Fermionic Quasi-free States and Maps in Information Theory
aa r X i v : . [ qu a n t - ph ] S e p Fermionic Quasi-free States and Maps in Information Theory
B. Dierckx ∗ and M. Fannes † Instituut voor Theoretische Fysica, KULeuven
M. Pogorzelska ‡ Institute of Theoretical Physics and Astrophysics, University of Gda´nsk
This paper and the results therein are geared towards building a basic toolbox forcalculations in quantum information theory of quasi-free fermionic systems. Variousentropy and relative entropy measures are discussed and the calculation of thesereduced to evaluating functions on the one-particle component of quasi-free states.The set of quasi-free affine maps on the state space is determined and fully charac-terized in terms of operations on one-particle subspaces. For a subclass of trace pre-serving completely positive maps and for their duals, Choi matrices and Jamiolkowskistates are discussed.
Keywords: quasi-free states, fermions, completely positive maps, Choi matrix, Jamiolkowskistate, entropy
1. INTRODUCTION
There are not too many classes of states or quantum operations that can be handled indetail. Well-known examples are gaussian structures in bosonic systems [1] and exchangeablestates and maps for spin systems [2, 5]. Although the systems under consideration can oftenbe quite large, even infinite, the computational difficulty of most associated quantities ismany orders of magnitude lower than what one encounters in more general systems. Inparticular gaussian states have been used in quantum optics to that effect for many years.Perhaps because of the close link between the two fields, gaussian states were also the firstof the aforementioned classes to come under consideration in quantum information theory.Recent years have seen a large amount of work done on their role in bosonic systems, seefor instance [7].This paper deals with quasi-free fermionic systems [1]. In field theory and statisticalmechanics such effective free evolutions and states have been used extensively as an approx-imation to interacting systems, a well-known example being the Hartree-Fock approximation.The main simplifying feature lies in the particular combinatorial properties of correlationfunctions and maps. In fact, states and maps are fully determined by one-particle opera-tors. As the dimension of the observables increases exponentially with the dimension of theone-particle space we obtain a very significant reduction of complexity.Fermionic systems and quasi-free states should be of particular interest to information ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] theorists. A qubit system can always be mapped on an interacting system of fermions bythe so-called
Jordan-Wigner isomorphism . A particular subset of quantum operationson qubits can then be identified with quasi-free evolutions of fermionic systems and in theseproblems quasi-free states play an important role. A recent article dealing with this dualitybetween fermions and qubits is [6].We attempted to write a rather self-contained paper, offering a toolbox for computationsand testing ideas in quantum information. Exponential elements instead of the standardcreation and annihilation operators, see [3], appear to be an efficient computational tool.Well-known objects, as quasi-free states, are reconsidered in these terms but the main goalare quasi-free quantum operations. As we restrict ourselves on purpose to finite dimensionswe only need linear algebra. This limitation can be overcome: infinite fermionic systems nolonger have a canonical representation and so one has to use appropriate representations.This typically involves introducing trace-class conditions on the one-particle operators.The paper is organized as follows: section 2 recalls some basic properties of fermionicFock space and creation and annihilation operators satisfying CAR (canonical anticommu-tation relations). In section 3 we introduce the basic exponential operators and study theirproperties. Section 4 reconsiders quasi-free states and introduces new calculation techniquesfor some entropic quantities. Section 5 deals with quasi-free quantum operations.
2. FERMIONIC FOCK SPACE
The quantum mechanical description of fermions is firmly connected to the mathematicalconcept of antisymmetric Fock space, especially in finite dimensions. Throughout the articlewe will regularly fall back on this to prove, calculate or physically motivate our expressions.In the following, H will denote a finite dimensional complex inner product space, which toalmost all intent and purpose, we can regard as a finite Hilbert space. Many of the resultscan be extended to infinite dimensional Hilbert spaces, modulo, of course, some suitableadditional conditions.The symbol ⊗ is well-known as a notation for the tensor product of two or more objects.In this, we will use the symbol ∧ to denote the antisymmetric tensor product of vectors,operators and even algebras. Although the meaning of the symbol will change dependingon the setting in which it is used, restricting ourselves to a single symbol, greatly simplifiesthe notation and looks, at least to a physicist’s eye, more elegant. As is often the case withdegenerate notations, the context should specify which version of the wedge we are talkingabout. We define the k -fold antisymmetric tensor product or wedge product of vectors ϕ , ϕ , . . . , ϕ k in H as ϕ ∧ ϕ ∧ . . . ∧ ϕ k := 1 √ k ! X σ ǫ ( σ ) ϕ σ (1) ⊗ ϕ σ (2) ⊗ . . . ⊗ ϕ σ ( k ) where σ runs over all permutations of the k indices and ǫ ( σ ) = ± depending on the parityof the permutation, + if even, − if odd.Let U σ be the unitary operator which implements the permutation σ on ⊗ k H . The fullyantisymmetric subspace of ⊗ k H consists of these vectors η ∈ ⊗ k H which satisfy U σ η = ǫ ( σ ) η .It is spanned by the k -fold antisymmetric vectors and we denote it as ∧ k H or H ( k ) . Thisspace has the same inner product as ⊗ k H , which can be written more succinctly as (cid:10) ϕ ∧ ϕ ∧ · · · ∧ ϕ k , ψ ∧ ψ ∧ · · · ∧ ψ k (cid:11) = Det (cid:16)(cid:2) h ϕ i , ψ j i (cid:3) i,j (cid:17) . This in turn makes it quite easy to transport a basis (cid:8) e , e , . . . (cid:9) of H to H ( k ) . To each suchorthonormal basis corresponds an orthonormal basis (cid:8) e Λ : Λ ⊂ { , , . . . } & k (cid:9) (1)of H ( k ) where e Λ := e i ∧ e i ∧ · · · ∧ e i k , i < i < · · · < i k & Λ = { i , i , . . . , i k } . The wedge product extends naturally to antisymmetric vector spaces. If ϕ ∈ H ( k ) and χ ∈ H ( ℓ ) then ϕ ∧ χ := 1 q(cid:0) k + ℓk (cid:1) X ǫ (Λ , M ) ϕ Λ ⊗ χ M is an element of H ( h + ℓ ) . The sum runs over the ordered partitions of { , , . . . , k + ℓ } insubsets Λ and M with k and M ) = ℓ , ϕ Λ and χ M are the ordered injections of ϕ and χ in the corresponding tensor products and ǫ (Λ , M ) is the parity of the permutation(Λ , M ). To clarify this a bit, assume that ϕ and χ are elementary antisymmetric tensors,i.e. ϕ = ϕ ∧ . . . ∧ ϕ k and χ = χ ∧ . . . ∧ χ ℓ . Then there is a canonical way to define the wedge product between these two vectors as ϕ ∧ χ = ϕ ∧ . . . ∧ ϕ k ∧ χ ∧ . . . ∧ χ ℓ = 1 p ( k + ℓ )! X σ ǫ ( σ ) U σ (cid:0) ϕ ⊗ . . . ⊗ ϕ k ⊗ χ ⊗ . . . χ ℓ (cid:1) . The above defined extension to wedge products of general antisymmetric vectors is then justthe linear extension of this and so it becomes easy to prove the following important propertyof the wedge.
Lemma 1.
The wedge operation is associative ( ϕ ∧ χ ) ∧ ψ = ϕ ∧ ( χ ∧ ψ ) . Although the k -antisymmetric vectors we introduced in the previous section are an aptdescription for the states or wave functions of k fermionic particles, we need more to properlydescribe a physical system of such particles. In nature there are processes which do notconserve the number of particles. So we need a setting in which we can jump betweendifferent k -antisymmetric vector spaces.Consider for example a fermion with d modes. The state space associated with it isthen also d -dimensional. If we add another identical particle to our system, the degreesof freedom do not go up as they do with qubits, rather, since we need to obey the Pauliexclusion principle, the dimension of the state space, which is the 2-antisymmetric space,is only (cid:0) d (cid:1) . In general, the dimension of the state space of k such particles is (cid:0) dk (cid:1) as canbe readily seen from (1), and in particular, the state space of d such particles is only one-dimensional. So we can only combine d fermions before we literally run out of space to putthem.To unify these concepts of particle creation/annihilation and the exclusion principle, theantisymmetric Fock space is introduced. The fermionic Fock space with one-particle space H is Γ( H ) := C ⊕ H ⊕ H (2) ⊕ · · · ⊕ C . It follows from the paragraph above that Γ( H ) has dimension 2 d where d = dim( H ). Thefirst term in the direct sum is the vacuum state and the last one is the completely filledFermi sea. Piecing two systems together
Given an orthogonal decomposition of H into subspaces H and H , there is a naturalisomorphism between Γ( H ) and the tensor product of the Fock spaces with H and H asone-particle spaces Γ (cid:0) H ⊕ H (cid:1) ∼ = Γ (cid:0) H (cid:1) ⊗ Γ (cid:0) H (cid:1) explicitly given by( ϕ ⊕ ψ ) ∧ · · · ∧ ( ϕ k ⊕ ψ k ) ∼ = P ⊕ (cid:0) ϕ i ∧ · · · ∧ ϕ i r (cid:1) ⊗ (cid:0) ψ j ∧ · · · ∧ ψ j s (cid:1) . The summation sign P ⊕ points at mixed sums and direct sums and the summation runs overall ordered partitions of { , , . . . , k } in two subsets { i , i , . . . , i r } and { j , j , . . . , j s } . Dueto the antisymmetry one has to pay attention to the order of the factors in this isomorphism. Elementary antisymmetric vectors
We would also like to point out a peculiar property of elementary vectors which will comein handy later on.
Lemma 2.
A nonzero vector ϕ ∈ H ( k ) is an elementary vector, i.e. can be written as ϕ = ψ ∧ . . . ∧ ψ k if and only if the space { χ ∈ H | χ ∧ ϕ = 0 } is k -dimensional.Proof. Consider an elementary vector ϕ in H ( k ) which can be written in the simple form ϕ = ψ ∧ . . . ∧ ψ k where { ψ i } i ∪ { χ ℓ } ℓ forms a basis for the generating space H and { ψ i } i ⊥{ χ ℓ } ℓ . From theabove, it is clear that a set of vectors is linearly dependent if and only if the elementarytensor constructed from them is zero. The χ ℓ are linearly independent of the ψ i and socannot contribute to the set in the lemma. Only vectors which are built up exclusively outof the ψ i contribute and the space generated by the ψ i is k -dimensional.Now assume that the set mentioned in the lemma is indeed k -dimensional. This impliesthat we can find ( d − k ) orthogonal vectors in the d -dimensional H which are linearly inde-pendent of this set and thus of the constituting vectors of ϕ . So only k linearly independentvectors can be involved in the creation of ϕ . But the only non-zero k -antisymmetric vectorswhich can be built out of k vectors, are proportional to each other and elementary. Fermionic Fock space can also be built in another way. Any vector ϕ of H induces alinear operator a ∗ ( ϕ ) from H ( k ) to H ( k +1) . We first define the action of a ∗ ( ϕ ) on elementaryvectors and then linearly extend it to the whole space. a ∗ ( ϕ )( ψ ∧ . . . ∧ ψ k ) := ϕ ∧ ψ ∧ . . . ∧ ψ k . The operator a ∗ ( ϕ ) is called a creation operator, its adjoint an annihilation operator and ineffect they emulate the creation or destruction of a Fermion in the state ϕ . By repeatedlyapplying creation operators to the vacuum vector, we can build up the entire Fock space.Although we will not use this language very frequently, the operators defined above satisfyexactly the CAR required for a quantum mechanical description of Fermions { a ( ϕ ) , a ( ψ ) } = 0 and { a ( ϕ ) , a ∗ ( ψ ) } = h φ , ψ i . (2)The algebra built on the creation and annihilation operators is called the CAR-algebrain reference to the important commutation relations (2). It coincides with the algebra oflinear transformations of the Fock space Γ( H ). This algebra is in fact a universal algebrabecause it is the unique algebra generated by a unit element and by { a ( ϕ ) : ϕ ∈ H } suchthat the operators a ( ϕ ) satisfy the CAR-conditions and that the map ϕ a ( ϕ ) is complexantilinear. We will denote it by A ( H ).The following proposition will be needed later on. It is well-known, so we state it withoutproof, see [1]. Proposition 1.
Suppose that ω is an even state, i.e. vanishes on monomials in creationand annihilation operators with odd number of factors, on A ( H ) and that σ is a state on A ( K ) ; then there exists a unique state ω ∧ σ on A ( H ) ∧ A ( K ) := A ( H ⊕ K ) , defined by ( ω ∧ σ ) ( xy ) := ω ( x ) σ ( y ) , x ∈ A ( H ) , y ∈ A ( K ) . Remark . The wedge product A ( H ) ∧ A ( K ) we implicitly defined in the above propositionis not the same as the tensor product of the two algebras as can be seen from the followingconstructive explanation.There is a natural embedding : A ( H ) ֒ → A ( H ⊕ H ) : ( a ( ϕ )) = a ( ϕ ⊕ , ϕ ∈ H and of course an analogous embedding of A ( H ). Clearly, ( A ( H )) and ( A ( H )) gen-erate A ( H ⊕ H ) but they do not sit in A ( H ⊕ H ) as tensor factors because { a ( ϕ ⊕ , a (0 ⊕ ϕ ) } = 0 instead of [ a ( ϕ ⊕ , a (0 ⊕ ϕ )] = 0 ,a denotes either a or a ∗ . Sometimes, A ( H ⊕ H ) is called the graded tensor product of A ( H ) and A ( H ).
3. THE GICAR ALGEBRA
For any one-particle basis { e i } , a special operator can be defined N := X i a ∗ ( e i ) a ( e i ) . (3)It is invariant under the gauge group of A ( H ), but also under any arbitrary basis transforma-tion of the one-particle space and so (3) defines a unique operator in the algebra. Considerthe action of N on an arbitrary (non-zero) k -particle vector N ϕ ∧ . . . ∧ ϕ k = k ϕ ∧ . . . ∧ ϕ k . So, N counts the number of particles in a given state and as such is called the numberoperator. Its eigenspaces are obviously the k -antisymmetric spaces and its spectrum consistsof the integers { , . . . , d } . The commutant of the number operator, is called the gaugeinvariant CAR-algebra or GICAR for short. It is the largest subalgebra of A ( H ) that isinvariant under the gauge group. It is also generated as the span of all monomials in a , a ∗ containing as many a ’s as a ∗ ’s. Exponential elements
With d = dim( H ) and M k the algebra of complex square matrices of dimension k , theGICAR-algebra can be written as A GICAR ( H ) = C ⊕ M d ⊕ M d ( d − / ⊕ · · · ⊕ C . The dimension of A GICAR ( H ), counted as a complex vector space, is then seen to be (cid:0) dd (cid:1) .Remark that the GICAR is the subalgebra of the CAR of block diagonal transformationsof Fock space, in particular it contains elements of the form E ( X ) := 1 ⊕ X ⊕ ( X ⊗ X ) (cid:12)(cid:12) H (2) ⊕ · · · ⊕ ( ⊗ d X ) (cid:12)(cid:12) H ( d ) or with an obvious notational meaning E ( X ) = 1 ⊕ X ⊕ ( X ∧ X ) ⊕ · · · ⊕ ( ∧ d X ) . Their spectrum σ (cid:0) E ( X ) (cid:1) can be computed to be σ (cid:0) E ( X ) (cid:1) = nY i ∈ Λ λ i : λ i ∈ σ ( X ) & Λ ⊂ { , . . . , Rank( X ) } o This property is easily verified by looking at the antisymmetric part of ⊗ k X . As a tensorproduct, the eigenvalues of this are exactly all possible monomials of the eigenvalues of X of length k . If we number the eigenvalues of X repeated according to their multiplicities as λ i , then a monomial of the λ i will contribute to the spectrum of ∧ k X if and only if each λ i appears exactly once or not at all.They also enjoy the following properties E ( ) = E ( X ) ∗ = E ( X ∗ ) E ( X ) E ( Y ) = E ( XY ) E ( X ) ≥ X ≥ E (cid:0) X ⊕ X (cid:1) ∼ = E (cid:0) X (cid:1) ⊗ E (cid:0) X (cid:1) Tr E ( X ) = Det ( + X ) A GICAR ( H ) = Span (cid:0) { E ( X ) } (cid:1) . (4)The first six properties can be easily checked by looking at the spectrum of the E -operatorsand the isomorphism we mentioned above. The last property follows from expanding λ E ( λX ) around λ = 0 and remarking that Span (cid:0)(cid:8) ⊗ k X (cid:12)(cid:12) H ( k ) (cid:9)(cid:1) coincides with the lineartransformations of H ( k ) . k -Particle projectors Another useful set of elements consists of certain one-dimensional projectors. Given a k -dimensional subspace K of H , all vectors ϕ ∧ · · · ∧ ϕ k with ϕ , . . . , ϕ k ∈ K are proportionaland span therefore a one-dimensional subspace of H ( k ) . The projector on that space will bedenoted by P ∗ ( K ). If we denote by [ K ] the projector on K then P ∗ ( K ) = [ K ] ⊗ · · · ⊗ [ K ] (cid:12)(cid:12)(cid:12) H ( k ) . It is convenient to associate the projector on the vacuum space with P ∗ (0) where 0 is thezero-dimensional vector space.These projectors are contained in the closure of Span (cid:0) { E ( X ) } (cid:1) and as such inherit allrelevant properties of the E ( X ) listed in (4). They arise as the limits of normalized E -operators, P ∗ ( K ) = lim X n → [ K ] Det ( − X n ) E (cid:16) X n − X n (cid:17) , ≤ X n < . Remark . The map ˜ E defined by˜ E ( X ) Det ( − X ) E (cid:16) X − X (cid:17) , ∀ ≤ X < is uniformly continuous and extends therefore continuously to [0 , ]. This extension shedssome light on how the n -particle projectors arise as limits of E -operators.Suppose that 1 is a k -degenerate eigenvalue of X . X can then be decomposed as a directsum of a projector P and a ( d − k )-dimensional object: X = P ⊕ ˜ X For any sequence 0 ≤ ( ǫ n ) n < E ( ǫ n P ⊕ ˜ X )are well-defined, bounded and as per (4) isomorphic to(1 − ǫ n ) k Det ( − ˜ X ) E (cid:16) ǫ n − ǫ n P (cid:17) ⊗ E (cid:16) ˜ X − ˜ X (cid:17) . By rearranging the factors in this expression, we get˜ E ( ǫ n P ) ⊗ ˜ E ( ˜ X ) . We can write out the first factor as(1 − ǫ n ) k n(cid:16) ⊕ j ≤ k (1 − ǫ n ) ( k − j ) ǫ jn ∧ j P (cid:17) ⊕ (cid:16) ⊕ j>k (cid:17)o , and then it is clear that this will converge to0 ⊕ . . . ⊕ (cid:0) ∧ k P (cid:1) ⊕ . . . ⊕ . We will often ignore the possibility that 1 is included in the spectrum of X when weare talking about expressions containing ˜ E ( X ). When 1 is contained in the spectrum of X , f (cid:0) ˜ E ( X ) (cid:1) should be interpreted aslim X n → X f (cid:0) ˜ E ( X n ) (cid:1) , ≤ X n < whenever f is a continuous function. Remark . We can express a general E ( X ) using the above projectors as E ( X ) = X Λ x Λ P ∗ (cid:0) H Λ (cid:1) where Λ plays the same role as in (4) and the x Λ are the products of the correspondingeigenvalues. So in essence, this gives us the eigendecomposition of E ( X ).
4. QUASI-FREE STATES
The chief objects under study in this article are the so-called quasi-free states and maps.These states are the fermionic counterparts of gaussian measures for classical systems orgaussian states for bosonic systems. They are sometimes referred to as determinantal statesor processes for reasons that will soon become obvious.A linear functional ω on the CAR algebra which assigns zero values to all monomials increation and annihilation operators except for ω (cid:0) a ∗ ( ϕ ) · · · a ∗ ( ϕ k ) a ( ψ k ) · · · a ( ψ ) (cid:1) = Det (cid:16)(cid:2)(cid:10) ψ i , Q ϕ j (cid:11)(cid:3)(cid:17) extends to a state on A ( H ) if and only if the linear one-particle space transformation Q satisfies 0 ≤ Q ≤ . Such an ω is called gauge invariant quasi-free and Q its corresponding symbol . The notation ω Q will be used to connect the state to its symbol.Using the language developed in section 3 we can calculate the density matrix ρ Q corre-sponding to a state ω Q . Lemma 3.
The density matrix ρ Q corresponding to a state ω Q with symbol Q can be writtendown explicitly as ˜ E ( Q ) = Det ( − Q ) (cid:26) ⊕ Q − Q ⊕ (cid:18) Q − Q ∧ Q − Q (cid:19) ⊕ . . . (cid:27) Proof.
Consider the symbol Q of a general quasi-free state ω . We can always find a 1Ddecomposition of H such that it is amenable with the eigendecomposition of Q , i.e. Q = X i q i | e i ih e i | , H i = C | e i i & ⊕ i H i = H . By straightforward computation we can check that ω Q = ∧ i ω Q i with Q i := q i | e i ih e i | andsince Γ( H ⊕ H ⊕ · · · ⊕ H n ) = Γ( H ) ⊗ Γ( H ) ⊗ · · · ⊗ Γ( H n ) , A ( ⊕ i H i ) ∼ = ⊗ i A ( H i ).For the one-dimensional Hilbert space H i , the Fock space Γ( H i ) is 2-dimensional and theassociated representation can be expressed on C as π Γ ( a ) = (cid:18) (cid:19) ; | Ω F i = (cid:18) (cid:19) . The relevant part of Q on this Fock space is then Q i = (cid:18) q
00 1 − q (cid:19) . Because of Proposition 1 and the uniqueness implied in there, the wedge state ∧ i ω Q i must be isomorphic to the product state ρ ˜ Q = ρ Q ⊗ . . . ⊗ ρ Q n . We can rewrite this in terms of exponential elements by ρ ˜ Q = (cid:18) q
00 1 − q (cid:19) ⊗ · · · ⊗ (cid:18) q d
00 1 − q d (cid:19) = Det ( − Q ) (cid:18) q − q
00 1 − q (cid:19) ⊗ · · · ⊗ (cid:18) q d − q d
00 1 (cid:19) , q i = 1= Det ( − Q ) E (cid:18) Q − Q (cid:19) ⊗ · · · ⊗ E (cid:18) Q d − Q d (cid:19) which is isomorphic to ρ Q = Det ( − Q ) E (cid:18) Q − Q (cid:19) . Finally, by continuity of the map ˜ E this results also holds for q i = 1 and so in particular forprojectors.0By remark 3, there is an alternative way to write this density matrix. Let 0 ≤ Q ≤ be alinear transformation of the one-particle space H and let (cid:8) e , e , . . . , e d (cid:9) be an orthonormalset of eigenvectors of Q , i.e. Q e j = q j e j , ≤ q j ≤ , j = 1 , , . . . , d. For a subset Λ of { , , . . . , d } define q Λ := Y r ∈ Λ q r Y s ∈{ ,...,d }\ Λ (1 − q s ) and H Λ := Span (cid:0)(cid:8) e r : r ∈ Λ (cid:9)(cid:1) then ρ Q = X Λ q Λ P ∗ (cid:0) H Λ (cid:1) . As the P ∗ ( H Λ ) project on mutually orthogonal subspaces, this decomposition shows that the q Λ are the eigenvalues of ρ Q . This allows to explicitly compute quantities as the Renyi andvon Neumann entropies of ρ Q .As every P ∗ ( K ) defines a pure state on A ( H ) that is gauge invariant and quasi-free, we seethat every gauge invariant quasi-free state is a convex mixture of pure gauge invariant quasi-free states. The projectors P ∗ are generally a small subset of the one-dimensional projectorsacting on H ( k ) . This can be seen by a simple parameter count. In order to parameterize ageneric m -dimensional complex subspace of C n , we need 2 m ( n − m ) real parameters. Asdim (cid:0) H ( k ) (cid:1) = (cid:0) dk (cid:1) with d = dim( H ) we need 2 (cid:0)(cid:0) dk (cid:1) − (cid:1) real parameters to specify a genericone-dimensional subspace of H ( k ) that is to say a k -particle pure state on A GICAR ( H ) whilewe need only 2 k ( d − k ) ≤ (cid:0)(cid:0) dk (cid:1) − (cid:1) real parameters to specify a k -dimensional subspaceof H which corresponds to a pure quasi-free state. Therefore, the convex hull of the gaugeinvariant quasi-free states is strictly smaller than the state space of A GICAR ( H ). The linearspan of the pure quasi-free states coincides however with all linear functionals on A GICAR ( H ).To illustrate that the quasi-free states do not form a convex set on their own, we providethe following proposition which will also be of use later on. Proposition 2.
Let ω Q and ω Q be quasi-free and let < λ < , then λ ω Q + (1 − λ ) ω Q is quasi-free iff Q − Q is of rank 0 or 1. Moreover, if the rank condition holds, λ ω Q + (1 − λ ) ω Q = ω λ Q +(1 − λ ) Q . For the proof of this we refer to [8].
The knowledge we have about the eigenvalue decomposition of a quasi-free state ω Q allows us to translate the general expressions for entropy related quantities of that stateto expressions on the one-particle density matrix. Especially calculations involving p -Renyientropies become very simple in this way. Proposition 3.
The p -Renyi entropy of a quasi-free state ω Q is H p ( ω Q ) = 11 − p Tr log (cid:0) ( − Q ) p + Q p (cid:1) . (5)1 Proof. H p ( ρ Q ) : = 11 − p log Tr ρ pQ = 11 − p log (cid:26) Det (cid:18) + (cid:18) Q − Q (cid:19) p (cid:19) Det ( − Q ) p (cid:27) = 11 − p log Det (cid:0) ( − Q ) p + Q p (cid:1) = 11 − p Tr log (cid:0) ( − Q ) p + Q p (cid:1) . Proposition 4.
The von Neumann entropy of a quasi-free state ω Q is S ( ω Q ) = − Tr (cid:16) Q log Q + ( − Q ) log( − Q ) (cid:17) . Proof.
The von Neumann entropy is the limit of the p -Renyi entropy when p ↓
1. As theexpression in (5) becomes indeterminate (0/0) we use de l’Hopital’s rule combined withJacobi’s formula for differentiating a determinant ddx
Det ( A ) = Tr (cid:16) A − dAdx (cid:17) Det ( A ) . This yields S ( ω Q ) = − lim p ↓ ddp log Det (( − Q ) p + Q p )= − lim p ↓ ddp Det (cid:0) ( − Q ) p + Q p (cid:1) = − Tr (cid:16) Q log Q + ( − Q ) log( − Q ) (cid:17) . Proposition 5.
Let Q and Q be two symbols such that ker Q ⊂ ker Q and ker( − Q ) ⊂ ker( − Q ) , then the relative entropy of ω Q with respect to ω Q is given by S (cid:0) ω Q ; ω Q (cid:1) = Tr n Q (cid:0) log Q − log Q (cid:1) + ( − Q ) (cid:0) log( − Q ) − log( − Q ) (cid:1)o . Proof.
As above, the computation uses an appropriate limit S ( ρ ; σ ) := Tr ρ (log ρ − log σ ) = lim p → ddp Tr (cid:16) ρ p +1 − ρ σ p (cid:17) . S (cid:0) ω Q ; ω Q (cid:1) = lim p → ddp n Det ( − Q ) p +1 Tr E (cid:16) Q p +11 ( − Q ) p +1 (cid:17) − Det ( − Q ) Det ( − Q ) p Tr E (cid:16) Q − Q Q p ( − Q ) p (cid:17)o = lim p → ddp n Det (cid:0) Q p +11 ( − Q ) p +1 (cid:1) − Det (cid:0) ( − Q ) ( − Q ) p + Q Q p (cid:1)o = Tr n Q (cid:0) log Q − log Q (cid:1) + ( − Q ) (cid:0) log( − Q ) − log( − Q ) (cid:1)o .
5. QUASI-FREE COMPLETELY POSITIVE MAPS
Let Λ be a completely positive map on M d . Dual to Λ is another completely positivemap Λ ∗ on M d Tr Λ( σ ) X = Tr σ Λ ∗ ( X ) , σ, X ∈ M d . A general quantum operation may be described either in Schr¨odinger or in Heisenberg pic-ture. In the first case we use completely positive maps Λ with the additional property thatTr Λ( σ ) = Tr σ . Such maps restrict to affine transformations of the state space of M d andare called trace-preserving completely positive (TPCP). Their duals Λ ∗ leave the identityuntouched and are therefore called unity-preserving completely positive (UPCP). We will consider here two families of UPCP maps on A ( H ) which generalize the expres-sions for quasi-free states [4]. Any unitary U on one-particle space defines an automorphismof A (( H ) through a ∗ ( ϕ ) a ∗ ( U ϕ ) . One can either check that the CAR are preserved or explicitly compute for an m -particlevector ψ E ( U ) a ∗ ( ϕ ) E ( U ) ∗ ψ = E ( U ) a ∗ ( ϕ ) (cid:0) U ∗ ⊗ · · · ⊗ U ∗ | {z } m times ψ (cid:1) = E ( U ) ϕ ∧ (cid:0) U ∗ ⊗ · · · ⊗ U ∗ ψ (cid:1) = E ( U ) (cid:0) U ∗ ⊗ · · · ⊗ U ∗ (cid:1)(cid:0) ( U ϕ ) ∧ ψ (cid:1) = ( U ϕ ) ∧ ψ = a ∗ ( U ϕ ) ψ. Such automorphisms are called quasi-free.Next consider a couple
A, B of linear transformations of H such that0 ≤ B ≤ − A ∗ A. (6)3The block matrix V := (cid:18) A √ − AA ∗ −√ − A ∗ A A ∗ (cid:19) is unitary on H ⊕ H and, using the constraint on ( A, B ), we can always find a linear map0 ≤ Q ≤ on H such that B = √ − A ∗ AQ √ − A ∗ A . Using these ingredients, weconstruct a UPCP map Λ ∗ A,B on A ( H ) by concatenating three UPCP maps: the injection a ∗ ( ϕ ) a ∗ ( ϕ ⊕ V and theprojection id ∧ ω Q from A ( H ⊕ H ) to A ( H ). An explicit computation yieldsΛ ∗ A,B (cid:0) a ∗ ( ϕ ) · · · a ∗ ( ϕ k ) a ( ψ ℓ ) · · · a ( ψ ) (cid:1) = X ǫ ω B (cid:16) a ∗ ( ϕ m ) · · · a ∗ ( ϕ m r ) a ( ψ n r ) · · · a ( ψ n ) (cid:17) × a ∗ ( A ϕ i ) · · · a ∗ ( A ϕ i k − r ) a ( A ψ j ℓ − r ) · · · a ( A ψ j ) . Here the summation is taken over all ordered partitions (cid:8) { i , . . . , i k − r } , { m , . . . , m r } (cid:9) and (cid:8) { j , . . . , j ℓ − r } , { n , . . . , n r } (cid:9) of { , . . . , k } and ǫ is the parity of the corresponding permutation. It is not hard to showthat the condition (6) is also necessary to have Λ ∗ A,B
UPCP. This follows already fromthe requirement that any quasi-free state on A ( H ) should be mapped into a state, whichincidentally will also be quasi-free. From the definition one sees that the map Λ ∗ restrictsto a UPCP map of A GICAR .To define the second family, we first need a complex conjugation: fix an orthonormalbasis { e , e , . . . , e d } in C d . The elements of the basis will be considered as real vectors andthe complex conjugation is the conjugate linear operator ϕ = X j c j e j ϕ := X j c j e j . From this definition we see that ( ϕ ) = ϕ and h ϕ , ψ i = h ψ , ϕ i . We also introduce theconjugate of a complex linear transformation A by Aϕ := A ϕ . The transformation A iscomplex linear and satisfies A + αB = A + αB, ( A ) = A, A ∗ = ( A ) ∗ , and AB = A B.
The entries of A in the distinguished basis { e , e , . . . , e d } are( A ) ij = h e i , Ae j i = h e i , Ae j i = h Ae j , e i i = h e i , Ae j i = A ij . In particular the conjugate coincides with the transpose for Hermitian elements.The second family of maps we will consider is of the formΓ ∗ A,B (cid:0) a ∗ ( ϕ ) · · · a ∗ ( ϕ k ) a ( ψ ℓ ) · · · a ( ψ ) (cid:1) = X ǫ ω B (cid:16) a ∗ ( ϕ m ) · · · a ∗ ( ϕ m r ) a ( ψ n r ) · · · a ( ψ n ) (cid:17) × a ( A ϕ i ) · · · a ( A ϕ i k − r ) a ∗ ( A ψ j ℓ − r ) · · · a ∗ ( A ψ j )4with the same summation convention as above, Γ ∗ A,B is UPCP if and only if 0 ≤ B ≤ − A T ( A T ) ∗ .Both Λ A,B and Γ
A,B map gauge-invariant quasi-free states into gauge-invariant quasi-freestates Λ
A,B ( ω Q ) = ω A ∗ QA + B and Γ A,B ( ω Q ) = ω − A T Q T ( A T ) ∗ + B + A T ( A T ) ∗ . We now compute the actions of Λ ∗ A,B and Γ ∗ A,B on elements of the type E . Lemma 4.
Suppose that ≤ B ≤ − A ∗ A and − B + XB invertible, then Λ ∗ A,B (cid:0) E ( X ) (cid:1) = Det ( − B + XB ) × E (cid:16) + A ( − B + XB ) − ( X − ) A ∗ (cid:17) = Det ( − B + BX ) × E (cid:16) + A ( X − )( − B + BX ) − A ∗ (cid:17) . Proof.
Suppose first that 0 ≤ B < − A ∗ A , the general case follows by continuity. Choosenow 0 ≤ Q < and put Q ′ := γ ( Q ) = A ∗ QA + B , then also 0 ≤ Q ′ < . Recall also thatthe density matrix ρ Q = Det ( − Q ) E ( Q/ ( − Q )) (and the analogous expression for ρ Q ′ ).We now compute using Det ( + CD ) = Det ( + DC ) , (7)Tr (cid:0) Λ A,B ( ρ Q ) E ( X ) (cid:1) = Det ( − A ∗ QA − B ) Det (cid:18) + A ∗ QA + B − A ∗ QA − B X (cid:19) = Det (cid:0) − A ∗ QA − B + ( A ∗ QA + B ) X (cid:1) = Det (cid:0) − B + BX + A ∗ QA ( X − ) (cid:1) = Det ( − B + BX ) Det (cid:0) + A ∗ QA ( X − )( − B + BX ) − (cid:1) = Det ( − B + BX ) Det (cid:0) + QA ( X − )( − B + BX ) − A ∗ (cid:1) = Det ( − B + BX ) Det ( − Q ) × Det (cid:16) − Q + Q − Q A ( X − )( − B + BX ) − A ∗ (cid:17) = Det ( + B ( X − )) Det ( − Q ) × Det (cid:16) + Q − Q (cid:0) + A ( X − )( − B + BX ) − A ∗ (cid:1)(cid:17) = Tr (cid:0) ρ Q Λ ∗ A,B ( E ( X )) (cid:1) . The second form of the expression follows from (7) and( − B + XB ) − ( X − ) = ( X − )( − B + BX ) − . Although the previous theorem completely defines the map Λ ∗ , it might be useful to statehow the action of this map looks on the density matrix of a quasi-free state.5 Proposition 6.
Let Q be the symbol of a quasi-free state ω Q and suppose that − Q + (2 Q − ) B is invertible, then Λ ∗ A,B ( ρ Q ) = Det (cid:0) − Q + (2 Q − ) B (cid:1) × E (cid:0) + A ( − Q + (2 Q − ) B ) − (2 Q − ) A ∗ (cid:1) . Proof.
It suffices to replace X with Q − Q in the proof of the previous proposition and addthe required normalization. The result also remains valid when Q becomes a projector, oreven more generally, when 1 ∈ σ ( Q ) by the continuity of the map ˜ E .The maps Γ ∗ can be handled in a similar way. Lemma 5.
Suppose that ≤ B ≤ − A T ( A T ) ∗ and that − B T − AA ∗ + X T B T + X T AA ∗ is invertible, then Γ ∗ A,B (cid:0) E ( X ) (cid:1) = Det (cid:0) − B T − AA ∗ + B T X T + AA ∗ X T (cid:1) × E (cid:16) + A ( − B T − AA ∗ + X T B T + X T AA ∗ ) − ( − X T ) A ∗ (cid:17) . Proposition 7.
Let Q be the symbol of a quasi-free state ω Q , ≤ B ≤ − A T ( A T ) ∗ andsuppose that − Q T + (2 Q T − )( AA ∗ + B T ) is invertible, then Γ ∗ A,B ( ρ Q ) = Det (cid:0) − Q T + (2 Q T − )( AA ∗ + B T ) (cid:1) × E (cid:0) + A (cid:0) − Q T + (2 Q T − )( AA ∗ + B T ) (cid:1) − ( − Q T ) A ∗ (cid:1) . The stability of the set of quasi-free states with respect to quasi-free CPTP maps canessentially be used as a characterization of such maps.
Proposition 8.
The set of quasi-free states is invariant under a linear map Γ if and onlyif the action of the map Γ can be expressed as Γ( ρ Q ) = ρ γ ( Q ) (8) where γ ( Q ) = ± A ∗ QA + B or γ ( Q ) = ± A ∗ Q T A + B . Furthermore γ ( Q ) = A ∗ QA + B isCP iff ≤ B ≤ − A ∗ A and γ ( Q ) = − A ∗ Q T A + B is CP iff A ∗ A ≤ B ≤ .Proof. Consider a map Γ : ( A GICAR ) ∗ → ( A GICAR ) ∗ with the following properties: • trace-preserving • C -linear • maps quasi-free states onto quasi-free states The ∗ in this formula denotes the dual of the algebra, i.e. the span of the state space. γ as in (8). Consider now a matrix Q whichis an element of the open interval [0 , [ in M ( C d ) and a one-dimensional projector P on thesame algebra. If ǫ is small enough (but non-zero) , Q + ǫP is still an element of the unitinterval. Since the difference of Q and Q + ǫP is of Rank 1 ω Q + λǫP = ω (1 − λ ) Q + λ ( Q + ǫP ) = (1 − λ ) ω Q + λω Q + ǫP . And because of the linearity of Γ, this gives us the following results about γ Rank { γ ( Q + ǫP ) − γ ( Q ) } = 0 or 1 , ∀ ǫ and (1 − λ ) γ ( Q ) + λγ ( Q + ǫP ) = γ ( Q + λǫP ) . We can rewrite this in a more useful form γ ( Q + ǫP ) = 1 λ { γ ( Q + λǫP ) − (1 − λ ) γ ( Q ) } = ǫ λǫ { γ ( Q + λǫP ) − γ ( Q ) } + γ ( Q )= γ ( Q ) + ǫd ( Q, P )where in the final line we have introduced the function d ( Q , Q ) with the property that if Q is a one-dimensional projector, the Rank of d is 0 or 1.As the maps Γ, ˜ E and ˜ E − are all Fr´echet differentiable on the unit interval, γ should alsobe differentiable. The function d we introduced above, is actually the derivative of γ andas such it is linear in the second argument. So clearly for any matrix 0 ≤ Q = P i q i P i ≤ such that the P i are one-dimensional projectors, γ ( Q ) = γ ( X i q i P i ) = X i q i d (0 , P i ) + γ (0)Because of the Rank conditions on d ( Q, P ) condition, we get γ ( Q ) = B ± A ∗ QA or γ ( Q ) = B ± A ∗ Q T A. The conditions 0 ≤ B ≤ − A ∗ A for the case γ ( Q ) = A ∗ QA + B and A ∗ A ≤ B ≤ forthe case γ ( Q ) = − A ∗ Q T A + B are necessary because states have to be mapped onto states,in particular the image of a symbol should remain a symbol. Conversely, if the conditionshold, then we may define the dual UPCP maps as in section 5.1 There are two ways of encoding a CP map, in fact of encoding any super-operator, goingunder the names of Jamio lkowski state and Choi matrix. Let { e i } be the standard basis of C d with associated matrix units e ij := | e i ih e j | . The Jamio lkowski state J (Γ) of Γ is definedas J (Γ) := (id ⊗ Γ) n d (cid:12)(cid:12)(cid:12)X i e i ⊗ e i EDX j e j ⊗ e j E(cid:12)(cid:12)(cid:12) = d X ij e ij ⊗ Γ( e ij ) . J (Γ) is a state whenever Γ is TPCP, i.e. a quantum operation inSchr¨odinger picture. The Choi matrix of a CP map Γ is a very similar object C (Γ) := X ij e ij ⊗ Γ( e ij ) , mostly used in Heisenberg picture. So if Γ is UPCP, the dual of a TPCP map, then theChoi matrix enjoys the property Tr C (Γ) = , where Tr denotes the partial trace over the first factor in C d ⊗ C d .We shall compute the Jamio lkowski states of quasi-free TPCP maps Λ A,B and Γ
A,B andthe Choi matrices of their duals. Both the Jamio lkowski state and the Choi matrix dependon the distinguished basis. A different choice of orthonormal basis in C d yields however aunitarily equivalent state and matrix. We can therefore equally well define J and C withrespect to a naturally chosen basis in fermionic context. Jamio lkowski state of a TPCP map.
Proposition 9.
The Jamio lkowski state of a quasi-free TPCP map Λ A,B or Γ A,B is unitarilyequivalent to a gauge-invariant quasi-free with symbol (cid:18) AA ∗ A ∗ A + 2 B (cid:19) for Λ A,B and (cid:18) − A − A ∗ A ∗ A + 2 B T (cid:19) for Γ A,B .Proof.
Consider first the case of a TPCP map Λ
A,B . In order to stay within the context ofgauge-invariant quasi-free states we embed A ( H ) in the usual way in A ( H ⊕ H ), identifying a ( ϕ ) with a (0 ⊕ ϕ ). The gauge-invariant quasi-free state on A ( H ⊕ H ) with symbol (cid:18) (cid:19) is pure and its marginal on A (cid:0) { } ⊕ H (cid:1) is totally mixed, hence it can be used to constructthe Jamio lkowski state.The algebra A ( H ⊕ H ) can be decomposed as A ( H ) ⊗ A ( H ) but the factors cannot simplybe chosen as A ( H ⊕
0) and A (0 ⊕ H ), indeed, a ( ϕ ⊕
0) anticommutes with a (0 ⊕ ϕ ). Thereexists in A ( H ) an element Θ such that Θ ∗ = Θ, Θ = and { Θ , a ( ϕ ) } = 0. In fact, Θ is upto a sign uniquely defined by these requirements and using the Fock space representationof the CAR, we easily see that Θ = ± E ( − ). Let us now embed A ( H ) in A ( H ⊕ H ) as ı ( a ( ϕ )) = Θ a (0 ⊕ ϕ ) where Θ is the element in A ( H ⊕
0) just described. It is then easilychecked that A ( H ⊕ H ) decomposes into the tensor product of A ( H ⊕
0) and the embedded8algebra A ( H ). Moreover, (cid:0) id ⊗ Λ ∗ A,B (cid:1)(cid:0) a ( ϕ ⊕ ϕ ) (cid:1) = (cid:0) id ⊗ Λ ∗ A,B (cid:1)(cid:0) a ( ϕ ⊕
0) + a (0 ⊕ ϕ ) (cid:1) = (cid:0) id ⊗ Λ ∗ A,B (cid:1)(cid:0) a ( ϕ ⊕
0) + Θ Θ a (0 ⊕ ϕ ) (cid:1) = (cid:0) id ⊗ Λ ∗ A,B (cid:1)(cid:0) a ( ϕ ⊕
0) + Θ ⊗ ı (cid:0) a ( ϕ ) (cid:1)(cid:1) = (cid:0) a ( ϕ ⊕
0) + Θ ⊗ ı (cid:0) a ( A ϕ ) (cid:1)(cid:1) = a ( ϕ ⊕ A ϕ ) . In this way, we see that id ⊗ Λ A,B is again quasi-free on A ( H ⊕ H ) with defining operators˜ A := (cid:18) A (cid:19) and ˜ B := (cid:18) B (cid:19) . (9)It is now obvious that the Jamio lkowski state will also be quasi-free with symbol ω n(cid:0) id ∧ Λ ∗ A,B (cid:1)(cid:0) a ∗ ( ϕ ⊕ ϕ ) a ( ψ ⊕ ψ ) (cid:1)o = D(cid:18) ψ ψ (cid:19) , (cid:18) AA ∗ A ∗ A + 2 B (cid:19) (cid:18) ϕ ϕ (cid:19)E . Remark that the positivity conditions for the symbol of the Jamio lkowski state preciselycoincide with the positivity requirement 0 ≤ B ≤ − A ∗ A for the TPCP map Λ A,B .The computation for a map Γ
A,B is similar. It is now convenient to compose id ∧ Γ A,B with the local automorphism id ∧ γ where γ ( a ( ϕ )) = a ∗ ( ϕ ). In this way we remain within theclass of gauge-invariant quasi-free states on the composite algebra A ( H ⊕ H ). The extendedmap is now of the form Λ ˜ A, ˜ B with˜ A := (cid:18)
00 ( A T ) ∗ (cid:19) and ˜ B := (cid:18) B (cid:19) . Choi matrix of a UPCP map.
Proposition 10.
The Choi matrix of a quasi-free UPCP map Λ ∗ A,B or Γ ∗ A,B is unitarilyequivalent to
Det ( B ) E h(cid:18) B − − B − A ∗ AB − + AB − A ∗ (cid:19)i for Λ A,B and
Det ( B ) E h(cid:18) B − − B − A T ( A T ) ∗ B − + ( A T ) ∗ B − A T (cid:19)i for Γ A,B . Proof. On C d ⊗ C d the matrix P ij e ij ⊗ e ij is equal to the dimension of the space times theprojector on a maximally entangled vector. We should therefore compute2 d (id ∧ Λ ∗ A,B ) ρ Q , where Q = (cid:18) (cid:19) . Using Proposition 6 we obtain2 d Det ( − Q + (2 Q − ) ˜ B ) = Det h (cid:18) − B − (cid:19)i = Det ( B )and + ˜ A ( − Q + (2 Q − ) ˜ B ) − (2 Q − )( ˜ A ) ∗ = (cid:18) B − − B − A ∗ AB − + AB − A ∗ (cid:19) , where ˜ A and ˜ B are as in (9).The computation for maps Γ ∗ A,B follows similar lines. Remark again that the positivityconditions for the Choi matrices coincide precisely with the conditions ensuring that Λ
A,B and Γ
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