generalized quantum observable, which will be formally introduced in 2.4,corresponds to a non-negative, hermitian operator of the form m m M M .

2.2. Projective measurements

Let us consider a projective measurement of a “standard” quantum observable A of the system S, which corresponds to a hermitian operator A A on the Hilbert space H of the system S.

The observable has spectral decomposition:

m

m m P A , (5)

Where m P is the projector onto the eigenspace of A with eigenvalue m . The set m P is a complete set of orthogonal projection operators with respect to the observable A,that is, the projectors m P are Hermitian operators:

m m P P , (6)satisfying the following two relations:

I P m

m (7)m mm m m P P P '' , (8)Where (7) is the completeness relation, and (8) is the requirement for the m P operators to be orthogonal and idempotent.

The possible outcomes of a projective measurement correspond to the eigenvalue m of the observable.

Upon measuring the system in state , the probability of getting the outcome m is: m P m p )(. (9)After the measurement, the system is left in the normalized state:

)('m p P m

. (10)2.3. Projective measurement as a special case of generalized measurements As we have seen in 2.1, the (generalized) measurement operators m M satisfy the completeness relation (1).

If the m M also satisfy the two additional relations:

m m M M (11)m mm m m M M M '' (12)They reduce to projection operators.

Thus, a projective measurement is a special case of generalized measurement.2.4. POVM: Positive Operator-Valued Measure

The postulate 2.1 of generalized measurement specifies both the probabilities of the different possible outcomes of the measurement, in (2), and the post-measurement state, in (3). In some cases, however, there is no necessity for knowing the post-measurement state, and interest focuses mainly on measurement statistics. Those cases are best analyzed by using the POVM formalism.A POVM operator is defined as:

m m m M M E (13)Where the m M , are (generalized) measurement operators introduced in 2.1.The POVM operators satisfy the following properties:i) The m E form the positive-definite partition of unity

m

m

m m m

I M M E

(14)

ii) The set m E is sufficient to determine the probabilities of the different measurement outcomes:

)()( m m E tr E m p (15)Where is the density matrix of the quantum system under consideration.iii) The m E , are positive operators (hence in particular Hermitian):02

m m m m M M M E (16)

For any H .

The m E as defined in (13) are the “generalized” quantum observables.

Notice that the spectral decomposition of a “standard” observable in (5) is replaced, in the generalized case, by the resolution of the identity operator I in (14).From i) it follows that the probabilities sum up to one:1

)( I E E m p m

m m m

m

From iii) it follows that the probabilities are positive:0

)(2

m m M E m p In case that the measurement operators m M reduce to projectors m P , then all the POVM elements m E coincide with the measurement operators m m P E , in fact:

m

m m m m P P P P E 2

2.5 Example of generalized measurement in quantum computing

Let us consider the most general two qubit state vector of 4C , in the computational basis:

111001003210a a a a Q (17)Where: C a i (i =0, 1, 2, 3) , 12

i

i

a .

Notice that the state in (17) is in general an entangled state.

If we measure the first bit of Q , there are two possible outcomes (m = 0, 1): either the first bit is zero (m = 0) or it is one (m =1).

In the first case, the corresponding generalized measurement operator is:

010100000 M (18)In the second case, the corresponding generalized measurement operator is:

111110101 M (19)The probability that the first bit of Q is zero is:

2

120000a a Q M M Q p (20)The post measurement state is:

2

1

20100000100'a a a a Q

M M Q Q

M Q

(21)

The probability that the first bit of Q is one is:

2

32

2111a a Q M M Q p (22)The post measurement state is:

2

3

223211111

10''a a a a Q M M Q Q M Q (23)

Notice that 01000P P M , 11101P P M , where 11100100,,,P P P P are the four projectors of 4C :

000000000000000100P , 000000000010000001P , 000001000000000010P ,

100000000000

000011P Where )4(11100100I P P P P , and )4(I is the identity operator of 4C .

0M ,1M are two bi-dimensional projectors in 22C C , in fact, rewritten as block

matrices they are:

000)2(0I M ,

)2(1000I M (24)where )2(I is the identity operator of 2C .

It holds: )4(1100I M M M M (25)In this case, the two POVM operators coincide with the generalized measurement operators:

0000M M M E , 1111M M M E (26) It holds:

)4(10I E E

(27)Analogously, if we want to measure the second bit of Q , there are two other possible

outcomes (m=2, 3): either the second bit is zero, or is one.

In the first case, the corresponding generalized measurement operator is: 101000002 M And, in the second case, it is:11

1110103 M Where 10002P P M ,11103P P M , and we have:

)

4(111001003322I P P P P M M M M But it is clear that these measurements with outcomes (m = 2, 3) are distinct from the previous ones (m = 0, 1).

2.6 Generalized measurement of a Bell state

An interesting example is the generalized measurement of a Bell state.

Bell states are the maximally bipartite entangled state and form the entangled basis of 4C .

The four Bell states are:

)1100(21

)1001(2

1

As an example, let us consider in particular the Bell state:

)1100(2

1

(28)

If we measure the first bit of in (28) there are two possible outcomes (m = 0, 1):either the first bit is zero, or it is one.

To m = 0 it corresponds the generalized measurement operator 0M in (18).

The probability that the first bit of

is 0 is:

2

1

000

M M p (29)The post measurement state is:

2

00

21'000

M M M (30)To m = 1 it corresponds the generalized measurement operator 1M in (19).

The probability that the first bit of

is 1 is:

2

1

111

M M p (31)The post measurement state is:

2

11

21'111

M M M (32)Notice that there are two POVM operators 0E , 1E given in (26) that are the generalized observables, and that Eq. (27) holds.

In Sect. 4.9 we will see how the interpretation of a Bell state changes when its measurement is conceived from the internal point of view.3. Inner Reversible Quantum Measurements

Postulates

i) The observer is considered internal to the closed quantum system if and only if

the space-time background is a quantum space isomorphic to the quantum system.

ii) The internal quantum measurement is described by a unitary operator U.From Postulate i), it follows the lemma:

Lemma 3.1

The quantum system S and the internal observer O do not form a composite system.Their Hilbert spaces, which are isomorphic, O S H H are identified:O S H H and do not form a tensor product, as it is:O S O S H H H ,.

Definition 3.2

A unitary operator is a bounded linear operator U on a Hilbert space, which satisfies the condition:

I UU U U , (33)Where I is the identity operator on H .This property implies the following:

Property 3.3

A unitary operator U preserves the inner product on the Hilbert space H . In fact, if we define: U ', we get:

I U U '', (34)

For all vectors , of the Hilbert space H .From postulate ii), it follows the lemma:

Lemma 3.4

An internal quantum measurement is a reversible quantum operation.In fact, any unitary operator U on a Hilbert space H performs a reversible

transformation on a quantum state of H . In fact, from definition 3.2, for every transformation U, there exists the inverse transformation )(1 U U , such that I U U 1.Let us define:

U '. (35)

The inverse transformation 1U on ' is ' U , which gives back the original state

:

U U U U ''1 (36)Where, in the last step definition 3.2 has been used.

From postulate ii) it also follows the lemma:

Lemma 3.5

In the case the closed quantum system is a quantum computer, the unitary operator U is a quantum logic gate, then an internal quantum measurement coincides with a quantum computational step.

Theorem 3.6

A reversible transformation performed by a unitary operator U, on a closed quantum system, which is in the quantum state , is a generalized quantum measurement.

Proof:

By definition 3.2, a unitary operator U satisfies the completeness equation (1), in the particular case that the collection m M of generalized measurement operators in postulate 2.1 consists of a single operator U M M 1. In this case, the unique outcome m=1 has probability one to occur:

1)1( U U p (37)After the measurement, the state of the system is:

U p U )

1(' (38)

Then, all the properties of a generalized quantum measurement required by postulate 2.1 are fulfilled by a unitary operator U. ?

Theorem 3.7

Given a collection m M (m = 0, 1, … N-1) of measurement operators, which satisfy the additional condition:

0 j i j i M M M M (j i ), N j i ,0 (39)

And a corresponding collection: m of complex numbers satisfying the condition:

12

m (40)Then, the linear superposition:

1

N m m m M M (41)

is a unitary operator and then, by theorem 3.6, it describes a reversible quantum measurement.

Proof:

We must show that M is unitary. To this aim, it is sufficient to show that I M M with M given in (41). In fact, as M is finite, and it is 1det 2

M

,

it follows that M is invertible, as 0det M

.

.

2

''

'*

I M M M M M M M M M M m m

m m m m

m m m m m m m m m m m m m (42)

where in the last three steps we have used equations (39), (40) and (1) respectively.

?

Corollary 3.8

Given a complete set of orthogonal projection operators, that is, a set m P such that the m P satisfy (6), (7) and (8), and a corresponding collection of complex numbers

m such that 12 m , the linear superposition:

1

N m m m P P (43)

is a unitary operator (which is a diagonal matrix with respect to the basis of the eigenvectors of the observable A whose spectral decomposition is given in (5)).P in (43) is the exponential operator: iA e P

, where A is the observable in (5), and the complex coefficients m in (27) are given in terms of the eigenvalues of A:m

i

m e

Then, by theorem 3.6, P

describes an inner, reversible, generalized quantum measurement.

Since the projectors m P are a special case of m M , the proof is the same of theorem 3.7. ?

Corollary 3.9

In the case of a reversible quantum measurement, the POVM consists of a single operator that is the identity operator, and the associated probability is one.

Proof:

As we have seen in Sect. 2, a POVM is a collection m E of positive-definite operators m E given in (13) and the associated probabilities )(m p are given in (15).

In the case that the collection m M consists of a single operator 1M , then, by theorem 3.6, 1M is unitary: U M 1. From the definition of m E it follows:

I U U M M E 111 (44)And the associated probability is one:

1)1(1 I E p . ?(45)

4. The Mirror Measurement

4.1. Theorem

A unitary transformation U on a vector state of a Hilbert space H leaves unchanged the probabilities of , if it commutes with a complete set of orthogonal projectors m P in H .

Proof :

Let us suppose we perform first a unitary transformation U on , given by (35).Thereafter, we perform a projective measurement on :

' U P P m m ' (46)The probability of getting the outcome m is: U P U P m p m m '')'(If it holds:

0, m P U (47)We get:)()'(m p P UP U m p m m . ?(48)

4.2. Corollary Any unitary operator on a Hilbert space H , which is of the kind P

given in (43),preserves the probabilities of any quantum state of H on which it acts.

Proof :

The proof is trivial, as P

commutes with the projector operators m P :

0,,,'

''''

'

m m m m m m m m m P P P P P P . (49)Then, by theorem 4.1, when P

acts on a quantum state , it preserves the

probabilities of . ?

Definition 4.3

A mirror measurement M

is a unitary transformation on a quantum state , which preserves the probabilities of .

Lemma 4.4

By definition 4.3 and from corollary 4.2, it follows that every unitary transformation

of the kind P introduced in 3.8, is a mirror measurement M

.

4.5 Example: mirroring one qubit state

Let us consider a diagonal 22 unitary matrix of the form:

)(1*0P P e U i D (50)

Where 1* , and 0P and 1P are the two orthogonal projection operators on the Hilbert space 2C .

Let us consider the generic one qubit state

10b a (51)With: C b a ,, and 1

2

2

b a The operator D U preserves the probabilities of in (51), as it can be easily shown: 1'0'

10)(1*0b a b a P P e U i

D (52)Where: a e a i

', b e b i

*' , then it is: 22'a a and 2

2'b b .

In [ ] we interpreted D U as a reversible measurement performed inside the quantum

computer and called it "mirror measurement"M

.

As we mentioned in the Introduction, the term "measurement" in [4] and [7] was justified in a geometrical setting but it was not formally defined. In this paper we fill this gap, as it is trivial to prove the following theorem:

Theorem 4.6

Any unitary diagonal matrix of the kind )(1*0P P e U i D on 2C (or its

extension to a higher dimensional Hilbert space) describes a generalized quantum

measurement, and in particular is a mirror measurement M

.

Proof:

Any mirror matrix )(1*0P P e U i

D is an operator P defined in (43),

with: i e 0, *1 i e , 12

1

2

. (53)

Then, by corollary 3.8, it follows that D U is a particular case of generalized quantum measurement.

Also, from lemma 4.4, it follows that D U is a mirror measurement. ?4.7 Example: Mirroring a general two qubit state

We briefly review the extension of mirror measurement in the Hilbert space 4C .Given two mirror- matrices of 2C : *1001 i e M ,

*2002 i e M (54)

The tensor product: 21M M M

, given by:

**0000

00000000 i e M (55)(21 , , * ) is a mirror-matrix of 4C . In fact, when applied to the

two qubit state Q in (17), it gives:

11'10'01'00''3210a a a a Q M Q

(56)Where: 00'a e a i

, 11'a e a i

….. And so on and probabilities are unchanged:

2020'a a , 2

121'a a ,….And so on.Notice that M

can be written as a linear superposition of the four projectors of 4C :)(11*10*0100P P P P e M i

(57)And it holds:)4(I M M

(58)And in this case, differently from the case in 2.5, there is only one POVM operator 1E ,which coincides with the identity:

)4(1I E (59)The difference from this case and the case in 2.5 is completely general; it does not depend on the dimensionality of the Hilbert space, but it is just a consequence of two different measurement attitudes: from the external and the internal points of view, as we will see in more detail in the next section.

4.8 Generalized measurement of one qubit from the outside and from the inside points of view

Let us consider the generic one qubit state vector in the Hilbert space 2C , given in (51).

From the outside point of view, the possible outcomes are two (m = 0, 1): either the measured bit is zero (m = 0), or it is one (m = 1).

The corresponding generalized measurement operators are:

000 M , 111 M (60)In the case of one qubit,0M and 1M then coincide with the two projectors of 2C , 00010P and 10001P respectively.

It holds:

)2(1100I M M M M (61) The two POVM operators also coincide with the two projectors:0000P M M E , 1111P M M E And it holds:

)2(1010I P P E E . (62)Then, the generalized measurement of one qubit state forcedly coincides with a

projective measurement. Consequently, there are two generalized observables:21,E E which coincide with the two standard observables, 0P and 1P respectively.This is the external point of view.

From the internal point of view, instead, there is a unique outcome, as to “measure” a qubit from inside, it is sufficient a unitary operator U of 2C . In fact, by theorem 3.6, a unitary transformation is a special case of generalized measurement.

In the particular case that U is a mirror measurement, as in 4.5, it “mimics” the

outside measurement, but it remakes it in parallel, in the sense that the two projectors are still present, but they are linked together by a linear superposition. And in this case there is only one POVM operator, which is the identity. The identity, in this case is a generalized observable.

Let us see the above in more detail.

Let it be U M 1. It is: I U U M M E 111If, in particular, it is:

*100 i e M (63)

With: 1* , we have:

1*0*11100P P M (64))2(102

111)(I P P M M E (65)

Algebraically, the difference between the outside and the inside points of view in the measurement of one qubit, is that in the former there are two distinct POVM

operators, or generalized observables, which coincide with the two projectors, while in the latter, there is only one POVM operator, which is the identity.

Geometrically, the outcomes of the measurement of one qubit can be interpreted, from

the outside point of view, as one single point 0x

(or 1x ) in the Euclidean space 3R ,corresponding to the projection of the North pole 0 (or South pole 1) of the Bloch sphere. Instead, the outcomes of the (mirror) measurement of one qubit are, from the inside point of view, the elements of a two point lattice, which is a subspace of the

fuzzy sphere [6] with n =2. In the case that the internal measurement was a general unitary operator of 2C , and not specifically a mirror measurement, the outcomes would be geometrically described as the two elementary cells of the n =2 fuzzy sphere.

Now, one might argue that two external observers A and B could perform the

measurement of the same qubit by acting together at the same time but this would be hopeless. In fact, at the very moment that A measures bit 0, for example, by the

projector operator 0P , the qubit is destroyed, but still observer B is measuring bit 1 by the projector 1P , and in fact he is destroying the qubit, leaving no chance to observer A.

Of course this leads to an absurd. The only way out for the external observers A and B would be making a copy of the original qubit, but this is forbidden by the no cloning theorem [11]. Or, they might become internal observers. Now, internal observers are equivalent to observers who inhabit parallel universes. In this case, the two parallel universes would constitute another quantum system isomorphic to the qubit, and the outcomes would be still superposed. In fact these two parallel universes would be centred at the two points lattice of the n =2 fuzzy sphere. Notice, however, that this picture is only similar, but not equivalent to the Many Worlds Interpretation of Quantum Mechanics by Everett [12]. In fact, in the latter, the observer is truly

external, and the total Hilbert space is the direct product of the Hilbert spaces of the original quantum system and of the apparatus. Instead, in our case the observers are living in a quantum space, which is isomorphic to the quantum system, thus they must be considered as internal to the system itself, and the dimensionality of the original Hilbert space is unchanged.

4.9 Mirroring a Bell state

Let us consider the Bell state in (28).

As we have seen in Sect. 2.5, the possible outcomes of an (external) generalized measurement of the first bit of are two: either the first bit is zero (m = 0) or is

one (m = 1).

Instead, from the internal point of view, there is only one outcome corresponding to a

unitary transformation U. If in particular, U is a Mirror-matrix M

of 4C , given in Sect. 4.7, we have:

)1100(

2

1'* i

e M (66)

Hence, a Bell state behaves as a single particle in the mirroring since ' has the

same structure of the “cat state” )10(2

1 after the mirroring, namely:

)10(2

1

'*

i

e (67)

The inner quantum logic of Bell states was investigated in [13].

The single particle behaviour of the mirrored Bell state is a feature arising from the internal measurement point of view. Instead, this feature is absent in the external

measurement point of view, as we have seen in 2.5. This discrepancy between the two points of view might be at the origin of paradoxes of EPR, in particular non-locality.

In fact, an internal observer does not attribute non-locality merely to the presence of Bell states, but to his own space-time background, which, being a non-commutative geometry, is non-local by definition.

In fact, the quantum system of a generic N = 2 qubit state is isomorphic to a fuzzy sphere with n = 4 elementary cells, each one encoding one of the four two-bits strings in the computational basis. In the case of a maximally entangled state (a Bell state),the fuzzy sphere has two cells, each one with doubled surface area, and encoding one of the two-bits strings 00 and 11 (or 01 and 10).

Algebraically, the difference between the internal and the external points of view (concerning a Bell state), stands in the fact that, while in the latter we have two generalized observables, which sum up to the identity:a) )

4(10I E E In the former, there is only one generalized observable:

b) )

4(1I M M E

From a) we can get only a partial knowledge (information) about the Bell state, while from b), we obtain the whole information at once. In this case, we will say that the single generalized observable is the Truth-Observable that is the topic of next section.

5. The Truth-Observable

The very word “truth” usually means “logical truth” (while for a physical truth one uses mostly “physical reality”), thus “truth” is in general a logical concept. On the other side, the word “observable” is a major physical concept. But in an inner

quantum world, the “truth” is a logical concept, as well as a physical observable, as it is the truth one “observes” from inside.

5.1 The inner quantum truth is a generalized observable

Let us consider the spectral decomposition in terms of projectors m P of the “standard”observable A in (5). The observable A reduces to the identity operator I in the n-dimensional Hilbert space H when all the eigenvalues of A are equal to 1, that is the eigenvalue 1 is n-degenerate. In this case Eq. (5) is identified with Eq. (7), which is the completeness relation for projectors m P .

More generally, the POVM m E realize the positive-definite partition of unity given in (14). The POVM elements m E are Hermitian operators, and are generalized observables (they do not satisfy the orthogonal relation (8) differently from

projectors). In the case that the family of generalized measurement operators m M consists of only one element 1M the latter is a unitary operator U, and the unique

associated generalized observable 1E is the identity operator )(n I in the n-dimensional Hilbert space:

)(111n I U U M M E . (68)Once U is given the meaning of a generalized measurement operator, as in Sect. 3, the identity operator acquires the status of generalized observable, which we will call Truth-Observable.

The Truth-Observable is measured when a quantum operation U is performed on a state of the system followed by the inverse operation:

1

'U U

(69)

In particular, if is a N-qubits state, Eq. (69) means that has been computed by a quantum logic gate U, which is a n n unitary matrix (where N n 2 ) and then un-computed by 1U .

Notice that (7) and (14) reflect the “interpretation” of the measurement of the truth-observable by an external observer, in terms of projectors and POVM respectively.However, in (7) and (14) the “truth” as a whole can be affirmed only in principle, as the external observer cannot perform all the m P (or m E ) measurements at the same time. Instead, in (68) the “truth” can be affirmed at once as a whole, as the Truth-Observable is measured in a single step (1E ) by the internal observer.

5.2 Logical and philosophical implications of the Truth-Observable

The Truth-Observable is a concept of a very weak logic, namely paraconsistent,

symmetric logic, which turns to be the inner logic of a closed quantum system, like a quantum computer.

The “inner-quantum logic” [7] is like basic logic [8] as it is paraconsistent [14] [15]and symmetric, but differently from basic logic, it has not the (usual) cut rule.

Paraconsistency, which implies (in addition to the invalidation of the excluded middle principle as in intuitionist logic: A A ) also the invalidation of the non contradiction principle:

A A & (70)Eq. (70), which we take as an axiom, as it follows from a mirror measurement, allows superposed states:

A A & (71)The absence of the rule of weakening:!

"!

"A , (72)

Is equivalent to “no erase” in quantum computing .Also, the absence of the contraction rule:!

"!

"A A A ,,, (73)

Means: “no cloning”.

Finally, the absence of the (usual) cut rule:B

B

A A !" ! ",, (74)

Means that projectors are not allowed.

Thus, by this logic, superposition of states can never be destroyed.

It is straightforward to show that the usual cut rule would destroy the superposed state in (71). In fact, by replacing A by A A &, B by A, and putting " ?, we would have:

)()(&&&cut A

L A

A A A

A A

A (75)

We introduce then the “branched” cut rule:

)(&&)

()

(&&&)()(&&&R A

A cut A

L A

A A A

A A A cut A L A A A A A A A (76)

This “branched” cut rule” does not destroy superposed states, as it corresponds to the linear superposition of two projectors, and, as we showed in Sect. 4.5, this is a

unitary, reversible measurement (in particular, it is a mirror measurement:M

)associated with the Truth-Observable )2(1I E .

We see now the relation between the paradigm of the Truth-Observable, and this new “branched” cut rule.

We believe then that the sequent calculus of the inner logic of a quantum computer should be performed in branches (work in progress).

The Truth-Observable can be considered as the inner, absolute truth. This is a novel Platonist approach to absolute truth conceived from an internal point of view.

Instead, the Platonist external view of absolute truth is a concept of a stronger logic,namely classical logic.

There is a main difference between the outsider and the insider Platonist views of absolute truth. While in the outsider view the absolute truth is an abstract concept, and an “act of faith”, in the insider view it can be proven, as it is a generalized observable,and can be measured. In a sense, and quite surprisingly, the inner Platonist view of absolute truth is an extreme product of constructive mathematics!

Notice that a weaker logic means a logic that has fewer structural rules and/or absence of some active contexts. For example, in our inner quantum logic there are not the weakening and the contraction rules, and there are no active contexts. In a very weak logic, there are many “constraints” in the logical derivation of theorems.

Now, we will discuss about the lack of (usual) sequent calculus in our extremely weak logic.

The external observer cannot recover the quantum “proof” of a theorem step by step.In fact, from a quantum computer, we can only know (by performing an external

measurement) whether a theorem is true or false but the proof will be never accessible to us, as it is lost when quantum parallelism is destroyed by the external measurement.To be more explicit, let us suppose that theorem “X” can be proven true by both a classical computer and a quantum computer. Also, suppose that an external observer wants to reproduce the proof by a sequent calculus in both cases. In the classical case he succeeds, while in the quantum case he does not, by previous arguments. So, there is not a usual sequent calculus which can reproduce a quantum proof from outside.Moreover, we think that a usual sequent calculus just cannot exist in inner quantum logic: it should be much more limited than usually, and structured in parallel branches.

A reasonable conclusion might be that usual sequent calculus is just an external logical procedure, that is, the logical reasoning of an observer external to the system about which he is giving judgements.

In particular, if one supposes that theorem “X” can be efficiently proved to be true only by a quantum computer, the fact that the external observer cannot reproduce the proof sounds like a quantum version of Goedel’s incompleteness theorem, as was pointed out in [10].

We think that a weaker logic would describe a physical reality with fewer degrees of freedom, as we will illustrate in the following.

One defines active contexts those which are close to the formula and passive context those which are separated from the formula by the sequent.

In our case, one might figure out passive contexts as those which belong to the computational state while active contexts will be those which do not, like external observers, measurement apparatus, the environment, in summary the external classical world.

Let us consider the reflection rule for & in basic logic, (valid also in our inner quantum logic):)(&&&L A B A A B B !

!

! ! (77)

And the same rule in Intuitionist logic, which has full context on the left only:)(&&,,&,,L B A B B A A !

"!

"! "! " (78)

Now, if we replace B with A in both of them, we get respectively:)(&&&L A A A A A A ! !

! ! (79)

And:

)(&&,,&,,L A A A A A A !

"!

"! "! " (80)

In (80) there is a chance that the superposed state A A & gets entangled with the environment ", instead in (79) there is not such a possibility. Then, logical “visibility”of formula as in basic logic (or absence of active contexts) would mean that by no means a superposed state can be entangled with the environment, unless a quantum measurement is performed. As we have seen, in fact, the usual cut rule (74) is admissible in basic logic, and corresponds to a projective measurement. Also, it should be noticed that, in basic logic, there is the possibility of having an active

context in the cut rule, as in (74) (of course this context can be considered empty, but in principle it is admissible). We will show now that in inner quantum logic this possibility is not always allowed.

For example, an active context in the branched cut rule is not allowed in the case of one qubit state, and in the case of Bell states.

Let us consider the example of a generic one qubit state, given by the superposition (71). Let us suppose that we can apply the branched cut rule (76) in presence of the active context !as in (71). We would get:

)(&&,)

(,&,&)(,&,&R A A cut A

A A A A A cut A A A A A A

!" !" ! " !" ! " (81)

But we cannot have the same active context !in the two separated branches, as those

branches correspond to two parallel worlds 1W ,2W , which cannot share any common information like !.

The same holds for the Bell state, since it behaves like a single “big cat state” after a mirror measurement, as in (66), and the corresponding cut rule would also get

branched in two parts as in (81). Instead, for not maximally entangled bipartite states,

the cut rule branches generically in four parts, and the two contexts H, K, will be different for the two pairs of parallel worlds:

H for00, K for 11

K for01, H for 10

So, in this case parallel worlds will have different contexts, and will not share any common information.

Instead, common information will be shared by pairs of communicating worlds:

H for both 00and 10

K for both01 and 11

Thus, in the case of non maximal entanglement, an active context would be allowed in a branched cut rule. The presence of an active context even in a branched cut rule can be regarded as a residual of classicality in the quantum computational state, and consequently in the associated quantum space. In fact, a true inner quantum space, is not simply a quantum space (non-commutative geometry), but a non-commutative geometry, which also reflects the actual computational state. When the quantum computational state (for more than one qubit) is not maximally entangled, the quantum space does not resemble a complete set of parallel universes, but a mixture of parallel and communicating universes.

Instead, in the case of one qubit state, or in the case of a (bipartite) maximally entangled state, the inner quantum space is given by a complete set of parallel universes, encoding quantum information. (This is the case of a n = 4 fuzzy sphere having two big entangled cells encoding 00 and11 (or01,10), in place of four smaller cells encoding all four computational states).

Thus, the two main novelties of the cut rule in inner quantum logic are the following: i) The cut rule is performed in branches.

ii) The active context is absent in the branched cut rule whenever the branches reflect a complete set of parallel universes.

Notice that quantum parallelism is always reflected also in the branching of the cut rule, which however does not always coincide with a complete set of parallel universes.

In summary, we argue then, that the total absence of any active context would correspond to the complete removal of the classicality (separability of quantum states or locality in the associated quantum space).

Our inner quantum logic describes a collapse of space-time inside the quantum computer. We like to interpret this as a kind of logical black hole: all the quantum information is “trapped” in the Truth-Observable.

6. Conclusions

We wish to conclude with the following remarks:

i) As we have seen, our approach to quantum measurement, finds its natural setting in the generalized formalism. However, the inner reversible measurement is conceptually very different from all kinds of measurements considered since now in the literature. In fact, it is the only kind of measurement which requires an isomorphism between the observed system and its space-time background. This isomorphism was illustrated geometrically in [4], logically in [7], and algebraically in the present paper. ii) It should be noticed that, while a generalized measurement deals with a composite system (original system plus apparatus) in a higher dimensional

Hilbert space, an internal reversible measurement only deals with a single isolated quantum system.

iii) In the case of the quantum computer, at least, we formally demonstrated the validity of Landauer's principle [16] stating that computation and measurement are fundamentally the same process. In fact, we showed that a quantum logic gate is a (generalized) measurement operator. This result might be of interest for the foundations of both quantum computation and quantum mechanics.

Landauer also raised the question whether the very concept of measurement necessarily requires an interaction between the system and the apparatus. In the case of the quantum computer, the "apparatus", made of quantum logic gates, interacts with the system, but it is itself part of the system. Quantum computation is then a kind of meta-measurement, in the sense that a quantum computer “measures” itself by its own computational process.

iv) The Truth-Observable is a new logical concept, by which truth judgements coincide with inner reversible measurements. Thus, not only "information is physical" (as Landauer said) but also "inner quantum logic is physical".

v) We showed that only an internal observer could assert the objective reality of a quantum state.

vi) In our view, the holistic cognitive attitude appears as a peculiarity of the internal observer. In fact, the inner quantum truth (the Truth-Observable) is a global truth. vii) With respect to the "measurement problem" the philosophical approach of this paper stands between Bohr and Einstein's views. In fact, Einstein's realist position relies on the belief that, under ideal conditions, measurements behave like mirrors [17] in the sense that they reflect an independently existing reality. To us, those ideal conditions are realized only in an inner reversible measurement. On the other hand, Bohr's position, which is neo-Kantian relationalism, stands on the assumption that the dependence of properties upon experimental conditions is relational, not causal. To Bohr, it is not meaningless to assign a value to a quantity in a quantum system unless it is measured (this was Heisenberg’s position). To us, Bohr's view means that the value of a certain quantity is reflected by an imperfect mirror, the imperfection of the resulting image depending on which kind of measurement is performed,. In the case of the inner reversible measurement, and only in that case, Einstein and Bohr views coincide, as the mirroring is perfect.

Acknowledgements

I am very grateful to Pieralberto Marchetti for many enlightening discussions and useful comments.

I also I wish to thank Giulia Battilotti and Dario Maguolo for useful discussions.

外国美术史--名词解释

外美史名词解释 1、意大利学院派: 1590年,意大利画家卡拉契兄弟创办了著名的波伦亚学院,旨在传承文艺复兴大师的传统,总结前人经验,培养年轻的美术家,并制定艺术法规,强调绘画标准, 避免样式主义和卡拉瓦乔的影响。由于这种办学原则明显缺乏创新精神,带有折衷主义的特点。故而题材较狭窄,多描绘宗教与神话,在技法上重素描轻色彩。可是对以后的古典主义美术的发展奠定了基础。 2、折衷主义: 这是17世纪意大利学院派美术的基本特色。主张绘画应有的最高标准是:米开朗基罗的人体、拉斐尔的素描、格累乔的典雅与风韵、威尼斯画派的色彩等等。这种企图培养新的艺术人才的办学思路与模式明显是综合了各家之长的折衷主义。 3、卡拉瓦乔主义: 17世纪由于卡拉瓦乔在当时欧洲画坛的巨大影响,吸引了相当多的敬慕者与追随者,他们的作品深受其艺术感染,无论是题材范围、画面明暗对比、人物写实手法等,多倾向于卡拉瓦乔的现实主义风格,故被称为卡拉瓦乔主义。 4、意大利地方画派: 17世纪意大利的现实主义画坛除了最具影响的卡拉瓦乔之外,还有颇具实力的有现实主义倾向的地方画派和画家:如热那亚画派(代表人物是斯特劳斯)、曼图亚画派(以多菲奇为代表)、那不勒斯画派(萨·洛撒为代表),另外还有威尼斯画派、佛罗伦萨画派和罗马画派等。 5、巴洛克艺术: 17世纪流行的艺术风格之一。这种风格最早产生于意大利的罗马,曾弥漫于欧洲的天主教国家。“巴洛克”这个派生词的由来说法不一:一说来自葡萄牙语或西班牙语,意指不圆的珠子;一说来自意大利语,有任意奇特、怪诞或推论上错误的含义。这个带有贬意的称呼是18世纪的古典主义理论家对这种艺术倾向的蔑视。 6、荷兰小画派: 17世纪的荷兰除了哈尔斯与伦勃朗这样伟大的现实主义绘画大师以外,还涌现了一批出色的画家。他们被称为“荷兰小画派”的原因有二:第一是画幅较小,适宜于市民阶层装饰居室;第二是不表现重大的社会题材,特别注重对生活细节的描绘,迎合了市民阶层的审美趣味。代表画家有:格拉尔德. 特鲍赫、皮特. 德. 霍赫、和加布里尔. 梅蒂绥等。 7、团体肖像画: 17世纪的荷兰美术除了单幅肖像画外还盛行团体肖像(群像)画,订件者往往要求画家对每个人都给以平等的表现机会,这就会使画家不能安排中心人物和一定的情节来统一画面与构图。但是,天才的肖像画家哈尔斯却突破了传统的呆板、平整的布局,尽量将人物安排的错落有致,营造出一种极其热烈的气氛。代表作品如《圣乔治射击手连军官的宴会》等。 8、西班牙“黄金时代”: 17世纪上半叶,西班牙美术人才辈出并出现了一个“黄金时期”(1560-1700)。他的繁荣主要得益于三个条件:一是西班牙文学的有力影响(文学家塞万提斯和戏剧家维加);二是地方画派中现实主义力量的存在和发展;三是意大利卡拉瓦乔艺术的影响。这个时期西班牙绘画的艺术特点是反映生活、表现时代和民族精神。代表画家有里韦拉、苏巴兰和委拉斯贵支。 9、“波德格涅斯”: 17世纪西班牙的塞维利亚城流行的一种画风(即西班牙的卡拉瓦乔主义)。波德格涅斯这一名词含有小酒店和小饭馆之意,由于古典主义的理论家们瞧不起描绘下层人民生活的风俗画,于是以嘲弄的口吻把这类作品通称为“波德格涅斯”的绘画。委拉斯贵支就曾受到过这种画风的影响。作品有《卖水的人》、《早餐》、《音乐师》等。 10、罗可可: 18世纪上半叶,法国出现的一种新的艺术样式。那种对岩洞、贝壳、钟乳石的模仿手法,被称为罗可可风格。它从巴洛克艺术的背景中演变出来,同时又反对巴洛克的豪华壮观,既繁华纤秀、浮华做作、矫揉妩媚,又愉悦亲切、轻快优雅,可

动物学名词解释。

1、物种:分类基本单位,种是具有一定的形态结构和生理特性以及一定自然分布区的生物种群,种内个体间可以彼此交配和产生后代,不同种之间存在生殖隔离。 2、双名法:对每种生物采用两个拉丁词或拉丁化的词的方法进行命名,第一个词为属名,第二个词为种加词。 7、出芽生殖:在亲体的一定部位长出与自身体形相似的个体,称为芽体。以后芽体可以脱离亲体发育成新个体或不脱离亲体而形成群体的生殖方式。 8、卵生::由母体产出的是受精卵或未受精卵,未受精卵则需在体外受精(孤雌生殖除外)。子代的胚胎发育在外界环境条件下进行,胚胎发育时所需营养物质由卵内所贮存的卵黄供给。 9、胎生:从母体内产出的是幼体。子代胚胎发育时所需的营养物质由母体供给。 10、卵胎生:从母体内产出的也是幼体。幼体胚胎发育时所需的营养仍由卵内所贮存的卵黄供给,母体的输卵管或孵育室仅提供子代胚胎发育的场所。 11、伸缩泡:原生动物所具有的泡状细胞器,能通过收缩和舒张排出体内多余的水分,也有部分的排泄功能。 12、刺丝泡:草履虫等表膜之下的小杆状结构,有孔开口在表膜上,当动物遇到刺激时,射出其内容物,遇水成为细丝,一般认为有防御功能。 13、变形运动:变形虫在运动时,其体表任何部位都可形成伪足,虫体不断向伪足伸出的方向移动,这种现象叫做变形运动。 14、伪足:肉足动物的足不固定,身体伸出的部分即代表足,有运动和取食功能。 15、接合生殖:草履虫等原生动物特有的一种有性生殖方式。生殖时两个虫体口沟贴合,表膜溶解,通过小核的分裂和部分交换,最终产生8个新个体的复杂过程。 16、裂体生殖:又叫复分裂。既细胞核首先分裂成很多个,称为裂殖体,然后细胞质随着核而分裂,包在每个核的外边,形成很多的小个体,称为裂殖子。是一种高效的分裂生殖方式。 17、寄生:一种生物生活在另一种生物的体内或体表,从中获取营养,并对该生物有害。 18、终末宿主:寄生虫成虫或有性生殖时期所寄生的寄主。 19、中间宿主:寄生虫幼虫或无性生殖时期所寄生的寄主。 20、胚层逆转:在胚胎发育中,大分裂球在外,小分裂球在内,与般多细胞动物相反。 24、生物发生律:生物的个体发育史是系统发展史的简单而迅速的重演。 25、世代交替:在动物的生活史中,无性世代和有性世代有规律地交替出现的现象。 26、辐射对称:通过身体的中轴有多个切面将身体分为大致相等的两部分。 27、消化循环腔:腔肠动物体壁围绕的中央腔既有消化功能又有循环功能。 28、网状神经系统:腔肠动物的神经细胞突起相互交织成网状结构。这是动物界首次出现的神经系统类型。网状神经系统无神经中枢,神经传导不定向,神经传导速度慢。 29、皮肌囊:扁形动物等的体壁,由皮肤和肌肉组成。起保护等作用。 30、两侧对称:通过身体的中央轴只有一个切面将身体分为大致相等的两部分的体制类型。 31、实质组织:在涡虫等动物的表皮、肌肉与内部器官之间填满了由中胚层来的实质,疏松地相互连接在一起,形成网状,可贮存养分。 32、不完全的消化系统:扁形动物等低等动物的消化管只有口,没有肛门,消化效率不高,称为不完全的消化系统。 33、原肾管:扁形动物等的排泄系统类型。在虫体两侧有一对弯曲、多次分支的纵行排泄管,每一小分支细管的末端连着焰细胞。通过焰细胞收集多余的水分和液体废物,经排泄管由体背面的排泄孔排出体外。 34、梯式神经系统:扁形动物的神经系统类型。身体前端有“脑”的雏形,由“脑”发出两条腹神经索,腹神经索发出神经分支彼此连接并分布到身体各部。

外国美术史口诀

外国美术史口诀 Document serial number【KK89K-LLS98YT-SS8CB-SSUT-SST108】

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明天教师节。 她大眼睛,红脸蛋。 三、宾语往往表示动作支配的对象,并且总是处在动词的后头。可由名词、代词、数词、名词化的形容词、不定式、动名词、宾语从句等来担任。 (1)名词性宾语。例如: 玫瑰花我给你们俩十朵,给你紫红的,给她粉红的。 (2)谓词性宾语。例如: 最有效的防御手段是进攻。 谁说女子不如男? 早上一起床,大家发现风停了,浪也静了。 四、定语是名词性词语的修饰成分。可以由名词,形容词和起名词和形容词作用的词,短语担任。如果定语是单个词,定语放在被修饰词的前面,如果是词组,定语放在被修饰词的后面。 (1)描写性定语,多由形容词性成分充当。例如: 弯弯曲曲的小河。青春气息。风平浪静的港湾。 (2)限制性定语:给事物分类或划定范围,使语言更加准确严密。例如: 晓风残月中的长城。野生动物。古城大理的湖光山色。 (3)助词“的”:定语和中心语的组合,有的必须加“的”,有的不能加“的”,有的可加可不加。 单音节形容词作定语,通常不加“的”,例如:红花、绿叶、新学校、好主意等。 双音节形容词作定语,常常加上“的”,特别是用描写状态的词,例如:晴朗的天、优良的传统、动听的歌声、粉红的脸等。 五、状语状语是动词性、形容词性词语的修饰成分。可以由副词、短语以及从句来担任。 (1)描写性状语:主要修饰谓词性成分,有的是描写动作状态,有些是限制或描写人物情态。例如: 他突然出现在大家面前。小李很高兴地对我说。 (2)限制性状语:主要表示时间、处所、程度、否定、方式、手段、目的、范围、对象、数量、语气等。例如: 午后,天很闷,风很小。白跑一趟。她的身上净是水。 (3)助词“地”:助词“地”是状语的标志。状语后面加不加“地”的情况很复杂。单音节副词做状语,一定不加,有些双音节副词加不加“地”均可,例如“非常热︰非常地热”。形容词里,单音节形容词做状语比较少,大都也不能加“地”,例如“快跑、苦练、大干”。多音节形容词有相当一部分加不加都可以,例如“热烈讨论︰热烈地讨论/仔细看了半天︰仔细地看了半天”

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动物学期末名词解释

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