QUANTUM COMPUTATIONAL STRUCTURES

GIANPIERO CATTANEO, MARIA LUISA DALLA CHIARA, ROBERTO GIUNTINI,AND ROBERTO LEPORINI

Abstract. Quantum computation has suggested new forms of quan-tum logic, called quantum computational logics ([CDCGL01]). The basicsemantic idea is the following: the meaning of a sentence is identifiedwith a quregister, representing a possible pure state of a compound phys-ical system, whose associated Hilbert space is an n-fold tensor product⊗

nC

2. The generalization to density operators, which might be use-ful to analyse entanglement-phenomena, is due to Gudder [Gu03]. Inthis paper we study structural properties of density operators systems,where some basic quantum logical gates are defined. We introduce thenotions of standard reversible and standard irreversible quantum com-putational structure. We prove that the second structure is isomorphicto an algebra based on a particular set of complex numbers.

1. Introduction

Quantum computation has recently suggested new forms of quantum logicthat have been called quantum computational logics ([CDCGL01]). Theselogics are based on the following semantic idea: unlike orthodox quantumlogic ([DCG02]), the meaning of a sentence is identified with a qubit or aquregister (a system of qubits) or, more generally, with a mixture of quregis-ters (shortly a qumix ). From a physical point of view, qubits represent pos-sible pure states of quantum systems, whose associated Hilbert space is C2.Quregisters represent pure states of compound systems, whose associatedHilbert space is an n-fold tensor product ⊗nC2, while qumixs correspond todensity operators.

The qubit semantics, presented in [CDCGL01], takes only in consider-ation qubits and quregisters. The generalization to qumixs, which mightbe useful to analyse entanglement-phenomena, is due to Gudder [Gu03].In this paper we will study structural properties of qumix systems, wheresome basic quantum logical gates are defined. The logics that are naturallycharacterized by such structures will be investigated in forthcoming papers.

2. Qubits, quregisters and qumixs

Consider the two–dimensional Hilbert space C2 (where any vector |ψ〉 is

represented by a pair of complex numbers). Let B(1) = {|0〉 , |1〉} be thecanonical orthonormal basis for C2, where |0〉 = (1, 0) and |1〉 = (0, 1).

Key words and phrases. quantum computation, quantum logic.

1

2 G. CATTANEO, M. L. DALLA CHIARA, R. GIUNTINI, AND R. LEPORINI

Definition 2.1 (Qubit). A qubit is a unit vector |ψ〉 of the Hilbert spaceC2.

Recalling the Born rule, any qubit |ψ〉 = c0 |0〉+c1 |1〉 (with |c0|2 + |c1|2 =1) can be regarded as an uncertain piece of information, where the answerNO has probability |c0|2, while the answer YES has probability |c1|2. Thetwo basis-elements |0〉 and |1〉 are usually taken as encoding the classicalbit-values 0 and 1, respectively. From a semantic point of view, they can bealso regarded as the classical truth-values Falsity and Truth.

An n-qubit system (also called n-quregister) is represented by a unit vec-tor in the n-fold tensor product Hilbert space ⊗nC2 := C

2 ⊗ . . . ⊗ C2

︸ ︷︷ ︸

n−times

(where

⊗1C2 := C2). We will use x, y, . . . as variables ranging over the set {0, 1}. At

the same time, |x〉 , |y〉 , . . . will range over the basis B(1). Any factorized unitvector |x1〉 ⊗ . . . ⊗ |xn〉 of the space ⊗nC2 will be called an n–configuration(which can be regarded as a quantum realization of a classical bit sequenceof length n). Instead of |x1〉⊗. . .⊗|xn〉 we will simply write |x1, . . . , xn〉. Re-call that the dimension of ⊗nC2 is 2n, while the set of all n–configurationsB(n) = {|x1, . . . , xn〉 : xi ∈ {0, 1}} is an orthonormal basis for the space⊗nC2. We will call this set a computational basis for the n–quregisters.Since any string x1, . . . , xn represents a natural number j ∈ [0, 2n−1] (wherej = 2n−1x1 + 2n−2x2 + . . . + xn), any unit vector of ⊗nC2 can be shortly

expressed in the following form:∑2n−1

j=0 cj ‖j〉〉, where cj ∈ C, ‖j〉〉 is the

n-configuration corresponding to the number j and∑2n−1

j=0 |cj |2 = 1.Consider now the two following sets of natural numbers:

C(n)1 := {i : ‖i〉〉 = |x1, . . . , xn〉 and xn = 1}

and

C(n)0 := {i : ‖i〉〉 = |x1, . . . , xn〉 and xn = 0}.

Let us refer to a generic unit vector of the space ⊗nC2:

|ψ〉 =2n−1∑

i=0

ai ‖i〉〉.

We obtain:

|ψ〉 =∑

i∈C(n)0

ai ‖i〉〉 +∑

j∈C(n)1

aj ‖j〉〉.

Let P(n)1 and P

(n)0 be the projections onto the span of

{

‖i〉〉 | i ∈ C(n)1

}

and{

‖i〉〉 | i ∈ C(n)0

}

, respectively. Clearly, P(n)1 + P

(n)0 = I(n), where I(n)

is the identity operator of ⊗nC2. Apparently, P(n)1 and P

(n)0 are density

operators iff n = 1. Let kn =1

2n−1be the normalization coefficient such that

knP(n)1 and knP

(n)0 are density operators. From an intuitive point of view,

QUANTUM COMPUTATIONAL STRUCTURES 3

knP(n)1 can be regarded as a privileged information corresponding to the

Truth, while knP(n)0 corresponds to the Falsity. In particular, P

(1)1 represents

the bit |1〉, while P(1)0 represents the bit |0〉. Let D(⊗nC2) be the set of all

density operators of ⊗nC2 and let D :=⋃∞

n=1 D(⊗nC2).

Definition 2.2 (Qumix). A qumix is a density operator in D.

Needless to say, quregisters correspond to particular qumixs that are purestates (i.e. projections onto one-dimensional closed subspaces of a given⊗nCn). Recalling the Born rule, we can now define the probability-value ofany qumix.

Definition 2.3 (Probability of a qumix). For any qumix ρ ∈ D(⊗nC2):

p(ρ) = tr(P(n)1 ρ).

¿From an intuitive point of view, p(ρ) represents the probability that theinformation stocked by the qumix ρ is true. In the particular case where ρcorresponds to the qubit

|ψ〉 = c0 |0〉 + c1 |1〉 ,

we obtain that p(ρ) = |c1|2.For any quregister |ψ〉, we will write p(|ψ〉) instead of p(P|ψ〉), where P|ψ〉 is

the density operator represented by the projection onto the one-dimensionalsubspace spanned by the vector |ψ〉.

3. Quantum Gates

In quantum computation, quantum logical gates (shortly gates) are uni-tary operators that transform quregisters into quregisters. Being unitary,gates represent characteristic reversible transformations. The canonical gates(which are studied in the literature) can be naturally generalized to qumixs.Generally, gates correspond to some basic logical operations that admit areversible behaviour. We will consider here the following gates: negation,the square root of negation, conjunction and disjunction.

Let us first refer to quregisters.

Definition 3.1 (The negation). For any n ≥ 1, the negation on ⊗nC2 is

the linear operator Not(n) such that for every element |x1, . . . , xn〉 of the

computational basis B(n):

Not(n)(|x1, . . . , xn〉) = |x1, . . . , xn−1〉 ⊗ |1 − xn〉 .

In other words, Not(n) inverts the value of the last element of any basis–vector of ⊗nC2.

Clearly:

Not(n) =

{

X, if n = 1;

I(n−1) ⊗ X, otherwise,

4 G. CATTANEO, M. L. DALLA CHIARA, R. GIUNTINI, AND R. LEPORINI

where X is the “first” Pauli matrix, i.e.,

X =

(0 11 0

)

Definition 3.2 (The Petri-Toffoli gate). For any n ≥ 1 and any m ≥ 1 the

Petri-Toffoli gate is the linear operator T (n,m,1) defined on ⊗n+m+1C2 suchthat for every element |x1, . . . , xn〉 ⊗ |y1, . . . , ym〉 ⊗ |z〉 of the computational

basis B(n+m+1):

T (n,m,1)(|x1, . . . , xn〉⊗|y1, . . . , ym〉⊗|z〉) = |x1, . . . , xn〉⊗|y1, . . . , ym〉⊗|xnym ⊕ z〉 ,

where ⊕ represents the sum modulo 2.

Clearly:

T (n,m,1) := (I(n+m) − P(n)1 ⊗ P

(m)1 ) ⊗ I(1) + P

(n)1 ⊗ P

(m)1 ⊗ X.

One can easily show that both Not(n) and T (n,m,1) are unitary operators.The quantum logical gates we have considered so far are, in a sense, “semi-

classical”. A quantum logical behaviour only emerges in the case where ourgates are applied to superpositions. When restricted to classical registers,such operators turn out to behave as classical (reversible) truth-functions.We will now consider a genuine quantum gate that transforms classical reg-isters (elements of B(n)) into quregisters that are superpositions.

Definition 3.3 (The square root of the negation). For any n ≥ 1, the

square root of the negation on ⊗nC2 is the linear operator√

Not(n)

suchthat for every element |x1, . . . , xn〉 of the computational basis B(n):

√Not

(n)(|x1, . . . , xn〉) = |x1, . . . , xn−1〉 ⊗

1

2((1 + i) |xn〉 + (1 − i) |1 − xn〉).

One can easily show that√

Not(n)

is a unitary operator. The basic

property of√

Not(n)

is the following:

for any |ψ〉 ∈ ⊗nC

2,√

Not(n)

(√

Not(n)

(|ψ〉)) = Not(n)(|ψ〉).In other words, applying twice the square root of the negation means negat-ing.

Clearly:√

Not(n)

:=

{

M, if n = 1;

I(n−1) ⊗ M, otherwise,

where

M :=1

2

(1 + i 1 − i1 − i 1 + i

)

.

Interestingly enough, the gate√

Not(n)

admits physical models and im-

plementations ([DEL00]). From a logical point of view,√

Not(n)

can beregarded as a “tentative partial negation” (a kind of “half negation”) that

QUANTUM COMPUTATIONAL STRUCTURES 5

transforms precise pieces of information into maximally uncertain ones. For,we have:

p(√

Not(1)

(|1〉)) =1

2= p(

√Not

(1)(|0〉)).

Consider now the set⋃∞

n=1 ⊗nC2 (which contains all quregisters |ψ〉 “liv-

ing” in ⊗nC2, for a given n ≥ 1). The gates Not,√

Not and T can beuniformly defined on this set in the expected way:

Not(|ψ〉) := Not(n)(|ψ〉), if |ψ〉 ∈ ⊗nC

2

√Not(|ψ〉) :=

√Not

(n)(|ψ〉), if |ψ〉 ∈ ⊗n

C2

T (|ψ〉 , |ϕ〉 , |χ〉) := T (n,m,1)(|ψ〉 , |ϕ〉 , |χ〉),if |ψ〉 ∈ ⊗n

C2, |ϕ〉 ∈ ⊗m

C2 and |χ〉 ∈ C

2.

On this basis, a conjunction And and a disjunction Or can be definedfor any pair of quregisters |ψ〉 and |ϕ〉:

And(|ψ〉 , |ϕ〉) := T (|ψ〉 , |ϕ〉 , |0〉);Or(|ψ〉 , |ϕ〉) := Not(And(Not(|ψ〉),Not(|ϕ〉))).

Clearly, |0〉 represents an “ancilla” in the definition of And.One can easily verify that, when applied to classical bits, Not, And and

Or behave as the standard Boolean truth-functions.At first sight, And and Or may look as irreversible transformations.

However, it is important to recall that, in this framework, And(|ψ〉 , |ϕ〉)should be regarded as a mere metalinguistic abbreviation for T (|ψ〉 , |ϕ〉 , |0〉)(where T is reversible). Similarly Or.

The gates considered so far can be naturally generalized to qumixs. Whenour gates will be applied to density operators, we will write: NOT,

√NOT, AND,

OR (instead of Not,√

Not, And, Or).

Definition 3.4 (The negation). For any qumix ρ ∈ D(⊗nC2),

NOT(n)ρ = Not(n)ρNot(n).

Definition 3.5 (The square-root of the negation). For any qumix ρ ∈D(⊗nC2), √

NOT(n)

ρ =√

Not(n)

ρ√

Not(n)∗

,

where√

Not(n)∗

is the adjoint of√

Not(n)

.

It is easy to see that for any n ∈ N+, both NOT(n)(ρ) and√NOT

(n)(ρ) are

qumixs of D(⊗nC2). Further: NOT(n)NOT(n) = I(n).

Definition 3.6 (The conjunction). Let ρ ∈ D(⊗nC2) and σ ∈ D(⊗mC2).

AND(n,m,1)(ρ, σ) = T (n,m,1)ρ ⊗ σ ⊗ P(1)0 T (n,m,1).

Like in the quregister-case, the gates NOT,√NOT, AND, OR can be uniformly

defined on the set D of all qumixs.The following theorems describe some basic properties of our gates.

6 G. CATTANEO, M. L. DALLA CHIARA, R. GIUNTINI, AND R. LEPORINI

Theorem 3.1. [Gu03]

(i) NotknP(n)0 Not = knP

(n)1 ;

(ii) NotknP(n)1 Not = knP

(n)0 ;

(iii) p(NOTρ) = 1 − p(ρ).

Consider now the “second” Pauli’s matrix:

Y =

(0 −ii 0

)

.

This matrix can be naturally generalized to an operator R(n) defined on⊗nC2 (for any n ∈ N+):

R(n) :=

{

Y, if n = 1;

I(n−1) ⊗ Y, otherwise.

Lemma 3.1. For any n ∈ N+, the following properties hold:

(i) tr(R(n)) = 0;

(ii) tr(R(n)P(n)1 ) = 0;

(iii) tr(R(n)P(n)0 ) = 0.

Proof. (i) Let n = 1. Then tr(R(1)) = tr(Y ) = 0. Let n > 1. Then,

tr(R(n)) = tr(I(n−1) ⊗ Y ) = tr(I(n−1))tr(Y ) = 0.

(ii) If n = 1, then tr(R(1)P(1)1 ) = tr(Y P

(1)1 ) = 0. If n > 1, then tr(R(n)P

(n)1 ) =

tr(I(n−1) ⊗ Y P(1)1 ) = 0.

(iii) It follows from the fact that tr(R(1)P(1)0 ) = tr(Y P

(1)0 ) = 0. ¤

Theorem 3.2.

(i)√NOT

√NOTρ = NOTρ;

(ii) p(√NOTρ) =

1

2− 1

2tr(R(n)ρ);

(iii) p(√NOTNOTρ) = p(NOT

√NOTρ);

(iv) ∀n ∈ N+: p(√NOTknP

(n)1 ) = p(

√NOTknP

(n)0 ) =

1

2.

Proof. The proof of (i) is contained in Gudder [Gu03].

(ii) Let n = 1. Then p(√NOTρ) = tr(M∗P

(1)1 Mρ) = tr

(1

2(I(1) − Y )ρ

)

=

1 − tr(Rρ)

2. Let n > 1. Then p(

√NOTρ) = tr(I(n−1) ⊗ M∗P

(1)1 Mρ) =

tr

(

I(n−1) ⊗ 1

2(I(1) − Y )ρ

)

=1

2− 1

2tr(R(n)ρ).

(iii)-(iv) It follows from (ii) and Lemma 3.1 (ii-iii). ¤

Theorem 3.3.

(i) p(AND(ρ, σ)) = p(ρ)p(σ);

(ii) p(√NOT(AND(ρ, σ))) =

1

2.

QUANTUM COMPUTATIONAL STRUCTURES 7

Proof.

(i) p(AND(ρ, σ)) =tr(P(n+m+1)1 T (n+m+1)ρ ⊗ σ ⊗ P

(1)0 T (n+m+1))

=tr((I(n+m) − P(n)1 ⊗ P

(m)1 )ρ ⊗ σ(I(n+m) − P

(n)1 ⊗ P

(m)1 )

⊗ P(1)1 P

(1)0 + P

(n)1 ρP

(n)1 ⊗ P

(m)1 σP

(m)1 ⊗ P

(1)1 XP

(1)0 X)

=tr((I(n+m) − P(n)1 ⊗ P

(m)1 )ρ ⊗ σ)tr(P

(1)1 P

(1)0 )

+ tr(P(n)1 ρ)tr(P

(m)1 σ)tr(P

(1)1 P

(1)1 )

=p(ρ)p(σ)

(ii) p(√NOT(AND(ρ, σ)))

=tr(P(n+m+1)1

√Not

(n+m+1)T (n+m+1)ρ ⊗ σ ⊗ P

(1)0 T (n+m+1)

√Not

(n+m+1)∗)

=tr((I(n+m) − P(n)1 ⊗ P

(m)1 )ρ ⊗ σ(I(n+m) − P

(n)1 ⊗ P

(m)1 ) ⊗ P

(1)1 MP

(1)0 M∗

+ P(n)1 ρP

(n)1 ⊗ P

(m)1 σP

(m)1 ⊗ P

(1)1 MP

(1)1 M∗)

=tr((I(n+m) − P(n)1 ⊗ P

(m)1 )ρ ⊗ σ)tr(P

(1)1 MP

(1)0 M∗)

+ tr(P(n)1 ρ)tr(P

(m)1 σ)tr(P

(1)1 MP

(1)1 M∗)

=(1 − p(ρ)p(σ))p(√NOTP

(1)0 ) + p(ρ)p(σ)p(

√NOTP

(1)1 ) =

1

2

¤

4. The standard reversible quantum computational structure

An interesting feature of the qumix system is the following: any real

number λ ∈ [0, 1] ⊂ R uniquely determines a qumix ρ(n)λ (for any n ∈ N+):

ρ(n)λ := (1 − λ)knP

(n)0 + λknP

(n)1 . (4.1)

Clearly, ρ(n)λ ∈ D(⊗nC2). From an intuitive point of view, ρ

(n)λ represents

a mixture of pieces of information that might correspond to the Truth withprobability λ.

¿From the physical point of view, ρ(n)λ corresponds to a particular prepara-

tion of the system such that the quantum system is in the state knP(n)0 with

probability 1−λ and in the state knP(n)1 with probability λ. It is worthwhile

recalling that the random polarized states of the photon are represented by

the density operator ρ(1)1/2 =

1

2I(1).

Lemma 4.1.

(i) ∀n ∈ N+ ∀λ ∈ [0, 1]: p(ρ(n)λ ) = λ;

(ii) p(√NOTρ

(n)λ ) =

1

2.

8 G. CATTANEO, M. L. DALLA CHIARA, R. GIUNTINI, AND R. LEPORINI

Proof. (i) Straightforward.

(ii) p(√NOTρ

(n)λ ) =

1

2− 1

2tr(R(n)ρ

(n)λ ) (Theorem 3.2 (ii))

=1

2− 1 − λ

2ntr

(

R(n)P(n)0

)

− λ

2ntr

(

R(n)P(n)1

)

=1

2(Lemma 3.1 (ii)-(iii))

¤

We will now introduce two interesting relations that can be defined onthe set of all qumixs. Both of them turn out to be a preorder-relation. Wewill speak of weak and of strong preorder, respectively.

Definition 4.1 (Weak preorder).

ρ ≤ σ iff p(ρ) ≤ p(σ).

Definition 4.2 (Strong preorder). ρ ¹ σ iff the following conditions hold:

(i) p(ρ) ≤ p(σ);(ii) p(

√NOTσ) ≤ p(

√NOTρ).

Clearly, ρ ¹ σ implies ρ ≤ σ, but not the other way around. One im-mediately shows that both ≤ and ¹ are reflexive and transitive, but notantisymmetric. Counterexamples can be easily found in D(C2).

Consider now the following structure:(

D ,¹ , AND , NOT ,√NOT , P

(1)0 , P

(1)1 , ρ

(1)1/2

)

. (4.2)

We will call such a structure the standard reversible quantum computationalstructure (shortly the RQC-structure).

In the following we will generally write I, P0, P1 and ρ1/2 instead of I(1),

P(1)0 ,P

(1)1 , ρ

(1)1/2. From an intuitive point of view, P0, P1 and ρ1/2 represent

privileged pieces of information that are true, false, indeterminate, respec-tively. Generally, our qumixs fail to satisfy Duns Scotus law : P0 and P1 arenot the minimum and the maximum element of the RQC-structure. Hence,in this situation, it is interesting to isolate the elements that have a Scotianbehaviour.

Definition 4.3 (Down and up scotian qumixs). Let ρ be a qumix of D.

(i) ρ is down Scotian iff P0 ¹ ρ;(ii) ρ is up Scotian iff ρ ¹ P1;(iii) ρ is Scotian iff ρ is both down and up Scotian.

Lemma 4.2.

(i) ρ ¹√NOTP1 iff p(ρ) ≤ 1

2;

(ii)√NOTP0 ¹ ρ iff p(ρ) ≥ 1

2.

QUANTUM COMPUTATIONAL STRUCTURES 9

Proof. (i) Suppose ρ ¹√NOTP1. By Theorem 3.2(iv), we obtain p(ρ) ≤

p(√NOTP1) =

1

2. Viceversa, suppose p(ρ) ≤ 1

2. Then, p(ρ) ≤ 1

2=

p(√NOTP1). Now,

√NOT

√NOTP1 = P0. Thus 0 = p(

√NOT

√NOTP1) ≤

p(√NOTρ). Hence: ρ ¹

√NOTP1.

(ii) Similar to the proof of (i), via Theorem 3.2(iv). ¤

Theorem 4.1.

(i) ρ is down Scotian iff p(√NOTρ) ≤ 1

2iff

√NOTρ ¹

√NOTP1;

(ii) ρ is up Scotian iff1

2≤ p(

√NOTρ) iff

√NOTP0 ¹

√NOTρ;

(iii) ρ is Scotian iff p(√NOTρ) =

1

2.

(iv) ∀n ∈ N+: knP(n)0 , knP

(n)1 , ρ

(n)1/2 are Scotian.

(v) For any ∈ N+, the set D(⊗nC2) contains uncountably many Scotiandensity operators.

Proof. The proof of (i)-(ii) follows from Lemma 4.2.The proof of (iii) follows from (i) and (ii).(iv) The proof follows from Lemma 4.1 and from (iii).(v) It is sufficient to show that D(C2) contains uncountably many Scotianelements. Let λ ∈ [−1, 1] ⊂ R. Consider the operator

ρ(λ) :=1

2

(1 λλ 1

)

Clearly, ρ(λ) ∈ D(C2). An easy computation shows that p(√NOTρ(λ)) =

1

2.

Thus, by (iii) we can conclude that ρ(λ) is Scotian. ¤

5. An irreversible operation: ÃLukasiewicz-sum

The gates we have considered so far represent typical reversible logicaloperations. From a logical point of view, it might be interesting to consideralso some irreversible operations. An important example is represented bya ÃLukasiewicz-like disjunction.

Definition 5.1 (The ÃLukasiewicz-disjunction). Let τ ∈ D(⊗nC2) and σ ∈D(⊗mC2).

τ ⊕ σ := ρ(1)p(τ)⊕p(σ),

where ⊕ in p(τ) ⊕ p(σ) is the ÃLukasiewicz “truncated sum” defined on thereal interval [0, 1] (i.e. p(τ) ⊕ p(σ) = min {1, p(τ) + p(σ)}) ([Za34]).

Lemma 5.1.

(i)

τ ⊕ σ =

ρ(1)p(τ)⊕p(σ), if p(τ) + p(σ) ≤ 1;

P(1)1 , otherwise;

10 G. CATTANEO, M. L. DALLA CHIARA, R. GIUNTINI, AND R. LEPORINI

(ii) p(τ ⊕ σ) = p(τ) ⊕ p(σ);

(iii) p(√NOT(τ ⊕ σ)) =

1

2.

Proof. (i) Straightforward.(ii) The proof follows from Lemma 4.1(i).(iii) The proof follows from Lemma 4.1(ii). ¤

Lemma 5.2. Let ρ ∈ D(⊗nC2).

(i) ∀n ∈ N+: ρ ⊕ knP(n)1 = P

(1)1 ;

(ii) ∀n ∈ N+: ρ ⊕ knP(n)0 = ρ

(1)p(ρ);

(iii) ρ ⊕ NOTρ = P(1)1 .

Proof. Straightforward. ¤

¿From Lemma 5.2 it follows that p(ρ⊕knP(n)1 ) = 1, p(ρ⊕knP

(n)0 ) = p(ρ)

and p(ρ ⊕ NOTρ) = 1.

6. The standard irreversible quantum computational algebra

The preorder ¹ permits us to define on the set of all qumixs an equivalencerelation in the expected way.

Definition 6.1 (The strong equivalence relation).

ρ u σ iff ρ ¹ σ and σ ¹ ρ.

Clearly, u is an equivalence relation. Let

[D]u := {[ρ]u | ρ ∈ D} .

We will omit u in [ρ]u if no confusion is possible.Unlike the qumixs (which are only preordered by ¹), the equivalence-

classes of [D]u can be partially ordered in a natural way.

Definition 6.2.

[ρ] ¹ [σ] iff ρ ¹ σ.

The relation ¹ (which is well defined) is a partial order.

Lemma 6.1.

(i) ∀n ∈ N+: [P1] =[

knP(n)1

]

;

(ii) ∀n ∈ N+: [P0] =[

knP(n)0

]

;

(iii) ∀n ∈ N+ ∀λ ∈ [0, 1]:[

ρ(1)λ

]

=[

ρ(n)λ

]

.

Proof. (i)-(ii) The proof follows from Theorem 3.2 (iv) and from the fact

that ∀n ∈ N+: p(P(1)1 ) = 1 = p(knP

(n)1 ).

(iii) The proof follows from Lemma 4.1. ¤

On this basis, one can naturally define on the set [D]u a conjunction, anegation, the square root of the negation, a ÃLukasiewicz-disjunction:

QUANTUM COMPUTATIONAL STRUCTURES 11

Definition 6.3. Let ρ ∈ D(⊗nC2) and σ ∈ D(⊗mC2).

(i) [ρ]AND[σ] = [AND(ρ, σ)];(ii) NOT[ρ] = [NOTρ];(iii)

√NOT[ρ] = [

√NOTρ];

(iv) [ρ] ⊕ [σ] = [ρ ⊕ σ].

Lemma 6.2. The operations of Definition 6.3 are well defined.

Proof. (i) Suppose ρ′ u ρ and σ′ u σ. We want to show that p(AND(ρ, σ)) =p(AND(ρ′, σ′)) and p(

√NOT(AND(ρ, σ))) = p(

√NOT(AND(ρ′, σ′))). The proof

follows from Theorem 3.3.(ii) The proof follows from Theorem 3.1(iii) and Theorem 3.2 (iii).(iii) The proof follows from Theorem 3.1(iii) and Theorem 3.2(i).(iv) Straightforward. ¤

Lemma 6.3.

(i) The operation AND is associative and commutative;(ii) The operation ⊕ is associative and commutative;(iii) NOT NOT[ρ] = [ρ];(iv)

√NOT

√NOT[ρ] = NOT[ρ];

(v)√NOT NOT[ρ] = NOT

√NOT[ρ].

Proof. Straightforward. ¤

Consider now the structure(

[D]u , AND ,⊕ , NOT ,√NOT , [P0]u, [P1]u , [ρ1/2]u

)

. (6.1)

We will call such a structure the standard irreversible quantum computa-tional algebra (shortly the IQC-algebra).

As happens in the case of ¹, also the weak preorder ≤ permits us to definean equivalence relation, which will be called weak equivalence relation.

Definition 6.4 (Weak equivalence relation).

ρ ≡ σ iff ρ ≤ σ and σ ≤ ρ.

Clearly, ≡ is an equivalence relation. Let

[D]≡ := {[ρ]≡ | ρ ∈ D} .

Also [D]≡ can be partially ordered in a natural way.

Definition 6.5.

[ρ]≡ ≤ [σ]≡ iff ρ ≤ σ.

One can easily show that the relation ≤ (which is well defined) is a partialorder.

A conjunction, a ÃLukasiewicz-disjunction, a negation (but not the squareroot of the negation!) can be naturally defined on [D]≡.

Definition 6.6. Let ρ ∈ D(⊗nC2) and σ ∈ D(⊗mC2).

12 G. CATTANEO, M. L. DALLA CHIARA, R. GIUNTINI, AND R. LEPORINI

(i) [ρ]≡ AND [σ]≡ = [AND(ρ, σ)]≡;(ii) NOT[ρ]≡ = [NOTρ]≡;(iii) [ρ]≡ ⊕ [σ]≡ = [ρ ⊕ σ]≡.

Lemma 6.4. The operations of Definition 6.6 are well defined.

Proof. (i) It is a consequence of Theorem 3.3(i).(ii) It is a consequence of Theorem 3.1(i).(iii) Straightforward. ¤

Unlike u, the relation ≡ is not a congruence with respect to√NOT. In fact,

the following situation is possible: [ρ]≡ = [σ]≡ and [√NOT ρ]≡ 6= [

√NOTσ]≡.

Consider for example the following unit vectors of C2: |ψ〉 :=

√2

2|0〉+

√2

2|1〉

and |ϕ〉 :=

√2

2|0〉 +

1 + i

2|1〉.

Let P|ψ〉 and P|ϕ〉 be the projections onto the unidimensional spacesspanned by |ψ〉 and |ϕ〉, respectively. It turns out that p(P|ψ〉) = p(P|ϕ〉) =1

2. Accordingly, [P|ψ〉]≡ = [P|ϕ〉]≡. However, p(

√NOTP|ψ〉) =

1

2and p(

√NOTP|ϕ〉) =

1

8+

(1

2− 1

2√

2

)2

≈ 0.146447. Consequently, [P|ψ〉]u 6= [P|ϕ〉]u.

An interesting relation between the weak and the strong preorder is de-scribed by the following theorem.

Theorem 6.1. For any ρ, σ ∈ D:

[ρ]≡ ≤ [σ]≡ iff [ρ]u AND [P1]u ¹ [σ]u AND [P1]u.

Proof. Suppose p(ρ) ≤ p(σ). By Theorem 3.3(i), we obtain

p(AND(ρ, P1)) = p(ρ) ≤ p(σ) = p(AND(σ, P1)). (6.2)

By Theorem 3.3(ii),

p(√NOT AND(ρ, P1)) =

1

2= p(

√NOT AND(σ, P1)). (6.3)

Thus, [ρ]u AND [P1]u ¹ [σ]u AND [P1]u.Viceversa, suppose [ρ]u AND [P1]u ¹ [σ]u AND [P1]u.Then,

p(ρ) = p(ρ)p(P1)) = p(AND(ρ, P1)) ≤ p(AND(σ, P1)) = p(σ). (6.4)

¤

Lemma 6.5.

(i) The structure ([D]≡ , AND , [P1]≡) is an Abelian monoid with neutralelement [P1]≡;

(ii) ([D]≡ ,⊕ , [P0]≡) is an Abelian monoid with neutral element [P0]≡;(iii) NOT NOT[ρ]≡ = [ρ]≡;

Proof. Easy. ¤

QUANTUM COMPUTATIONAL STRUCTURES 13

7. The Poincare irreversible quantum computationalstructures

We will now restrict our analysis to qumixs living in the two-dimensionalspace C2. As is well known, every density operator of D(C2) has the follow-ing matrix representation:

1

2(I + r1X + r2Y + r3Z) , (7.1)

where r1, r2, r3 are real numbers such that r21 + r2

2 + r23 ≤ 1 and X, Y, Z are

the Pauli matrices:

X =

(0 11 0

)

Y =

(0 −ii 0

)

Z =

(1 00 −1

)

.

It turns out that a density operator1

2(I + r1X + r2Y + r3Z) represents

a pure state (a qubit) iff r21 + r2

2 + r23 = 1. Consequently,

• Pure density operators are in 1 : 1 correspondence with the pointsof the surface of the Poincare sphere;

• Proper mixtures are in 1 : 1 correspondence with the inner points ofthe Poincare sphere.

Let ρ be a density operator of D(C2). We will denote by ρ the point of thePoincare sphere that is univocally associated to ρ.

Let (r1, r2, r3) be a point of the Poincare sphere. We will denote by(r1, r2, r3) the density operator univocally associated to (r1, r2, r3).

Lemma 7.1. Let ρ ∈ D(C2) such that ρ = (r1, r2, r3). The following con-ditions hold:

(i) p(ρ) =1 − r3

2;

(ii) p(√NOT ρ) =

1 − r2

2.

Proof. Easy computation. ¤

An irreversible conjunction can be now naturally defined on the set of allqumixs of D(C2).

Definition 7.1. (The irreversible conjunction)Let τ, σ ∈ D(C2).

IAND(τ, σ) = ρ(1)p(τ)p(σ) (7.2)

Interestingly enough, the density operator IAND(τ, σ) can be described interms of the partial trace. Suppose we have a compound physical systemconsisting of three subsystems, and let

H = (⊗nC

2) ⊗ (⊗mC

2) ⊗ (⊗rC

2)

14 G. CATTANEO, M. L. DALLA CHIARA, R. GIUNTINI, AND R. LEPORINI

be the Hilbert space associated to our system. Then, for any density oper-ator ρ of H, there is a unique density operator tr1,2(ρ) that represents thepartial trace of ρ on the space ⊗rC2 (associated to the third subsystem).The two operators ρ and tr1,2(ρ) are statistically equivalent with respect to

the third subsystem. In other words, for any self-adjoint operator A(r) of⊗rC2:

tr(tr1,2(ρ)A(r)) = tr(ρ I(n) ⊗ I(m) ⊗ A(r)).

The density operator tr1,2(ρ), obtained by “tracing out” the first andthe second subsystem, is also called the reduced state of ρ on the thirdsubsystem.

One can prove that:

IAND(τ, σ) = tr1,2(AND(τ, σ)).

In other words, IAND(τ, σ) represents the reduced state of AND(τ, σ) onthe third subsystem.

An interesting situation arises when both τ and σ are pure states. Forinstance, suppose that:

τ = P|ψ〉 and σ = P|ϕ〉,

where |ψ〉 and |ϕ〉 are proper qubits. Then,

AND(τ, σ) = PT (1,1,1)(|ψ〉⊗|ϕ〉⊗|0〉),

which is a pure state. At the same time, we have:

IAND(τ, σ) = tr1,2(PT (1,1,1)(|ψ〉⊗|ϕ〉⊗|0〉)),

which is a proper mixture. Apparently, when considering only the propertiesof the third subsystem, we loose some information. As a consequence, weobtain a final state that does not represent a maximal knowledge. As iswell known, situations where the state of a compound system represents amaximal knowledge, while the states of the subsystems are proper mixtures,play an important role in the framework of entanglement-phenomena.

Lemma 7.2.

(i) IAND is associative and commutative;(ii) IAND(ρ, P0) = P0;(iii) IAND(ρ, P1) = ρp(ρ);(iv) p(IAND(ρ, σ)) = p(ρ)p(σ);

(v) p(√NOT IAND(ρ, σ)) =

1

2.

Proof. Easy computation. ¤

Consider now the structure(

D(C2) , IAND ,⊕ , NOT ,√NOT , P0, P1 , ρ1/2

)

. (7.3)

We will call such a structure the Poincare irreversible quantum computa-tional algebra (shortly the Poincare IQC-algebra).

QUANTUM COMPUTATIONAL STRUCTURES 15

We can now refer to the relation ¹u, representing the restriction of u toD(C2). For any ρ ∈ D(C2), let

[ρ]¹u :={σ ∈ D(C2) | ρ u σ

}. (7.4)

Further define

[D(C2)]¹u :={[ρ]¹u | ρ ∈ D(C2)

}. (7.5)

The operations IAND ,⊕ , NOT ,√NOT and the relation ¹ can be defined on

[D(C2)]¹u in the expected way.On this basis we obtain the following quotient-structure

(

[D(C2)]¹u , IAND ,⊕ , NOT ,√NOT , [P0]¹u , [P1]¹u , [ρ1/2]¹u

)

.

We will call such a structure the contracted Poincare irreversible quantumcomputational algebra (shortly the contracted Poincare IQC-algebra).

Theorem 7.1. The contracted Poincare IQC-algebra is isomorphic to theIQC-algebra, via the map g : [D(C2)]¹u → [D]u such that ∀ρ ∈ D(C2):

g([ρ]¹u) = [ρ]u. (7.6)

Further, for any ρ , σ ∈ D(C2): [ρ]¹u ¹ [σ]¹u iff g( [ρ]¹u) ¹ g([σ]¹u).

Proof. One can readily see that g preserves the operation NOT,√NOT and ⊕.

By Theorem 3.3 and Lemma 7.2(iv-v), g preserves also the operation IAND.Clearly, the map g is injective. Let us prove that g is also surjective. To thisaim, it is sufficient to show that for any n ∈ N+ and for any ρ ∈ D(⊗nC2),there exists a density operator ρ′ ∈ D(C2) such that:

(i) p(ρ) = p(ρ′);(ii) p(

√NOTρ) = p(

√NOTρ′).

Let ρ ∈ D(⊗nC2) and let ρ′ be the reduced state of ρ on C2. Accordingly,for any self-adjoint operator A of C2, we have:

tr(I(n−1) ⊗ A ρ) = tr(A ρ′). (7.7)

Thus, p(ρ) = tr(P(n)1 ρ) = tr(I(n−1) ⊗ P

(1)1 ρ) = tr(P

(1)1 ρ′).

We now prove (ii).

p(√NOTρ) = tr(P

(n)1 (I(n−1) ⊗ M)ρ(I(n−1) ⊗ M∗))

= tr(I(n−1) ⊗ M∗P(1)1 Mρ)

= tr(M∗P(1)1 Mρ′) (7.7)

= p(√NOTρ′).

¤

16 G. CATTANEO, M. L. DALLA CHIARA, R. GIUNTINI, AND R. LEPORINI

8. The complex quantum computational algebra

An interesting algebraic property of the contracted Poincare IQC-structureis the following: our structure turns out to be isomorphic to a structure basedon a particular subset of the set C of all complex numbers. Let

C1 :={(a, b) | a, b ∈ R and (1 − 2a)2 + (1 − 2b)2 ≤ 1

}.

Note that for all pairs (a, b) ∈ C1, both elements a, b belong to the realinterval [0, 1].

Let 0 :=

(

0,1

2

)

, 1 :=

(

1,1

2

)

, 1/2 :=

(1

2,1

2

)

.

The following operations (IANDC1 , NOTC1 ,√NOT

C1 , ⊕C1) can be definedon C1.

Definition 8.1.

(i) (a1, a2)IANDC1(b1, b2) =

(

a1b1,1

2

)

;

(ii) NOTC1(a1, a2) = (1 − a1, 1 − a2);

(iii)√NOT

C1(a1, a2) = (a2, 1 − a1);

(iv) (a1, a2) ⊕C1 (b1, b2) =

(

a1 + b1,1

2

)

, if a1 + b1 ≤ 1;

1, otherwise.

One can easily see that C1 is closed under the operations of Definition 8.1.

Lemma 8.1.

(i) The operations IANDC1 and ⊕C1 are commutative and associative;(ii) (a1, a2)IAND

C10 = 0;

(iii) (a1, a2) ⊕C1 0 =

(

a1,1

2

)

;

(iv) (a1, a2)IANDC11 =

(

a11

2

)

;

(v) (a1, a2) ⊕C1 1 = 1;(vi) NOTC1NOTC1(a1, a2) = (a1, a2);

(vii)√NOT

C1NOTC1(a1, a2) = NOTC1

√NOT

C1(a1, a2);

(viii)√NOT

C1√NOT

C1(a1, a2) = NOTC1(a1, a2);

(ix) (a1, a2) is a fixed point of NOTC1 iff (a1, a2) is a fixed point of√NOT

C1

iff (a1, a2) = 1/2.

Proof. Easy computation. ¤

Definition 8.2.

(a1, a2) ¹ (b1, b2) iff a1 ≤ b1 and b2 ≤ a2.

Consider now the structure(

C1 , IANDC1 ,⊕C1 , NOTC1 ,√NOT

C1 , 0 , 1 , 1/2)

.

We will call such a structure the complex quantum computational algebra(shortly the C1QC-algebra).

QUANTUM COMPUTATIONAL STRUCTURES 17

We will prove that the contracted Poincare IQC-algebra and the C1QC-algebra are isomorphic.

Let (a, b) ∈ C1 and let ρ(a, b) be the density operator associated to thetriple (0, 1 − 2b, 1 − 2a). Thus,

ρ(a, b) := (0, 1 − 2b, 1 − 2a).

Hence:

ρ(a, b) =

1 − a −i

(1

2− b

)

i

(1

2− b

)

a

Lemma 8.2.

(i) ρ((a1, a2)IAND

C1(b1, b2))

= IAND(ρ(a1, a2), ρ(b1, b2));

(ii) ρ(NOTC1(a1, a2)) = NOT(ρ(a1, a2));

(iii) ρ(√NOT

C1(a1, a2)) =√NOT (ρ(a1, a2));

(iv) ρ((a1, a2) ⊕C1 (b1, b2)) = ρ(a1, a2) ⊕ ρ(b1, b2).

Proof. Easy computation. ¤

Theorem 8.1. The C1QC-algebra(

C1 , IANDC1 ,⊕C1 , NOTC1 ,√NOT

C1, 0 , 1 , 1/2

)

is isomorphic to the contracted Poincare IQC-algebra(

[D(C2)]¹u , IAND ,⊕ , NOT ,√NOT , [P0]¹u , [P1]¹u , [ρ1/2]¹u

)

.

Proof. Let h be the map of C1 into [D(C2)]¹u such that ∀(a, b) ∈ C1:

h((a, b)) := [ρ(a, b)]¹u.

That h is a homomorphism follows from Lemma 8.2. We now prove thath is injective. Suppose (a, b) 6= (c, d). Suppose, by contradiction, thath((a, b)) = h((c, d)). Then, [ρ(a, b)]¹u = [ρ(c, d)]¹u. Thus,

p(ρ(a, b)) = p(ρ(c, d)) and p(√NOT ρ(a, b)) = p(

√NOT ρ(c, d)).

By Lemma 7.1, we obtain

p(ρ(a, b)) = a = c = p(ρ(c, d))

andp(√NOT ρ(a, b)) = b = d = p(

√NOT ρ(c, d)).

Hence: (a, b) = (c, d), contradiction.We now prove that h is surjective. Let ρ be a density operator of D(C2)and let (a, b, c) be the point of the Poincare sphere associated to ρ. Thus,

(a, b, c) = ρ. Take

(1 − c

2,1 − b

2

)

∈ C1. By Lemma 7.1, [ρ

(1 − c

2,1 − b

2

)

]¹u =

[ρ]¹u. Consequently, [ρ]¹u = h

((1 − c

2,1 − b

2

))

. ¤

18 G. CATTANEO, M. L. DALLA CHIARA, R. GIUNTINI, AND R. LEPORINI

As a consequence of Theorem 7.1 and of Theorem 8.1, we obtain that theIQC- algebra and the C1QC-algebra are isomorphic.

References

[CDCGL01] G. Cattaneo, M. L. Dalla Chiara, R. Giuntini and R. Leporini, “An unsharplogic from quantum computation”, e-print: quant-ph/0201013.

[DCG02] M. L. Dalla Chiara and R. Giuntini, “Quantum logics”, in G. Gabbay and F.Guenthner (eds.), Handbook of Philosophical Logic, vol. VI, Kluwer, Dordrecht, 2002,pp. 129–228.

[DGLL02] M. L. Dalla Chiara, R. Giuntini, A. Leporati and R. Leporini, “Qubit semanticsand quantum trees”, quant-ph/0211190.

[DEL00] D. Deutsch, A. Ekert, and R. Lupacchini, “Machines, logic and quantumphysics”, Bulletin of Symbolic Logic, 3, 2000, pp. 265–283.

[Gu03] S. Gudder, “Quantum Computational Logic”, preprint[Pe67] C. A. Petri, “Grundsatzliches zur Beschreibung diskreter Prozesse”, In Proceedings

of the 3rd Colloquium uber Automatentheorie (Hannover, 1965), Birkhauser Verlag,Basel, 1967, pp. 121–140. English version: “Fundamentals of the Representation ofDiscrete Processes”, ISF Report 82.04 (1982), translated by H.J. Genrich and P.S.Thiagarajan.

[To80] T. Toffoli, “Reversible computing”, in J. W. de Bakker and J. van Leeuwen (eds.),Automata, Languages and Programming, Springer, 1980, pp. 632–644. Also availableas TechnicalMemo MIT/LCS/TM-151, MIT Laboratory for Computer Science, Feb-ruary 1980.

[Za34] Z. Zawirski, “Relation of many–valued logic to probability calculus” (in Polish,original title: Stosunek logiki wielowartosciowej do rachunku prawdopodobienstwa),Poznanskie Towarzystwo PrzyjacioÃl Nauk, 1934.

(G. Cattaneo) Dipartimento di Informatica, Sistemistica e Comunicazione (DISCo),Universita degli Studi di Milano – Bicocca, Via Bicocca degli Arcimboldi 8,I-20126 Milano, Italy

E-mail address: cattang@disco.unimib.it

(M. L. Dalla Chiara) Dipartimento di Filosofia, Universita di Firenze, viaBolognese 52, I-50139 Firenze, Italy

E-mail address: dallachiara@unifi.it

(R. Giuntini) Dipartimento di Scienze Pedagogiche e Filosofiche, Universitadi Cagliari, via Is Mirrionis 1, I-09123 Cagliari, Italy

E-mail address: giuntini@unica.it

(R. Leporini) Dipartimento di Informatica, Sistemistica e Comunicazione (DISCo),Universita degli Studi di Milano – Bicocca, Via Bicocca degli Arcimboldi 8,I-20126 Milano, Italy

E-mail address: leporini@disco.unimib.it