Entropic characterization of quantum operations
aa r X i v : . [ qu a n t - ph ] J a n Entropic characterization of quantum operations
W. Roga , M. Fannes and K. ˙Zyczkowski , Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagiello´nski, PL-30-059 Krak´ow, Poland Instituut voor Theoretische Fysica, Universiteit Leuven, B-3001 Leuven, Belgium Centrum Fizyki Teoretycznej, Polska Akademia Nauk, PL-02-668 Warszawa, Poland
Abstract:
We investigate decoherence induced by a quantum channel in terms of minimal output entropyand of map entropy. The latter is the von Neumann entropy of the Jamio lkowski state of thechannel. Both quantities admit q -Renyi versions. We prove additivity of the map entropyfor all q . For the case q = 2, we show that the depolarizing channel has the smallest mapentropy among all channels with a given minimal output Renyi entropy of order two. Thisallows us to characterize pairs of channels such that the output entropy of their tensor productacting on a maximally entangled input state is larger than the sum of the minimal outputentropies of the individual channels. We conjecture that for any channel Φ acting on a finitedimensional system there exists a class of channels Φ sufficiently close to a unitary map suchthat additivity of minimal output entropy for Ψ ⊗ Ψ holds.PACS: 02.10.Ud (Mathematical methods in physics; Linear algebra), 03.67.-a (Quantum me-chanics, field theories, special relativity; Quantum information), 03.65.Yz (Decoherence; opensystems; quantum statistical methods) In quantum information any experimentally realizable set-up that processes states of an n -level system is modelled by a quantum operation, also called quantum channel. This is acompletely positive affine transformation of the state space. The set of quantum operations1as a real dimension n ( n −
1) and its structure is far from trivial. Even for the simplestcase n = 2 the structure of the 12 dimensional convex set of qubit operations is only partiallyunderstood [2].The information encoded in a given quantum state is quantified by its von Neumann entropyor by some similar quantity such as its Renyi entropy of order q . The randomizing action of agiven quantum channel Φ can then be characterized by the minimal output entropy S min q (Φ):this is the minimal Renyi entropy of order q of an output state of the channel where theminimization is over the entire set of quantum input states. Finding out whether the minimaloutput entropy is additive with respect to the tensor product of channels was considered tobe one of the key questions of quantum information theory. Although Hastings [3] recentlyshowed that in general additivity does not hold, finding explicit counterexamples in low di-mensions is still an open problem. An even more relevant question is to specify classes ofmaps for which additivity holds [4].The decoherence induced in an n -level system by a channel may alternatively be characterizedby the map entropy S map (Φ). This quantity, defined as the entropy of the correspondingJamio lkowski state [6], varies from zero for a unitary channel to 2 log n for the completelydepolarizing channel. The entropy of a coarse graining channel with respect to a given basis,Φ CG ( ρ ) = diag ( ρ ), is equal to log n . If two quantum maps are close in the sense that thetrace distance between the corresponding states is small, then they have similar map entropies[5]. The map entropy is easier to determine than the minimal output entropy as there is nominimization to be performed.The aim of this work is to investigate links between both entropic characterizations of quan-tum maps. We prove additivity of the map entropy with respect to the tensor productand generalize this result to arbitrary Renyi entropies. To establish relations between theminimal output entropy and the map entropy we investigate the structure of the set of allquantum operations projected onto the plane (cid:0) S map (Φ) , S min (Φ) (cid:1) . For qubit channels we findthe boundaries of this projection and obtain in this way bounds between both quantities.For the Renyi entropy of order two we show that, for any dimension n , the upper boundary2f this projection corresponds to the family of depolarizing channels. As for these channelsboth entropies are explicitly known we obtain inequalities between S map2 (Φ) and S min2 (Φ).Applying these results to composite channels and using the additivity of the map entropywe prove a bound for the output entropy of a composite channel minimized over the set ofmaximally entangled states. This allows us to conjecture that for any two quantum channelsof sufficiently different degree of decoherence, e.g. S map (Φ ) ≫ S map (Φ ), the minimal outputentropy of their product is additive, S min (Φ ⊗ Φ ) = S min (Φ ) + S min (Φ ).This paper is organized as follows. In Section 2 we introduce some notation and necessaryconcepts. Some properties of the map entropy, including its additivity with respect to thetensor product, are discussed in Section 3. In Section 4 we derive bounds between the minimaloutput entropy and the map entropy and we characterize sets of maps for which the additivityof the minimal output entropy can be conjectured. The case of qubit maps is treated inSection 5 where the projection of the entire set of bistochastic quantum operations onto theplane spanned by both entropies is worked out. Some auxiliary material concerning propertiesof qubit maps is presented in the Appendix. A quantum state of an n -level system can be identified with a density matrix ρ of dimension n , i.e. a positive definite and normalized matrix: D n = (cid:8) ρ : C n → C n | ρ ≥ , Tr ρ = 1 (cid:9) . (1)A quantum operation or quantum channel describes a discrete evolution of the quantum states,it is a linear map Φ : D n → D n that is trace preserving (Tr Φ( ρ ) = Tr ρ ) and completelypositive . Complete positivity means that the map Φ ⊗ id m transforms a positive operator intoa positive operator for every dimension m of the extended space. Kraus’s theorem [7] saysthat a map is completely positive if and only if it is of the form Φ( ρ ) = P rα =1 K α ρK † α . Thetrace preserving property is equivalent with P rα =1 K † α K α = .3he Jamio lkowski isomorphism [12] represents a quantum map on D n by a state in D n : σ Φ := (cid:0) Φ ⊗ id (cid:1) ( | φ + ih φ + | ) with | φ + i = 1 √ n n X i =1 | i i ⊗ | i i = 1 √ n n X i =1 | ii i . (2)The matrix D Φ := nσ Φ , acting on the doubled space H A ⊗ H B , is called dynamical matrix or Choi matrix [8]. Positivity of the Choi matrix is equivalent with complete positivity ofthe corresponding channel Φ, while the partial trace condition Tr B D Φ = is equivalent withΦ preserving the trace. The rank of the Choi matrix D Φ is equal to the minimal number ofterms needed in a Kraus decomposition. This number is also called the Kraus rank of Φ.The Renyi entropy of order q of a state ρ is defined by S q ( ρ ) := 11 − q log Tr ρ q . (3)In the limit q →
1, the Renyi entropy tends to the von Neumann entropy S ( ρ ) = lim q → S q ( ρ ) = − Tr ρ log ρ. (4)For any map Φ acting on the set D n of quantum states one introduces the minimum outputentropy , S min q (Φ) := min ρ S q (cid:0) Φ( ρ ) (cid:1) , (5)where the minimum is taken over all states in D n . The interesting question then arises whetherthe minimal output entropy of the tensor product of two quantum operations is equal to thesum of minimal output entropies of these operations [4]. The additivity of minimal outputentropy is equivalent to the additivity of channel capacity [9]. For some special classes ofquantum operations additivity of minimal output entropy holds but it is known that it failsin general. The first proof by Hastings [3] was based on random operations acting on highdimensional state spaces and was not constructive. Later some concrete counterexamples toadditivity were presented in [11].Another characteristic of the decoherent behaviour of a quantum channel is the map entropy of the channel which is the entropy of the rescaled dynamical matrix σ Φ [6]: S map q (Φ) := 11 − q log Tr( σ Φ ) q . (6)4he map entropy is equal to 0 if and only if Φ is a unitary operation. It reaches its maximum,2 log n , at the maximally depolarizing channel Φ ∗ which transforms any initial state into themaximally mixed state ρ ∗ := n . The map entropy was considered earlier in the contextof quantum capacity: the quantum capacity of a bistochastic qubit channel of Kraus ranktwo was shown to be equal to its map entropy [13]. Several properties of this entropy wererecently discussed in [14, 16]. This quantity can be used to bound the Holevo information ofoutput states of a measurement apparatus [15] defined by the Kraus operators of a quantumchannel. The map entropy is as a special instance of the exchange entropy: it is the entropyof the environment, initially in a pure state, after an action of the quantum operation onthe maximally mixed state. For bistochastic channels, i.e. channels preserving the maximallymixed state, the map entropy is subadditive with respect to concatenation [14]: S map (Φ ◦ Φ ) ≤ S map (Φ ) + S map (Φ ) . (7)A generalization of this relation to general quantum maps was also found. Further propertiesof the map entropy and its relation to the minimal output entropy are discussed in thesubsequent sections. Further on depolarizing channels play a distinguished role, they form a one parameter familyof quantum operations Λ n on D n [10]:Λ n ( ρ ) := λρ + (1 − λ ) n where λ ∈ [ − n − , . (8)The constraint on λ ensures the complete positivity of Λ n . The minimal Renyi output entropyof such a channel can be computed explicitly [10] by considering the image of an arbitrarypure state: S min2 (Λ n ) = − log (cid:16) n − λ n (cid:17) . (9)5ccording to (2) the normalized dynamical matrix of a depolarizing channel reads σ Λ n = 1 n (cid:16) X ij λ | i ih j | ⊗ | i ih j | + 1 − λn δ ij ⊗ | i ih j | (cid:17) , (10)where δ ij denotes the Kronecker delta. Therefore the map Renyi entropy of order two is givenby S map2 (Λ n ) = − log (cid:16) n − λ n (cid:17) . (11)Note that both the ranges of values of the minimal Renyi output entropy and of the mapRenyi output entropy coincide with the full ranges that such entropies can attain. Proposition 1.
Among all channels with a given minimal Renyi output entropy of order twothe depolarizing channel has the smallest map Renyi entropy.Proof.
Putting S min2 (Λ n ) = − log(1 − ǫ ) S map2 (Λ n ) = − log (cid:16) − ǫ ( n + 1) n (cid:17) . (12)The aim is to prove that the map entropy of a quantum operation Φ on D n is not less thanthe map entropy of a depolarizing channel with the same minimal Renyi output entropy.Equivalently we want to show thatTr (cid:16) Φ( | ϕ ih ϕ | ) (cid:17) ≤ − ǫ = ⇒ Tr (cid:0) σ Φ (cid:1) ≤ − ǫ ( n + 1) n , (13)where D Φ = nσ Φ is the Choi matrix of Φ.Using a Kraus decompositionΦ( ρ ) = X α K α ρK † α , X α K † α K α = (14)we find Tr Φ( | ϕ ih ϕ | ) = X α,β h ϕ ⊗ ϕ , K † α K β ⊗ K † β K α ϕ ⊗ ϕ i (15)6nd Tr (cid:0) σ Φ (cid:1) = 1 n X α,β (cid:12)(cid:12) Tr K α K † β (cid:12)(cid:12) . (16)Now we use the following result: let µ be the Haar measure on the unitary matrices U n ofdimension n and let A be a matrix of dimension n , then [18] Z U n µ ( dU ) U ⊗ U A U † ⊗ U † = (cid:16) Tr An − − Tr AFn ( n − (cid:17) − (cid:16) Tr An ( n − − Tr AFn − (cid:17) F. (17)Here F denotes the swap operation: F ( ϕ ⊗ ψ ) = ψ ⊗ ϕ . We apply this result to find Z U n µ ( dU ) h U ϕ ⊗ U ϕ , A ( U ϕ ⊗ U ϕ ) i = 1 n ( n + 1) (Tr A + Tr AF ) . (18)This allows us to write the inequality1 n ( n + 1) X α,β (cid:0)(cid:12)(cid:12) Tr K † α K β (cid:12)(cid:12) + Tr K α K † α K β K † β (cid:1) ≤ − ǫ. (19)Now, by Schwarz’s inequality for the Hilbert-Schmidt inner product n = (cid:16) Tr X α K α K † α (cid:17) ≤ n X α,β Tr K α K † α K β K † β (20)and (19) implies 1 n X α,β (cid:12)(cid:12) Tr K † α K β (cid:12)(cid:12) ≤ − ǫ ( n + 1) n , (21)which proves (13).For any dimension n ≥ S min2 (cid:16) S map2 (Λ n ) (cid:17) = − log (cid:16) n e − S map2 (Λ n ) n + 1 (cid:17) . (22)This implies that the following statement is also true: among all maps of a same map entropyof order two the depolarizing channel has the largest minimal output entropy. In other words,representing in the (cid:0) S map2 (Φ) , S min2 (Φ) (cid:1) -plane the set of all quantum operations, there are nopoints above the line corresponding to the depolarizing channels. This result holds in anydimension. 7 .2 Additivity of the map entropy Proposition 2.
Let Φ and Φ be trace preserving, completely positive maps. For any q ≥ the Renyi map entropy satisfies the additivity relation: S map q (Φ ⊗ Φ ) = S map q (Φ ) + S map q (Φ ) . (23) Proof.
We show that D Φ ⊗ Φ is unitarily equivalent with D Φ ⊗ D Φ from which additivityof the map entropies follows. To do so, it is convenient to equip the n -dimensional matriceswith the Hilbert-Schmidt inner product h A , B i h := Tr A † B. (24)In this space the matrix units (cid:8) | i ih j | (cid:12)(cid:12) i, j = 1 , , . . . , n (cid:9) form an orthonormal basis. We usethe notation | i ih j | := | ij i h . A channel Φ is now represented by a matrix ˆΦ: h ij , ˆΦ kℓ i h = Tr (cid:16) | j ih i | Φ( | k ih ℓ | ) (cid:17) , (25)hence Φ( | k ih ℓ | ) = X i,j h ij , ˆΦ kℓ i h | i ih j | . (26)Therefore, the entries of the dynamical matrix (2) can be obtained by permuting the entriesof the matrix ˆΦ: h ab , D Φ cd i h = h ac , ˆΦ bd i h . (27)We define an unnormalized maximally entangled state | Ψ + i := P i,ℓ | iℓ i ⊗ | iℓ i and computedirectly the entries of D Φ ⊗ Φ : h abcd , D Φ ⊗ Φ ef gh i = h abcd , (cid:2) (Φ ⊗ Φ ) ⊗ id (cid:3)(cid:0) | Ψ + ih Ψ + | (cid:1) ef gh i = X i,ℓ,j,m h abcd , (cid:2) (Φ ⊗ Φ )( | iℓ ih jm | ) ⊗ | iℓ ih jm | (cid:3) ef gh i . (28)Now we use (26) and obtain: h abcd , D Φ ⊗ Φ ef gh i = X α,β,γ,δ h αβ , c Φ ij i h h γδ , c Φ ij i h h abcd , αγiℓ i h βδjm , ef gh i . (29)8fter summation over the Greek indices we get: h abcd , D Φ ⊗ Φ ef gh i = h ac , D Φ eg i h bd , D Φ f h i = h acbd , D Φ ⊗ D Φ egf h i . (30)The matrix D Φ ⊗ Φ is not equal to D Φ ⊗ D Φ . However, both are related by a unitary per-mutation matrix which exchanges the second and the third indices: U = P a,b,c,d | abcd ih acbd | .Therefore both matrices have the same spectra and hence the same entropies.We present some applications of Propositions 1 and 2 in the next section. The additivity conjecture states that sending an entangled state through a product channelΦ ⊗ Φ yields an output state with entropy not less than the smallest output entropy ofinput states with a product structure. A counterexample to this conjecture was given e.g.in [3], where estimating the entropy of an output state arising from a maximally entangledinput state plays an important role. This convinces us that it is useful to find methodsfor estimating the output entropy of maximally entangled input states. We use entropiccharacteristics to provide a typical estimation. We also use the propositions of the previoussection to characterize a class of channels for which we conjecture additivity of minimal outputentropy. Proposition 3.
For any maximally entangled state | ψ + i the following inequality for von Neu-mann entropy holds: (cid:12)(cid:12) S map (Φ ) − S map (Φ ) (cid:12)(cid:12) ≤ S (cid:0) (Φ ⊗ Φ )( | ψ + ih ψ + | ) (cid:1) ≤ S map (Φ ) + S map (Φ ) . (31) Proof.
Lindblad’s inequality [21] states that (cid:12)(cid:12) S ( ρ ) − S ( ς (Φ , ρ )) (cid:12)(cid:12) ≤ S (Φ( ρ )) ≤ S ( ρ ) + S ( ς (Φ , ρ )) , (32)9here the state ς (Φ , ρ ) is the output state of the channel Φ ⊗ id acting on a purification of ρ . The quantity S ( ς (Φ , ρ )) is called the exchange entropy and does not depend on the chosenpurification. We apply Lindblad’s inequality to S (cid:0) (Φ ⊗ Φ )( | ψ + ih ψ + | ) (cid:1) = S (cid:16) (Φ ⊗ id) (cid:0) (id ⊗ Φ )( | ψ + ih ψ + | ) (cid:1)(cid:17) . (33)Note that by the definition of the dynamical matrix one has(id ⊗ Φ )( | ψ + ih ψ + | ) = σ Φ = D Φ /n. We get (cid:12)(cid:12) S map (Φ ) − S (cid:0) ς (Φ ⊗ id , σ Φ ) (cid:1)(cid:12)(cid:12) ≤ S (cid:0) (Φ ⊗ Φ )( | ψ + ih ψ + | ) (cid:1) ≤ S map (Φ ) + S (cid:0) ς (Φ ⊗ id , σ Φ ) (cid:1) . (34)The exchange entropy S (cid:0) ς (Φ ⊗ id , σ Φ ) (cid:1) is equal to S (cid:0) ς (Φ , Tr σ Φ ) (cid:1) because a purificationof σ Φ is a special case of a purification of Tr σ Φ . Because Φ is a trace preserving mapTr σ Φ = ρ ∗ [17]. Moreover, ς (Φ , ρ ∗ ) = σ Φ . This completes the proof.Since Lindblad’s inequality (32) is based on subadditivity of entropy, Proposition 3 can begeneralized to other entropies which satisfy this property. Renyi entropy of order 2 is notsubadditive in contrast to Tsallis q -entropy: T q ( ρ ) := 1(1 − q ) (cid:0) Tr ρ q − (cid:1) , (35)which is sub-additive for q > S of a product channel acting on a maximally mixed initial state: − log (cid:16) − (cid:12)(cid:12) e − S map2 (Φ ) − e − S map2 (Φ ) (cid:12)(cid:12)(cid:17) ≤ S (cid:0) (Φ ⊗ Φ )( | ψ + ih ψ + | ) (cid:1) . (36)We can now characterize pairs of channels Φ on D n and Φ on D m for which the maximallyentangled state is certainly not the minimizer of the output Renyi entropy S . Althoughthis is not a necessary condition for channels for which the additivity holds, it suggests pairs10f maps for which additivity may hold. The maximally entangled state is certainly not theminimizer of ρ S (cid:0) (Φ ⊗ Φ )( ρ ) (cid:1) if the lower bound in (36) is larger than the minimaloutput entropy of a depolarizing channel Λ nm which satisfies S map2 (Λ nm ) = S map2 (Φ ⊗ Φ ). Asufficient, but not necessary, condition on pairs of channels (Φ , Φ ) for which a maximallyentangled state is not the minimizer of the output entropy can be written using (9) and (11)1 − nm + 1 nm (cid:12)(cid:12) e − S map2 (Φ ) − e − S map2 (Φ ) (cid:12)(cid:12) ≤ e − S map2 (Φ ⊗ Φ ) = e − (cid:0) S map2 (Φ )+ S map2 (Φ ) (cid:1) . (37)Figure 1: R is the region in the (cid:0) S map2 (Φ ) , S map2 (Φ ) (cid:1) -plane for which additivity of minimaloutput entropy may hold. Its boundary is determined by inequality (37) with m = n . Itcharacterizes a class of channels Φ ⊗ Φ for which any maximally entangled input state doesnot decrease the output entropy S below the smallest value obtained by states with a tensorproduct structure. For higher dimensions the allowed region may be enlarged. The dashed linecharacterizes pairs of complex conjugate channels used in [3] to show violation of additivityof S min .In Fig. 1 the region (cid:0) S map2 (Φ ) , S map2 (Φ )) wherein inequality (37) holds is plotted for m = n = 2 , ,
4. For any channel one can choose another one of sufficiently small S map2 to obtaina pair for which no maximally entangled state minimizes the output entropy. For such pairsof channels we may thus conjecture additivity of the minimum output entropy. The map11ntropy provides only sufficient information to recognize whether two channels belong to thisset. The set R , for which additivity of S min2 can be conjectured, consists of two regions closethe axes and is symmetric with respect to the diagonal. It consists of pairs of maps such thatthe decoherence induced by one map, as measured by the entropy, is much smaller than thedecoherence induced by the other one: S map2 (Φ ) ≤ α n S map2 (Φ ) . (38)The coefficient α n := 12 log n log (cid:16) n ( n + 2) n ( n + 1) + 1 (cid:17) (39)is the slope of the line joining the origin with the point A n from the boundary of R suchthat S map2 = 2 log n . The counterexamples to additivity used in [3] are conjugated channels,they have therefore a same map entropy and belong to the diagonal S map2 (Φ ) = S map2 (Φ ) inFig. 1. Note that for large n the coefficient α n tends to zero implying that the maps for whichadditivity may hold are atypical. Proposition 1 determines the upper boundary of the projection of the set of all quantumchannels on the (cid:0) S map2 , S min2 (cid:1) -plane. For bistochastic qubit maps the remaining boundariescorrespond to quantum maps at the edges of the tetrahedron of bistochastic qubit channels. Consider the set of all bistochastic quantum channels on D . Up to two unitary rotations,they are convex combinations of unitary channels determined by Pauli operators and aretherefore called Pauli channels :Φ ~b ( ρ ) = X i =0 b i σ i ρσ i , b i ≥ , and X i =0 b i = 1 . (40)12ere { σ i | i = 0 , , , } denotes the identity matrix and the three Pauli matrices. Thisconvex set is a tetrahedron and its four vertices correspond to the identity and to threeunitary rotations generated by Pauli matrices with respect to three perpendicular axes. Theset is shown in Fig. 2 a ).Figure 2: a ) The tetrahedron ∆ is the set of bistochastic qubit channels. An asymmetrictetrahedron K inside ∆ is magnified in panel b ). Any point in K is a convex combination (42)of the vertices W i with weights a i .Using an appropriate permutation of the vertices of the tetrahedron we may restrict ourattention to the asymmetric part K of ∆ , defined as the convex hull of four vectors: A = W = (1 , , , ,B = W = ( , , , ,C = W = ( , , , , and D = W = ( , , , ) . (41)Any point V ∈ K is a convex combination of the vertices W i V = X i a i W i , a i ≥ ∀ i , and X i a i = 1 . (42)The channel corresponding to a point inside K transforms the Bloch ball into an ellipsoid withaxes of ordered lengths | λ | ≤ | λ | ≤ | λ | ≤
1, see the Appendix. Comparing (42) with (51)13ields the coefficients a i in terms of the λ i : a = ( λ + λ ) ,a = ( λ − λ ) ,a = ( λ − λ ) , and a = (1 + λ − λ − λ ) . (43)The extreme points W i represent quantum maps of different ranks: identity ( a = 1, all λ i = 1), coarse graining ( a = 1, λ = 1, λ = λ = 0), a depolarizing channel ( a = 1, − λ = λ = λ = ) and the completely depolarizing channel ( a = 1, λ = λ = λ = 0).The convex combinations of these four maps exhaust all possible shapes of ellipsoids which areimages of the Bloch ball under bistochastic channels, see the Appendix. Due to the specificchoice of K the longest axis of the ellipsoid is parallel to the z -axis. The position of theellipsoid with respect to the axes of the Bloch ball has, however, no influence neither on theminimal output entropy nor on the map entropy.The minimal output entropy of the quantum map Φ corresponding to a given point V is S min2 ( λ , λ , λ ) = − log (1 + λ ) . (44)As V is a vector of eigenvalues of the dynamical matrix D Φ , see the Appendix, the Renyientropy of Φ equals S map2 (Φ) = − log X i =0 | b i | = − log k V k . (45)Both these entropies depend only on the values λ i . The lines representing the edges of theasymmetric tetrahedron in the (cid:0) S map2 , S min2 (cid:1) -plane are shown in Fig. 3 and we show in the nextsection that they correspond indeed to the boundaries of the allowed region in the entropyplane (cid:0) S map2 , S min2 (cid:1) . 14igure 3: Boundaries of the set ∆ of bistochastic qubit channels projected on the (cid:0) S map2 , S min2 )-plane, the tetrahedron ∆ , and its asymmetric part K . Proposition 4.
The boundaries of the set of Pauli channels projected on the (cid:0) S map2 , S min2 (cid:1) -plane correspond to the edges of the asymmetrical tetrahedron K ⊂ ∆ .Proof. Consider figure 2 b ). Take a = 0 and a = 0 so that a = 1 − a and use (54) to seethat S min2 = 0. This is the smallest possible value of minimal output entropy. The line AB inFig. 3, corresponding to the dephasing channels, describes such maps. The proof that the line AD , characterizing depolarizing channels, is a boundary of the set is given in Section 3.1 ingeneral, not necessarily for qubits or for bistochastic channels. The line BD , correspondingto a = a = 0 in the tetrahedron, represents classical bistochastic maps, characterized bya diagonal dynamical matrix. All bistochastic qubit channels which have the same minimaloutput entropy have the same the longest axis | λ | . They are situated on the horizontal line15n the (cid:0) S map2 , S min2 (cid:1) plot. The dynamical matrix of such maps reads:12 D Φ = σ Φ = 14 λ λ + λ − λ λ − λ λ − λ − λ λ + λ λ . (46)The dynamical matrix of a classical bistochastic qubit map Φ c of the same minimal outputentropy contains only diagonal elements of this matrix D Φ c = diag( D Φ ). Due to the majoriza-tion theorem the spectrum of a density matrix majorizes its diagonal. Schur concavity of theRenyi entropy S q for q ≥
1, see e.g. [20], implies that S q ( D Φ ) ≤ S q (diag( D Φ )). Thereforeclassical bistochastic channels have the greatest map entropy among all bistochastic mapswith the same minimal output entropy. This completes the proof.Qubit stochastic channels can occupy also the space on the right from the line BD . In this work we prove in general additivity of map entropies with respect to the tensor productof channels. We also analyse the relation between two entropic characteristics of a quantumchannel: the map entropy S map2 and the minimal output entropy S min2 . This approach allowsus to distinguish a class of product channels for which additivity of minimal output Renyientropy of order 2 is conjectured. The relation between the minimal output entropy andthe map entropy distinguishes the depolarizing channels as those which form a part of theboundary of the set of all quantum maps projected on the (cid:0) S map2 , S min2 )-plane. The image ofthe bistochastic qubit channels on this plane was analysed using the asymmetric tetrahedronof Pauli channels.A similar projection of the set of operations determined by the von Neumann entropy insteadof the second Renyi entropy does not yield a convex set, see Fig. 4. Moreover, the family16igure 4: Boundaries of the set of bistochastic qubit channels projected on the plane ofvon Neumann entropies ( S map , S min ).of depolarizing channels does not correspond in this case with the upper boundary of theset. Thus the reasoning in Section 4 about the set of maps for which additivity of minimaloutput Renyi entropy S min2 is conjectured cannot be directly transferred to the von Neumannentropy. On the other hand the Renyi entropy is a non-increasing function of the parameter q , in particular S ( ρ ) ≥ S ( ρ ). Moreover, in finite dimensions the Renyi entropy dependscontinuously on its parameter. It is therefore likely that a similar statement holds also forthe von Neumann entropy: For two channels Φ and Φ , acting on a space of n -dimensionaldensity matrices, such that S map (Φ ) ≫ S map (Φ ) we conjecture the additivity of the minimaloutput entropy, S min (Φ ⊗ Φ ) = S min (Φ ) + S min (Φ ).The above statement also suggests that one should consider in low dimensions two channelswith a same map entropy in order to find a counterexample to the additivity conjecture ofminimal output entropy. This is precisely the case for Hastings’s counterexample [3] in whicha random channel and its conjugate were used.17 ppendix. Qubit channels Any qubit density matrix ρ can be decomposed in the basis of the identity and three Paulimatrices. This decomposition is called the Bloch representation: ρ = ( + ~ w · ~ σ ) . (47)Since a density matrix is Hermitian the Bloch vector ~ w is real while positivity of ρ is equivalentwith k ~ w k ≤
1. The set of all such vectors ~ w is the Bloch ball . Therefore one can representany affine transformation of the qubit states, a fortiori a qubit channel, by a 4 × , ~ w ) T . One can choose a basis such thatΦ = t λ t λ t λ . (48)The channel transforms the Bloch ball into the ellipsoid (cid:16) x − t λ (cid:17) + (cid:16) y − t λ (cid:17) + (cid:16) z − t λ (cid:17) ≤ . (49)The ellipsoid has three main axes of half lengths {| λ i | | i } and its centre is translated withrespect to the centre of Bloch ball by the vector ~ t = ( t , t , t ). The positivity of the mapguarantees that the ellipsoid lies inside the ball.The corresponding normalized dynamical matrix in Bloch parametrization (48) is12 D Φ = σ Φ = 14 λ + t t + i t λ + λ − λ + t λ − λ t + i t t − i t λ − λ − λ − t λ + λ t − i t λ − t . (50)The eigenvalues v i of the dynamical matrix are connected with the parameters λ i in the Bloch18epresentation of a channel (48) by: v = (1 + λ + λ + λ ) ,v = (1 − λ − λ + λ ) ,v = (1 − λ + λ − λ ) , and v = (1 + λ − λ − λ ) . (51)The vector ~ v := ( v , v , v , v ) corresponds to the vector ~b from (40), see [17]. The minimaloutput entropy of a bistochastic channel is the minimal entropy of an output state for a pureinput state. The output state obtained by acting with the operation (48) on a state withBloch vector ~ w with k ~ w k = 1 has a Renyi entropy S map2 ( λ , λ , λ ) := − log (cid:0) λ + w ( λ − λ ) + w ( λ − λ ) (cid:1) . (52)Because | λ | ≤ | λ | ≤ | λ | the coefficients of w and w are non positive. Hence, S map2 ( λ , λ , λ )reaches its minimum when w = w = 0 and the minimum Renyi output entropy dependsonly on the longest axis S min2 ( λ , λ , λ ) = − log (cid:0) λ (cid:1) . (53)By solving the equations (51, 42) we obtain minimal output entropy of Φ as a function of theweights S min2 ( a , a , a ) = − log (cid:0) a + 2 a − a ) (cid:1) . (54) Acknowledgements:
We acknowledge financial support by the grant number N202 090239of the Polish Ministry of Science, by the Belgian Interuniversity Attraction Poles ProgrammeP6/02, and by the FWO Vlaanderen project G040710N.
References [1] A. Peres,
Quantum Theory: Concepts and Methods , Springer (1995)192] M.B. Ruskai, S. Szarek, and E. Werner, An analysis of completely positive trace preserv-ing maps on 2 × Linear Algebra Appl. , 159 (2002)[3] M. Hastings, Superadditivity of communication capacity using entangled inputs,
NaturePhysics , 255 (2009)[4] C. King and M.B. Ruskai, Minimal entropy of states emerging from noisy quantumchannels, IEEE Trans. Info. Theory , 192–209 (2001)[5] M. Fannes, A continuity property of the entropy density for spin lattice systems, Com-mun. math. Phys , 291 (1973)[6] K. ˙Zyczkowski and I. Bengtsson, On duality between quantum maps and quantum states, Open Systems & Information Dynamics , 3 (2004)[7] K. Kraus, General state changes in quantum theory, Ann. Phys. , 311–35 (1971)[8] M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra andIts Applications , 285 (1975)[9] P. Shor, Equivalence of additivity questions in quantum information theory, Commun.Math. Phys. , 453 (2004)[10] C. King, The capacity of the quantum depolarizing channel,
IEEE Trans. Inf. Theory , 221–229 (2003)[11] M. Horodecki, On Hastings’ counterexamples to the minimum output entropy additivityconjecture, Open Systems & Information Dynamics , 31 (2010)[12] A. Jamio lkowski, Linear transformations which preserve trace and positive semidefinite-ness of operators, Rep. Math. Phys. , 275 (1972)[13] F. Verstraete, H. Verschelde, On quantum channels, arXiv:quant-ph/0202124 (2003)2014] W. Roga, M. Fannes, and K. ˙Zyczkowski, Composition of quantum states and dynamicalsubadditivity, J. Phys. A: Math. Theor. , 035305 (2008)[15] W. Roga, M. Fannes, and K. ˙Zyczkowski, Universal bounds for the Holevo quantity,coherent information, and the Jensen-Shannon divergence, Phys. Rev. Lett. , 040505(2010)[16] M. Ziman, Incomplete quantum process tomography and principle of maximal entropy,
Phys. Rev. A , 032118 (2008)[17] I. Bengtsson and K. ˙Zyczkowski, Geometry of Quantum States: An Introduction to Quan-tum Entanglement , Cambridge University Press, Cambridge (2006)[18] W. Thirring,