You are an expert at summarizing long articles. Proceed to summarize the following text: 
when a pure state , represented by a density matrix @xmath2 , is transmitted along a noisy channel , it is mapped into a mixed state @xmath3 . the entropy of the initial pure state is necessarily zero , i.e. , @xmath4 since @xmath5 and so the only eigenvalues of @xmath2 are @xmath6 and @xmath7 . however , the entropy @xmath8 $ ] of the mixed state which emerges need not be zero . one seeks states @xmath2 which minimize the effect of the noise in the sense of minimizing the entropy @xmath8 $ ] of the state that emerges from the channel . there are a number of reasons for studying such states , most notably the connection between minimizing entropy and maximizing channel capacity , which will be discussed in section [ subsect : prelim.cap ] . however , in this paper we focus attention on the entropy .    the noise , which results from interactions between the states in a hilbert space @xmath9 and the environment , is represented by the action of a completely positive , trace - preserving map @xmath10 on the trace class operators in @xmath11 . we use the term _ stochastic _ to describe such maps . ( following a similar use by alberti and uhlmann @xcite , this terminology was used by petz , @xcite and reflects the fact that @xmath10 is the non - commutative analogue of the action of a column stochastic matrix on a probability vector . ) we restrict attention to two - level quantum systems in which case @xmath12 or tensor products of @xmath1 . a stochastic map @xmath10 acting on states in @xmath1 , can be naturally extended to tensor products , e.g. , @xmath13 acting on states on @xmath14 etc . , and leads to questions about the additivity of the minimal entropy and capacity of product channels . in particular , shor has conjectured that the minimal entropy is additive . we were led independently to this conjecture because it would imply additivity of channel capacity for _ unital _ stochastic maps , i.e. , maps which take the identity operator to itself so that @xmath15 . although we have a convincing argument that our results for entropy , unlike those for channel capacity , extend to non - unital maps , we focus most of our attention on unital maps . in the last section we briefly consider non - unital maps . recall that every completely positive map @xmath10 can be represented ( non - uniquely ) in the kraus form [ eq : kraus ] ( ) = _ k a_k^ a_k . every map @xmath10 representing noisy evolution in a quantum channel must preserve the trace of @xmath2 , since @xmath3 is also a state . in terms of the kraus operators , the condition that @xmath10 be stochastic , that is completely positive and trace preserving , is [ eq : cond.tracepres ] ( ) = & & _ k=1^n a_k a_k^ = i . the map @xmath10 is _ unital _ if @xmath15 , that is if @xmath10 maps the identity operator to itself . in terms of the kraus operators , the condition for @xmath10 to be completely positive and unital is [ eq : cond.unital ] ( i ) = i   & & _ k=1^n a_k^ a_k = i . a sufficient condition that a stochastic map be unital is that the kraus operators are self - adjoint , i.e. , @xmath16 . this condition is not necessary ; for example , a double - stochastic matrix which is not symmetric corresponds to a unital stochastic map which is not self - adjoint . henceforth in this paper , `` unital map '' will mean `` unital stochastic map '' unless otherwise stated . it is worth noting that the kraus operators are self - adjoint if and only if @xmath10 is self - adjoint with respect to the hilbert - schmidt inner product @xmath17 . the dual or adjoint map @xmath18 is then defined by the condition @xmath19^{\dagger } q = \tr   p^{\dagger } \phi(q)$ ] . it is easy to see that if @xmath10 is given by ( [ eq : kraus ] ) then @xmath20 . in addition , since the dual of any trace - preserving map satisfies @xmath21 , any stochastic map ( considered as a linear map on the space of hilbert - schmidt operators ) has eigenvalue @xmath7 , and hence a fixed point @xmath22 such that @xmath23 .    for unital maps , the identity is an eigenvector whose orthogonal complement is the set of operators with trace zero . hence , a unital map also defines a linear map on the traceless part of a density matrix . by contrast , a non - unital map is only affine on the set of traceless matrices . this distinction is easily seen for the @xmath1 case when the bloch sphere representation is used as described in section [ sect : notation ] . appendix c contains a list of examples of unital and non - unital maps . recall that the entropy of a density matrix can be written in terms of its eigenvalues @xmath24 , namely @xmath25 . the minimal entropy @xmath26 occurs if and only if one eigenvalue of @xmath2 is @xmath7 and all others @xmath6 ; the maximal entropy ( in @xmath27 dimensions ) of @xmath28 occurs if and only if all eigenvalues are @xmath29 so that @xmath30 . thus , if @xmath31 , one must have one eigenvalue close to @xmath7 and the others near @xmath6 . hence , states with small entropy are those for which @xmath32 . thus , in seeking pure states @xmath2 which have minimal entropy @xmath8 $ ] after emerging from a noisy channel , we are led to seek states for which @xmath33 is maximal . in section [ sect : norm.bnd ] we give a precise definition of the maximal norm and show that it is multiplicative when ( at least ) one channel is a unital map on @xmath1 . our results suggest that multiplicativity of the maximal norm may hold for general channels ; in fact , we can extend our result to some non - unital channels ( see remarks at the end of section [ sect : norm.bnd ] ) . in any case , our results provide strong support for the additivity of minimal entropy for unital channels . another property of unital channels is that the entropy of a state is non - decreasing under the action of a unital stochastic map . this follows easily from the fact that the relative entropy h(p , q ) = p [ p - q ] decreases under stochastic maps , i.e. , [ eq : relent.dec ] h[(p),(q ) ] h(p , q ) since @xmath34 if @xmath10 is unital , it follows from ( [ eq : relent.dec ] ) that @xmath8 \geq s(\rho)$ ] . for a non - unital map , the entropy of a pure state can not decrease ; however , one can have mixed states for which the entropy actually decreases . we now discuss the information capacity of a noisy quantum channel @xcite , @xcite used for what is sometimes called `` classical '' communication , i.e. , communications in which signals are sent using quantum particles but without additional or prior entanglement between sender and receiver . in the simplest case where no entanglement is used in either the transmission or the measurement , each letter @xmath35 from the source alphabet is represented by a pure state which we represent by its density matrix @xmath36 on a quantum hilbert space . during transmission the channel transforms this state into @xmath37 , where @xmath10 implements the noisy interaction between states and the environment . the map @xmath10 is a completely positive , trace - preserving map on the set of states . the resulting state @xmath38 is measured , and the outcome determines a letter from the output alphabet . in the general case this measurement is effected by a positive operator - valued measurement ( povm ) namely , there is a positive operator @xmath39 assigned to each letter @xmath40 of the output alphabet , which together satisfy the constraint @xmath41 . when the measurement is performed on a state @xmath2 , the result will be @xmath40 with probability @xmath42 . several definitions of channel capacity have been proposed , corresponding to whether or not entangled states are used for transmission , and whether or not entangled measurements are made by the receiver . bennett and shor @xcite identify four possibilities , which we denote @xmath43 and @xmath44 where the subscripts @xmath22 and @xmath45 refer to product and entangled processes respectively .    in the process described above , with no entanglement at either end , the channel is equivalent to a classical noisy channel with transition probabilities @xmath46 . therefore its maximum rate of information transmission is given by [ shannon ] c_pp ( ) = c_shan ( ) = _ , , e _ i _ j _ i p_ij ( p_ij _ k _ k p_kj ) , where we take the @xmath47 over all probability distributions @xmath48 for the input alphabet , as well as all choices of input states and measurements . we call this the _ shannon capacity _ of the channel , since it bears closest resemblance to the classical case . it is reasonable to expect that by transmitting entangled states and by using entangled measurements it may be possible to exceed the shannon capacity for a noisy channel . the holevo - schumacher - westmoreland theorem @xcite , @xcite provides a closed form expression for the capacity in the case where product states are transmitted at the input and entangled measurements of arbitrary length are allowed at the output : [ hsw ] c_pe = c _ ( ) . here @xmath49 is the _ holevo capacity _ of the channel : [ holevo ] c _ ( ) = _ , ( s ( ) - _ i _ i s(_i ) ) , where @xmath50 and @xmath51 . the well - known holevo bound states that c _ ( ) c _ ( ) . holevo @xcite provided examples of channels in which this inequality is strict , i.e. , @xmath52 . furthermore , it has been shown @xcite that a necessary and sufficient condition for strict inequality is that the output states @xmath53 do not commute . one important open question is whether or not the holevo capacity can be exceeded when entangled states are used at the input , that is whether @xmath44 exceeds @xmath54 . this would be equivalent to the superadditivity of the holevo capacity . in this paper we address this question in the case of messages which are entangled over _ two inputs _ only . this is equivalent to the question whether @xmath55 exceeds @xmath56 . holevo @xcite has shown that @xmath57 for the quantum binary channel , but to the best of the authors knowledge there is no known example of a superadditive channel for the holevo capacity . bruss et al @xcite showed that @xmath58 for the depolarising channel , which is an example of a unital channel . as we will show , our results strongly suggest that _ if _ the holevo capacity is superadditive then the channel must be non - unital . we prove several theorems about the minimal entropy and the maximal norm for states of the form @xmath59 , where @xmath10 and @xmath60 are unital stochastic maps on @xmath61 and @xmath62 is an entangled state . in addition , we explain how these results provide evidence for the conjecture that _ minimal entropy is additive _ for all stochastic maps on @xmath61 . we also show that this conjecture has important implications for the capacity of unital quantum channels . in particular , we show that the conjecture implies that if @xmath10 is unital , then the holevo capacity is additive over two inputs , that is @xmath63 .    our first theorem concerns the maximal value of @xmath64 as @xmath2 varies over states on @xmath65 . we will consider the general possibility of two stochastic maps @xmath10 and @xmath60 on @xmath61 and denote their maximal values by @xmath66 and @xmath67 respectively . in theorem [ thm : normbnd ] we prove , under mild conditions on one of these maps , that the maximal value of @xmath68 is @xmath69 , as @xmath62 varies over states on @xmath61 . that is , the norm of @xmath70 achieves its maximal value on product states @xmath71 , rather than on entangled states .    in theorem [ thm : mixing ] , we prove a similar , though slightly weaker , result for the minimal entropy of @xmath70 . namely , we restrict @xmath62 to the family of entangled states whose reduced density matrices @xmath72 and @xmath73 are such that @xmath74 and @xmath75 have minimal entropy . then we prove that the minimal entropy of @xmath76 , as @xmath62 varies over this family , is the sum of the minimal entropies of @xmath74 and @xmath75 . that is , the entropy also achieves its minimal value on product states . it seems extremely unlikely that we could further decrease the entropy of @xmath70 by using an entangled state whose reduced density matrices do not have minimal entropy themselves . therefore we believe that the conclusion of theorem [ thm : mixing ] also holds as @xmath62 varies over all entangled states on @xmath61 . in fact , similar arguments and numerical evidence support an even stronger conclusion , namely that minimal entropy is additive for _ all _ stochastic maps on @xmath61 . this is the content of the conjecture below . [ conj : minent ] if @xmath10 and @xmath60 are stochastic maps on @xmath61 , then _   s= _ s.    we will discuss the evidence for this conjecture in detail in section [ sect : entanal ] and section [ subsect : nonu.min.ent ] . we note that p. shor @xcite earlier made a similar conjecture , and together with j. smolin obtained numerical evidence which supports it . the unital case is particularly important because it yields the following results as immediate corollaries .    if @xmath10 is unital , then the holevo capacity is additive , i.e.,@xmath77 . if @xmath10 is unital , then the holevo capacity can be achieved with orthogonal states . if @xmath10 is unital , then @xmath78 and @xmath79 . if @xmath10 is unital , then the shannon capacity is also additive , i.e. , @xmath80 .    in section [ subsect : prelim.cap ] we will explain in detail how these corollaries follow if conjecture [ conj : minent ] holds for unital maps . this paper is organized as follows . in section [ sect : prelim ] we introduce the notation we will use for the stokes parametrization for representing both states and maps in a basis consisting of the identity and pauli matrices , and show how the various conditions of unital , trace - preserving , and complete positivity can be expressed in this representation . with this background , we conclude section [ sect : prelim ] by presenting the arguments leading to the corollaries above . in section [ sect : norm.bnd ] we prove the multiplicativity of the maximum value of @xmath33 . section [ sect : entanal ] contains the heart of the paper in which we give the details of the proof of our theorem about additivity of minimal entropy . in section [ sect : non.unit ] we discuss some of the features of non - unital maps using a special subclass and then present the evidence for additivity of minimal entropy in general . finally , in section [ sect : conc ] we summarize our results and discuss their implications for channel design . we also include three appendices . appendix a gives some important background on singular value decompositions and the details needed for the diagonal representation introduced in section [ sect : notation ] . appendix b gives the details needed to verify the complete positivity conditions of section [ sect : singv.cond ] . in appendix c we provide a number of examples of different types of channels and show how some familiar examples appear in the representation and notation we use . recall that the identity and pauli matrices form a basis for @xmath81 so that any @xmath82 matrix @xmath83 can be written as @xmath84 where @xmath85 denotes the vector of pauli matrices and @xmath86 then for @xmath83 = @xmath84    * @xmath83 is self - adjoint @xmath87 is real , i.e. , @xmath88 and @xmath89 ; * @xmath90 ; and * @xmath91    thus , @xmath92 also form a basis for the _ real _ vector space of self - adjoint matrices in @xmath81 and every density matrix can be written in this basis as @xmath93 $ ] with @xmath89 and @xmath94 . furthermore    * @xmath2 is a one - dimensional projection ( or pure state ) @xmath95    every linear map @xmath96 can be represented in this basis by a unique @xmath97 matrix @xmath98 , and @xmath10 is trace - preserving if and only if the first row satisfies @xmath99 , i.e. , @xmath100 where @xmath101 is a @xmath102 matrix ( and @xmath103 and @xmath104 are row and column vectors respectively ) so that @xmath105 the matrix @xmath98 is self - adjoint if and only if @xmath10 is self - adjoint as an operator on @xmath81 with respect to the hilbert - schmidt inner product . we are interested in those @xmath10 which map a subspace of self - adjoint matrices into itself , which holds if and only if @xmath106 is real . the map @xmath10 is unital if and only if @xmath107 . thus , any unital stochastic map @xmath10 acting on density matrices on @xmath1 can be written in the form [ eq : t3rep ] ( ) = , where @xmath101 is a real @xmath102 matrix . using the singular value decomposition ( see appendix a ) , we can write [ svd ] = r s where @xmath108 is a rotation and @xmath109 is self - adjoint . define the map @xmath110 by _ s ( ) = as explained in appendix a , the rotation @xmath108 defines a unitary operator @xmath111 such that for any state @xmath2 , [ eq : def.u ] ( ) = u u^ in this paper we are interested only in the critical values of certain functions of the spectrum of @xmath3 , as @xmath2 varies over the space of states  the maximum value of the norm , the minimum value of the entropy . since a unitary transformation leaves the spectrum unchanged , these are the same for @xmath10 and @xmath110 . also , since @xmath109 is self - adjoint it can be diagonalized by a change of basis . hence without loss of generality we need henceforth consider only unital stochastic maps whose matrix @xmath101 defined in ( [ eq : trep ] ) is diagonal , with eigenvalues @xmath112 . as a shorthand , we will denote this diagonal map by @xmath113 $ ] . the image of the set of pure state density matrices @xmath114 $ ] ( with @xmath115 ) under the action of @xmath113 $ ] is the ellipsoid [ eq : ellip.unit ] ( ) ^2 + ( ) ^2 + ( ) ^2 = 1 , and the image under the action of @xmath10 is obtained by a further rotation of the ellipsoid , corresponding to the operator @xmath111 in ( [ eq : def.u ] ) . similar reasoning applies when @xmath10 is non - unital . using ( [ eq : t.svd.nonpos ] ) and ( [ phi.selfadj ] ) from appendix a , the map @xmath10 can be written in the form @xmath116 where @xmath117 are unitary , @xmath118 is diagonal and @xmath119 is represented by the matrix [ eq : t.nonunit ] = the vector @xmath120 is equal to @xmath121 in the notation of ( [ eq : t.svd.nonpos ] ) . in this case , the image of the set of pure state density matrices @xmath114 $ ] ( with @xmath115 ) under the action of @xmath119 is the translated ellipsoid [ eq : ellip.nonunit ] ( ) ^2 + ( ) ^2 + ( ) ^2 = 1 , and again the image under @xmath10 is a rotation of this .    it will be useful to write out explicitly the action of the diagonal unital map @xmath113 $ ] on a density matrix in the form [ phi ] ( ) & = & + & = & . the requirement that @xmath10 be stochastic imposes a number of constraints on the matrix @xmath98 . we describe these in appendix b in which we give explicit formulas for the matrix elements of @xmath98 in terms of the stokes parameterization of the operators @xmath122 . these formulas in turn imply constraints on the eigenvalues @xmath112 described in the previous section . let @xmath123 denote the elements of @xmath106 using the convention that @xmath124 . then the point with coordinates @xmath125 must lie inside a tetrahedron with corners at @xmath126 . these conditions are equivalent to four linear inequalities which can be written compactly as @xmath127 . )    in the special case where @xmath10 is unital , ( [ eq : fuji.dcond ] ) implies that the eigenvalues ( which are necessarily real ) satisfy latexmath:[\ ] ] using convexity of the function @xmath358 , we can bound ( [ approx ] ) from below by @xmath359 , which demonstrates positivity for @xmath360 small . similar results are obtained for the other cases .      in this section we extend theorem [ thm : main ] to pure states @xmath361 formed from any entanglement of states of minimal entropy . after a precise statement of this and proof of this extension , we discuss its interpretation and the evidence for more general validity of conjecture [ conj : minent ] . let @xmath10 be a unital stochastic map . as explained in appendix a , we can write @xmath362 where @xmath111 is a unitary operator and @xmath110 is self - adjoint . let @xmath363 . define [ def : l ] l ( ) = \ { = ( i + n ) | _ s(n ) = n } . in words , @xmath364 is the collection of density matrices which lie in the direction of the largest eigenvalue of @xmath110 . if this largest eigenvalue is non - degenerate , then @xmath364 is a line segment between antipodal points on the bloch sphere . in case of degeneracy it may be a disk , or even the entire bloch sphere . if @xmath62 is a density matrix on @xmath365 , we denote by @xmath366 and @xmath367 the reduced density matrices on @xmath1 obtained by taking the indicated partial traces . [ thm : mixing ] let @xmath10 and @xmath60 be unital stochastic maps . let @xmath62 be a density matrix on @xmath365 , such that @xmath368 lies in @xmath364 , and @xmath369 lies in @xmath370 . then s ( ) ( _ 12 ) _ s [ ( ) ] + _ s [ ( ) ] . * proof * : wwe assume wlog that @xmath10 and @xmath60 are diagonal maps in the form @xmath371 $ ] and @xmath372 $ ] . furthermore we can arrange that @xmath373 is the largest eigenvalue , so that @xmath374 $ ] lies in @xmath364 . similarly we can arrange that @xmath375 is the largest eigenvalue , so that @xmath376 $ ] also lies in @xmath370 . wlog we can assume that @xmath377 is a pure state . in the bases which diagonalize @xmath10 and @xmath60 , we have | = a_00 | 00 + a_01 | 01 + a_10 | 10 + a_11 | 11 define the matrix @xmath378 to be a = ( ) then as shown in appendix a , the reduced density matrices are _ 1 = a a^ , _ 2 = ( a^ a ) ^t we obtain the `` schmidt decomposition '' by applying the singular value decomposition to @xmath378 . the result is a new basis in which @xmath379 has the diagonal form assumed in theorem [ thm : main ] . this decomposition is obtained by finding unitary operators @xmath380 which diagonalize @xmath381 and @xmath382 respectively , so that @xmath383 is also diagonal . by assumption @xmath368 lies in @xmath364 , and hence so does @xmath381 . if @xmath199 is non - degenerate , then @xmath364 is the line segment consisting of the diagonal density matrices . therefore @xmath381 is also diagonal , so @xmath384 is equal to the identity , up to a phase . if @xmath200 is also non - degenerate then @xmath385 is also proportional to the identity , and hence @xmath62 is already in diagonal form . the result follows immediately by applying theorem [ thm : main ] .    in general either or both of @xmath199 and @xmath200 may be degenerate . for example , if @xmath199 is 2-fold degenerate , then @xmath364 is a disk containing the @xmath386-axis in the bloch sphere . so @xmath381 lies in this disk , and hence it is diagonalized by a rotation of the bloch sphere which preserves this disk . by definition , such a rotation commutes with the action of @xmath10 , since the plane which contains this disk is an eigenspace of @xmath10 . hence the unitary operator @xmath384 commutes with @xmath10 . similarly if @xmath199 is 3-fold degenerate , then every unitary operator commutes with @xmath10 . to apply the argument to @xmath60 , note that by assumption @xmath369 lies in @xmath370 . if @xmath200 is non - degenerate , or is 3-fold degenerate , the same argument applies . if @xmath200 is 2-fold degenerate , then @xmath370 is a disk . the transpose operation on the bloch sphere is the reflection in the xz - plane , and this does not in general preserve a disk containing the z - axis . however we have assumed that @xmath60 is diagonal , and hence @xmath370 either lies in the xz - plane or the yz - plane . in both cases the transpose leaves @xmath370 invariant , and hence the same argument can be applied to deduce that @xmath385 also commutes with @xmath60 . it follows that @xmath387 is unitarily equivalent to @xmath388 where @xmath389 has the form assumed in theorem [ thm : main ] , and hence the result follows . * remarks : *    1 . the set @xmath364 contains the states of minimal entropy for @xmath10 . theorem [ thm : mixing ] shows that by entangling the minimal entropy states of the individual channels we can not decrease the entropy of the product channel . since it seems unlikely that entangling states of higher entropy will improve the situation , we present this as strong evidence for our conjecture . 2 .   we illustrate theorem [ thm : mixing ] in the case where @xmath113 $ ] is self - adjoint and diagonal , with @xmath390 . if the @xmath391 is real , i.e. , the matrix @xmath392 is real , then the @xmath82 unitary matrices @xmath384 and @xmath385 which diagonalize @xmath381 and @xmath382 can be chosen real and orthogonal , in which case we emphasize this by writing them as @xmath393 and @xmath394 . now suppose that our original orthogonal basis @xmath395 on @xmath1 corresponds to the eigenvectors of @xmath396 so that the corresponding pure state projections are @xmath397 $ ] , with @xmath398 corresponding to the `` north pole '' of a sphere . each unitary @xmath82 matrix @xmath111 can be associated with a real orthogonal @xmath102 matrix , and the effect of @xmath111 on the basis vectors corresponds to a rotation on the sphere or the action of a real orthogonal @xmath102 matrix on @xmath399 when the original @xmath82 matrix is real orthogonal , the corresponding rotation on the sphere reduces to a rotation in the xz - plane . + if we now write the unital stochastic map @xmath10 in the form : = ( ) = , then the change of basis is equivalent to replacing @xmath400 by @xmath401 where @xmath402 denotes the @xmath102 orthogonal matrix associated with @xmath403 . thus , for example , when @xmath404 ,   @xmath405 . if @xmath400 is diagonal with eigenvalues @xmath406 , then @xmath407 hence every state @xmath391 with real coefficients can be diagonalized , and the matrix @xmath400 is unchanged . 3 .   suppose that @xmath10 has one very large singular value and two small ones . then the unit sphere corresponding to the set of density matrices is mapped into an ellipsoid shaped like a football , and the states of minimal entropy lie at the ends of the football ( see figure [ fig : ellip1 ] ) . we can interpret theorem [ thm : main ] as saying that entangling these minimal entropy states will not decrease the entropy below the sum of the minimum entropies . + now suppose that we always keep two eigenvalues equal , but vary the parameters so that the ends of the football move in until it becomes a sphere and then a `` flying saucer '' ( see figure [ fig : ellip2 ] ) . the ends of the football have moved in to states corresponding to maximal entropy . the minimal entropy states now form a great circle . as we explained above , our special form for @xmath408 allows a general entanglement of states corresponding to these great circles of minimal entropy . yet even this more general entanglement does not decrease the entropy below that of product states . the discussion above shows that , at least in the case of unital maps , if conjecture [ conj : minent ] does not hold , then the entanglements which use states of higher entropy would achieve a lower entropy on the product space than entanglements of states of minimal entropy . in addition , such entanglements would have to lower the entropy without increasing the largest eigenvalue of @xmath409 beyond the product value given in theorem [ thm : normbnd ] . we do not find this plausible . in this section we give some heuristic evidence to support conjecture [ conj : minent ] . before doing so , we illustrate the differences between unital and non - unital maps by discussing some of the properties of a special class of maps on @xmath81 . a non - unital map is one for which @xmath410 . this means that it takes a randomly distributed alphabet to a non - random distribution . one would intuitively expect that the maximum capacity would then be achieved for alphabets which are _ not _ evenly distributed . although this is true classically , it need not be true for quantum stochastic maps as shown by the example below . it follows from equation ( [ eq : trep ] ) that a unital stochastic map defines a linear map on the subspace of matrices with trace zero . however , a non - unital map is affine when restricted to this subspace . we now consider non - unital maps which correspond , in the notation of section [ sect : notation ] to the matrix [ eq : phi.fuchs ] = with @xmath411 . this is easily seen to yield the map [ eq : spec.nonunit ] ( ) = . if @xmath412 and @xmath413 , this is equivalent to the `` splaying '' channel introduced by fuchs @xcite to demonstrate that there exist stochastic maps for which the holevo capacity ( [ holevo ] ) is achieved only by non - orthogonal states . the case @xmath414 was considered in @xcite in a different context . the case @xmath415 is essentially classical , i.e. , if @xmath2 is restricted to the subset of states of the form @xmath416 $ ] , the action of @xmath417 is equivalent to the action of the column stochastic matrix @xmath418 on the probability vector @xmath419    since equality in ( [ eq : af.nonunit ] ) holds for fuchs example , it can be regarded as an extreme case . because @xmath420 for this class of maps , they map the unit sphere of density matrices into the ellipse in the x - z plane satisfying the equation [ eq : ellipse ] + = 1 in the special cases , @xmath415 and @xmath414 , these ellipses become vertical and horizontal line segments respectively .    in the classical case ( @xmath415 ) it is not hard to show that the maximal capacity is _ never _ achieved for @xmath421 . one has only two pure states @xmath422 $ ] ( which _ are _ orthogonal ) for which @xmath423 $ ] is not identical . by contrast , for the non - unital quantum case with @xmath424 , it appears , in general , that maximal capacity is achieved at @xmath421 and with non - orthogonal states . moreover , these non - orthogonal states need not correspond to the minimal entropy states . some insight into these observations can be obtained by looking at the ellipse ( [ eq : ellipse ] ) in figure [ fig : fuchs ] corresponding to fuchs channel . ( fuchs @xcite showed explictly that non - orthogonal states are required to achieve maximum capacity , and that the maximal capacity achievable with orthogonal states occurs for @xmath421 . ) the endpoints of the ellipse ( denoted @xmath425 ) correspond to @xmath426 \big ) $ ] and have entropy @xmath427 $ ] while the minimal entropy states ( denoted @xmath428 ) correspond to the images of @xmath429 \right)$ ] and have entropy @xmath430 $ ] . note that this is the point at which the ellipse meets the circle @xmath431 , which is a level set for the entropy on the bloch sphere . the states @xmath432 $ ] are the only pair of orthogonal states with identical entropy . if one tries to move from @xmath433 toward @xmath434 to lower the entropy from one of a pair of orthogonal states , the other orthogonal state must move along the ellipse away from @xmath435 , down and closer to the origin and hence has a higher entropy than @xmath435 . explicit computation shows that the entropy price paid by moving away from @xmath435 is greater than that gained by moving @xmath433 toward @xmath434 . for any pair of states @xmath436 , the state @xmath437 which occurs in ( [ holevo ] ) lies along the line segment between @xmath438 and @xmath439 . for the points @xmath425 it is easy to see that the maximum capacity will occur for @xmath421 . if one does not require orthogonal states , then simultaneously moving both @xmath425 toward @xmath428 to decrease the entropy seems advantageous . however , this will also decrease the entropy of the convex combination @xmath440 which _ increases _ the capacity . hence , maximal capacity does not occur at the minimal entropy states but at states on the ellipse which lie between @xmath428 and @xmath425 . ( as long as the move is symmetric , maximal capacity will occur at @xmath421 . symmetry suggests this is optimal , but that does not seem to have been proved explicitly . ) one expects similar behavior for any map @xmath10 of the form ( [ eq : phi.fuchs ] ) for which @xmath441 . when @xmath442 the major axis of the ellipse lies along the x - axis , there is only one state of minimal entropy at the `` top '' of the ellipse , and one expects the channel to behave more and more like a classical channel as the ratio @xmath443 increases .    when @xmath414 , the ellipse becomes a horizontal line so that the minimal entropy states and the endpoints of the ellipse coincide . hence , the maximal capacity is again achieved for orthogonal states . in the limiting case @xmath444 , the endpoints of the ellipse lie on the unit circle of pure states and hence , have entropy zero . however , the capacity does _ not _ achieve its maximum value of @xmath445 but instead has the value @xmath446 .    as was noted earlier , @xmath10 always has a fixed point . for channels of the form ( [ eq : phi.fuchs ] ) , this fixed point is at @xmath447 . however , we have been unable to attach any significance to the fixed point . ( for fuchs channel , the fixed point is at @xmath448 $ ] ; however , this seems coincidental . )    at first glance , it might seem that the price paid for the versatility of non - unital channels is too great . if we fix @xmath449 and @xmath280 , then when @xmath411 the requirement ( [ eq : cpmcond.nonunit ] ) implies that @xmath194 must be smaller ( i.e. , `` noisier '' ) than for the corresponding unital channel . for example , using fuchs values @xmath450 , one finds that @xmath451 is optimal and corresponds to the least `` noisy '' direction . however , if @xmath452 , one could increase this to @xmath453 with corresponding decrease in noise so that the minimal entropy ( which comes from the states @xmath454 $ ] ) is @xmath455 . for the non - unital channels these same states would yield an entropy of only @xmath456 . however , the states @xmath457 $ ] will emerge with entropy @xmath458 . the decrease in eigenvalues of the part of @xmath10 corresponding to the restriction to matrices of trace zero , is overcome by the contribution to the emerging state of the non - unital part of the map . we expect that this behavior is generic for non - unital maps .      recall the discussion in section [ sect : notation ] . for unital maps , the generic situation is that there are two states of minimal entropy corresponding to the endpoints of the major axis . in the generic situation for a non - unital map , unless @xmath459 is perpendicular to the major axis , the ellipsoid in ( [ eq : ellip.nonunit ] ) will have _ only one _ state of minimal entropy . hence _ any _ entanglement will require mixing with a state of higher entropy . in the case of a map ( such as the fuchs map ( [ eq : spec.nonunit ] ) with @xmath460 ) for which the translation @xmath459 is perpendicular to the ( non - degenerate ) major axis of the ellipsoid , there will be two states of minimal entropy . however , because the translated ellipsoid is not centered at the origin , these two states will not be the images of two orthogonal states , but rather the images of two non - orthogonal pure states corresponding to vectors @xmath461 and @xmath462 . this suggests that the best candidate for a minimal entropy entangled state should have the form @xmath463 . however , after changing to orthogonal bases and using the svd decomposition ( [ eq : schmidt ] ) , this can be rewritten in the form @xmath464 where @xmath465 and , at most , only _ one _ state in each of the pairs @xmath466 and @xmath467 can equal either @xmath468 or @xmath469 . hence any entanglement must include states which are mapped into states of higher entropy under the action of @xmath470 . thus one expects states of minimal entropy under the action of @xmath471 to have the product form @xmath472 .    in the case of two - fold degeneracy ( of the form @xmath473 ) , a shift orthogonal to both major axes will yield a circle of states of minimal entropy ; however , that circle will _ not _ correspond to a `` great circle '' but rather a circle of constant latitude . such a circle never includes the image of two orthogonal states and hence the argument above still holds . in the case of three - fold degeneracy , the image will be a sphere and _ any _ shift will again yield only a single state of minimal entropy . since the maps @xmath10 and @xmath470 differ only by rotations of the bloch sphere , the same conclusions hold for @xmath10 . moreover , our analysis suggests that for _ any _ pair of maps @xmath10 and @xmath60 , the states which yield minimum entropy under @xmath474 will be simple product states , regardless of whether one or both are non - unital . our main result in this paper is that channel capacity is additive for unital channels , where _ unital _ means that the channel maps the totally random input state ( whose density matrix is proportional to the identity ) into itself ( for example , every self - adjoint channel is unital ) . specifically , we present strong evidence that both the shannon capacity ( no entanglement for input states or output measurements ) and the holevo capacity ( inputs unentangled , but output measurements may be entangled ) are additive over two uses of a two - dimensional unital channel . this is the first result that establishes additivity of channel capacity for a broad class of quantum channels . we show that the problem reduces to finding the states of _ minimal entropy _ for two uses of the channel . if these minimal entropy states are _ product states _ , then this implies additivity of capacity . we then prove that one of two possibilities occurs : either these minimal entropy states are product states , or else they are entangled states whose reduced density matrices have greater than minimal entropy . we argue that the latter case is very unlikely ( numerical experiments confirm this ) , and so conclude that the former is true . as further supporting evidence , we prove that the maximal norm of states which emerge from the channel is multiplicative over two channel uses . our results rely heavily on the stokes parametrization and the properties of the image of the bloch sphere under stochastic maps . in order to extend them to higher dimensions , we would need an effective parameterization of the subspace of matrices of trace zero in higher dimensions . one can always write a density matrix in @xmath475 as @xmath476 $ ] where @xmath477 . however , for @xmath478 , we do not know what the analog of the bloch sphere looks like . we know only that its boundary corresponds to those @xmath479 with eigenvalues @xmath480 which is not the analogue of the surface of a sphere . without knowing the geometry of this region , we can hardly hope to answer the important question of how it transforms under maps of the form @xmath156 . thus , we have been forced to use indirect methods to reach conclusions about the states of minimal entropy emerging from @xmath156 . our results have implications for the design of communication channels . ideally , one wants to eliminate all noise . however , this will not be practical and one wants to know how best to allocate resources .    in the case of unital channels , minimal entropy and maximal capacity are achieved if signal codes are chosen to correspond to the least noisy `` direction '' or polarization . ( here , we use `` direction '' in the sense of the bloch sphere or maximum @xmath24 in our notation . this is unrelated to direction of signal transmission . ) hence , if maximizing capacity is the primary goal , then it would seem sufficient to minimize noise in only one direction . even if the orthogonal directions are extremely noisy , signals sent using optimal codes will not be affected . however , in this case `` classical communication '' becomes truly classical . if codes are restricted to one direction , then one is back in the classical situation with one choice for encoding @xmath6 and @xmath7 . one has effectively lost the versatility of rotating the code basis as a tool for such purposes as signal encryption . non - unital channels have far more versatility , some aspects of which were discussed briefly in section [ sect : nonunit.sub ] . much more work needs to be done analyzing the properties of non - unital channels . thus far , most authors have looked for examples of particular maps which illustrate particular facets of stochastic maps ( such as fuchs @xcite example demonstrating the possibility of maximizing capacity with non - orthogonal states ) . our approach has been to try to find parameters which characterize subclasses of stochastic maps with certain properties . as summarized in appendix c , most of the known examples of noisy channels can easily be shown to belong to one of the groups discussed in this paper . a complete analysis of non - unital maps would seem to require an extension of conditions of the type ( [ eq : af.nonunit ] ) to general maps of the form ( [ eq : t.nonunit ] ) with @xmath481 . in this paper we make repeated , and sometimes subtle , use of the singular value decomposition ( svd ) of matrices . in view of this , and of some confusion in the literature about the connection between the svd and what is often referred to as the `` schmidt '' decomposition , we provide a brief summary and review of these closely related decompositions and their connection to the better known polar decomposition . we begin with the polar decomposition which can be extended to bounded , and even some unbounded , operators @xcite on a hilbert space . [ thm : polar ] _ ( polar decomposition ) _ any @xmath482 matrix @xmath378 can be written in the form @xmath483 where the @xmath184 matrix @xmath484 is positive semi - definite and the @xmath482 matrix @xmath111 is a partial isometry .    the term _ partial isometry _ means that @xmath485 ( or , equivalently , @xmath486 ) is a projection . in general , @xmath111 need not be unique but can be uniquely determined by the condition @xmath487 . if @xmath378 is a square @xmath184 matrix , then @xmath111 can instead be chosen ( non - uniquely ) to be unitary . since @xmath488 is self - adjoint , it can be written as @xmath489 where @xmath118 is a diagonal matrix with non - negative entries and @xmath490 is unitary . inserting this in theorem [ thm : polar ] with @xmath111 chosen to be unitary yields the svd since @xmath491 is also unitary . [ thm : svd ] _ ( singular value decomposition ) _ any @xmath184 matrix @xmath378 can be written in the form @xmath492 with @xmath493 unitary and @xmath118 a positive semi - definite diagonal matrix . the non - zero elements of @xmath118 are called the _ singular values _ of @xmath378 . they are easily seen to be the eigenvalues of @xmath488 and , hence , their squares yield the non - zero eigenvalues of @xmath494 . as an immediate corollary , one finds that @xmath494 and @xmath495 are unitarily equivalent and that @xmath490 and @xmath496 are , respectively , the unitary transformations that diagonalize @xmath494 and @xmath495 . these results can be extended to non - square matrices if the requirement that @xmath497 be unitary is relaxed to partial isometry .    using the notation of section [ sect : notation ] , we can apply the svd to the @xmath102 matrix @xmath101 which corresponds to the restriction of the stochastic map @xmath10 to the subspace of matrices with trace zero . because @xmath101 is real , the matrices @xmath497 can be chosen to be real orthogonal so that we can write [ eq : t.svd ] = o_1 _ 2^t where @xmath498 , @xmath499 are orthogonal and the superscript @xmath400 denotes transpose . now , every @xmath102 orthogonal matrix is either a rotation , or the product of a rotation with the inversion @xmath500 . hence we can rewrite ( [ eq : t.svd ] ) as [ eq : t.svd.nonpos ] = r_1 ( d ) r_2^t = ( r_1 r_2^t ) r_2 ( d ) r_2^t where @xmath501 and @xmath502 are rotations , and we conclude that @xmath101 can be written as [ eq : t.selfadj ] = r s where @xmath109 is self - adjoint and @xmath108 is a rotation . if @xmath10 is unital , define the map @xmath470 by [ eq : phi.selfadj ] _ s ( w_0 i + ) = w_0 i + s since every rotation is implemented by a unitary transformation on @xmath1 , there is a unitary operator @xmath111 such that for any state @xmath2 , ( ) = u _ s ( ) u^    for non - unital maps , a similar argument can be used to show that any stochastic map has the form ( [ eq : trep ] ) where the restriction of @xmath10 to the matrices with trace zero has the form ( [ eq : t.selfadj ] ) and @xmath503 , i.e. @xmath504 where [ phi.selfadj ] _ s ( w_0 i + ) = w_0 i + ( r^t * t * + s ) and @xmath108 is the rotation on @xmath505 corresponding to @xmath111 .    by construction , either @xmath109 or @xmath506 is positive semi - definite . however , if @xmath507 , then composing @xmath508 with the diagonal map @xmath509 merely changes the signs of two of the diagonal elements of @xmath118 . since @xmath510 is also a unitary map , we can drop the restriction that @xmath109 be semi - definite by modifying @xmath111 if necessary . this is useful because , as we will see in the next section , the general conditions on the eigenvalues of a matrix @xmath109 corresponding to a self - adjoint map @xmath10 include the possibility of negative and positive eigenvalues . we can summarize this discussion in the following    [ thm : polar.phi ] any stochastic map @xmath10 on @xmath81 can be written in the form @xmath504 where @xmath111 is unitary and @xmath470 is a stochastic map whose restriction to matrices with trace zero is self - adjoint .    it may be worth noting that we can apply the polar decomposition theorem directly to a completely positive map @xmath10 . if we use @xmath511 to denote the adjoint with respect to the hilbert - schmidt inner product , then theorem [ thm : polar ] implies that we can write [ eq : polar.phi ] = || where @xmath512 and @xmath513 is a partial isometry . if @xmath10 takes an algebra ( e.g. , @xmath164 ) to itself , then @xmath514 can be chosen to be an isometry , i.e. , @xmath515 . however , neither @xmath516 nor @xmath513 need be stochastic in general ( even though their composition is ) . on the contrary , if @xmath470 has an odd number of negative eigenvalues , then @xmath517 where @xmath518 changes the sign of an odd number of eigenvalues and , hence , is _ not _ a completely positive map . ( see appendix c for further discussion and explicit examples . ) what we have shown in the argument above is that for unital maps on @xmath81 , the isometry @xmath519 can always be implemented by a unitary transformation , possibly composed with a map @xmath518 that takes @xmath520 \raw [ w_0 i - \bw \dtsig ] $ ] , i.e. , there is a unitary matrix @xmath111 such that @xmath519 has the form @xmath521 u^{\dg}$ ] where one and only one sign holds . if @xmath10 is a non - unital stochastic map , then @xmath516 will not even be trace - preserving and the isometry @xmath519 will correspond to a change of basis that mixes the identity @xmath522 with the three @xmath523 matrices ( in contrast to the unital case in which the change of basis affects only the subspace of traceless matrices spanned by the three pauli matrices ) . hence for non - unital maps , the full polar decomposition ( [ eq : polar.phi ] ) of a stochastic map @xmath10 may be less useful than the polar decomposition on the restriction to matrices of trace zero . for unital maps on @xmath164 with @xmath524 it would be interesting to know how much @xmath519 can differ from a map of the form @xmath525 where @xmath111 is an @xmath184 unitary matrix .    in order to see the connection between the svd and the so - called `` schmidt decomposition '' , consider a wave function or vector of the form [ eq : wavefctn ] = _ jk a_jk _ j _ k with @xmath526 and @xmath527 orthonormal . it is not hard to see that there is an isomorphism between such vectors and operators of the form [ def : kpsi ] k_= _ jk a_jk| _ j _ k | and that @xmath528 is a hilbert - schmidt operator if and only if @xmath529 is square - integrable ( in the case of wave functions ) . moreover , if @xmath530 , then _ 1 t_2 ( _ 12 ) & = & k_k_^ + _ 2 t_1 ( _ 12 ) & = & ( k_^ k_)^t where @xmath531 and @xmath532 are the reduced density matrices obtained by taking the indicated partial traces @xmath533 and @xmath534 . the `` schmidt decomposition '' is an immediate consequence of the application of the svd to the operator @xmath535 given by ( [ def : kpsi ] ) which implies the following result .    _ ( schmidt ) _ any wave function of the form ( [ eq : wavefctn ] ) can be rewritten as [ eq : schmidt ] = _ k _ k _ k _ k where @xmath536 are the singular values of the matrix @xmath378 , the bases @xmath537 and @xmath538 are orthonormal and related by @xmath539 with @xmath528 given by ( [ def : kpsi ] ) . it follows immediately that the reduced density matrices @xmath531 and @xmath532 have the same non - zero eigenvalues @xmath540 and @xmath537 and @xmath538 are the eigenvectors of @xmath531 and @xmath532 respectively . there is an interesting history associated with both the svd and schmidt decompositions , as well as the attachment of schmidt s name to ( [ eq : schmidt ] ) in the physics literature . in chapter 3 of @xcite , horn and johnson give a detailed account of the history of the svd which goes back to beltrami and jordan who independently obtained the svd for real @xmath184 matrices in the 1870 s . in 1902 autonne obtained the svd for general nonsingular complex @xmath184 matrices and later made explicit the straightforward generalization to singular matrices in a long paper in 1915 which seems to have been subsequently overlooked by many researchers . independently , schmidt obtained analogous results in 1907 for operators associated with integral kernels . when quantum chemists became interested in density matrices in the 1960 s , carlson and keller @xcite rediscovered some of his results . however , john coleman @xcite soon pointed out the connection with schmidt s much earlier work . coleman s observation probably made physicists and chemists aware of that work although he did not use the term `` schmidt decomposition '' which has recently become popular in the quantum computing literature . physical chemists were initially interested in the reduced density matrices which arise from a multi - particle wave function of the form @xmath541 where @xmath542 denotes the space and spin coordinates associated with the i - th particle . it was natural to decompose these coordinates into two subsets @xmath543 and @xmath544 so that the density matrices @xmath531 and @xmath532 actually correspond to p - th and ( n - p)-th order reduced density matrices of @xmath529 . it is noteworthy that if the original wave function @xmath529 has some symmetry , then the functions @xmath537 and @xmath538 can always be chosen so that ( [ eq : schmidt ] ) has the same symmetry . in particular , if @xmath529 is antisymmetric , ( as required by the pauli exclusion principle for fermions ) then these bases can be chosen so that ( [ eq : schmidt ] ) is also antisymmetric ; i.e. , there is no need to apply an additional antisymmetrizer .    in the case of multi - particle wave functions @xmath545 is readily interpreted as the kernel of an integral operator ( which corresponds to @xmath546 defined above ) acting on an infinite dimensional hilbert space . hence , it is quite natural to attribute the results to schmidt when used in this context . however , in quantum computation , one only considers the `` spin '' part of the wave function . since this can always be represented by a finite dimensional matrix , the `` diagonalization '' of @xmath547 in ( [ eq : wavefctn ] ) can be obtained directly from the svd ; there is no need to detour into infinite dimensional hilbert spaces to use results for integral operators . moreover , since the matrix form of the svd preceded schmidt s work , it seems natural to use the term svd decomposition . to prove the eigenvalue conditions ( [ eq : fuji.cond ] ) , we rewrite the kraus operators defined by ( [ eq : kraus ] ) in the form a_k = v_k0 i + _ k where @xmath548 is a vector in @xmath155 . we will let @xmath490 denote the @xmath549 matrix with elements @xmath550 , @xmath551 its columns as vectors in @xmath552 and @xmath553 . using the relation [ eq : matprod ] ( ai + ) ( bi + ) & = & ( ab + ) i + ( a + b + i ) one finds a_k a_k^ & = & _ j=0 ^ 3    a_k^ a_k & = & _ j=0 ^ 3 |v_kj|^2 i + ( v_k0 _ k + _ k0 _ k - i _ k _ k ) so that _ k a_k a_k^ & = & ( _ j=0 ^ 3 |v_j|^2 ) i + 2 ( v_0 , + _ k _ k _ k ) + _ k a_k^ a_k & = & ( _ j=0 ^ 3 |v_j|^2 ) i + 2 ( v_0 , - _ k _ k _ k ) [ eq : ident.form ] where @xmath554 denotes the standard inner product in @xmath552 and @xmath553 . hence if @xmath10 is _ either _ unital or trace - preserving , then [ eq : icond ] _ j=0 ^ 3 |v_j|^2 = v^ v = 1 . in addition , it follows from ( [ eq : cond.tracepres ] ) and ( [ eq : cond.unital ] ) that [ eq : vcond.tracepres ] ( ) = & & v_0 , + _ k _ k _ k = 0 + ( i ) = i   & & v_0 , - _ k _ k _ k = 0 [ eq : vcond.unital ] . this implies that @xmath10 is both unital and trace - preserving if and only if , in addition to ( [ eq : icond ] ) , v_0 , & = & 0 [ iden2 ] + _ k _ k _ k & = & 0 [ iden3 ] where ( [ iden3 ] ) can be rewritten as [ iden4 ] v_j , v_k & = & 0 ,   jk 1,2,3 by defining @xmath555 to be the matrix obtained by replacing the first column of @xmath490 by @xmath556 , one can rewrite conditions ( [ iden2 ] ) and ( [ iden3 ] ) as the single requirement that that @xmath557 is diagonal or , equivalently , @xmath558 .    we will now derive the general form of the real @xmath102 matrix @xmath101 defined by ( [ eq : t3rep ] ) so that [ eq : matrix.el ] _ jk = ( _ j ( _ k ) ) in general @xmath101 is not symmetric ; in fact , it is symmetric if and only if every operator @xmath122 is self - adjoint . one finds after straightforward calculation that [ diags ] t_jj & = & v_0,v_0 + v_j , v_j - _ i j v_i , v_i + [ offdiags ] t_ij & = & 2 v_i , v_j 2 v_0 , v_k where the @xmath559 holds in ( [ offdiags ] ) if @xmath560 is an even or cyclic permutation of @xmath561 and the @xmath562 sign if it is an odd permutation . thus , for example , t_11 & = & v_0,v_0 + v_1,v_1 - v_2 , v_2 - v_3,v_3 + t_12 & = & 2 v_1,v_2 - 2 v_0,v_3    in the special case where @xmath400 is diagonal its eigenvalues can easily be obtained from ( [ diags ] ) . let @xmath112 be the eigenvalues , and define @xmath563 . then ( [ diags ] ) becomes _ 1 & = & q_0 + q_1 - q_2 - q_3 + _ 2 & = & q_0 - q_1 + q_2 - q_3 + _ 3 & = & q_0 - q_1 -q_2 + q_3 together with the condition ( [ eq : icond ] ) which can be written as 1 & = & q_0 + q_1 + q_2 + q_3 this implies that the point with coordinates @xmath112 must lie inside the tetrahedron with corners at @xmath126 . furthermore by taking @xmath564 and choosing the vectors @xmath565 to be orthogonal we see that every point in this tetrahedron defines a triplet of eigenvalues which can arise from a unital stochastic operator . these conditions are equivalent to four linear inequalities which must be satisfied by the eigenvalues , namely @xmath566 which is equivalent to the more compact ( [ eq : fuji.cond ] ) . these conditions were obtained earlier by algoet and fujiwara @xcite . the expression in ( [ diags ] ) and ( [ offdiags ] ) also hold for non - unital @xmath10 . to calculate the non - zero elements in the first column extend ( [ eq : matrix.el ] ) to @xmath567 and observe that the trace - preserving condition ( [ eq : vcond.tracepres ] ) implies @xmath568 . using this in ( [ eq : ident.form ] ) yields t_j = t_j0 = _ j ( i ) = 4 v_0 , we now give some examples of unital and non - unital maps which illustrate some of the features discussed earlier and show the correspondence between our parameterizations and some well - known examples which are usually described by their kraus operators . following the notation of section [ sect : notation ] , we will let @xmath569 $ ] denote a diagonal unital map .    before doing so we note that the classic example of a linear , positivity preserving map which is _ not _ completely positive is the transpose , which corresponds to @xmath570 $ ] . we also note that if @xmath508 denotes the maps @xmath507 , the composition @xmath571 $ ] changes the sign of the two eigenvalues whose subscript is not @xmath572 . ( for example @xmath573 = \phi[-\lambda_1,\lambda_2,-\lambda_3 ]   $ ] . ) thus the map @xmath574 $ ] which takes @xmath575 \raw \half[i - \bw \dtsig]$ ] is also _ not _ completely positive . * depolarizing channel : @xmath576 $ ] @xmath577 * bfs @xcite two - pauli channel : @xmath578 $ ] @xmath579 * phase - damping channel : @xmath580 $ ] @xmath581 $ ] * rotation : a single rotation is the simplest example of a unital , non - self - adjoint map . @xmath582 where @xmath111 is unitary and @xmath583 . + a convex combination of such rotations using kraus operators @xmath584 where @xmath585 yields a more general example of a unital map which is not self - adjoint , namely , @xmath586 matrix @xmath587 where @xmath588 is the rotation corresponding to @xmath589 . the bfs two - pauli channel was introduced by bennett , fuchs , and smolin @xcite to demonstrate that entangled states could reduce the probability of error . it is worth noting that this map corresponds to the extreme points studied in detail in section [ sect : entanal ] . hence , it is particularly noteworthy that our analysis provides particularly strong evidence that entanglements do _ not _ decrease the entropy for these maps . it is also worth noting that when @xmath590 this map becomes @xmath591 $ ] . thus , although @xmath592 $ ] and @xmath574 $ ] are _ not _ completely positive , both @xmath593 $ ] and @xmath594 $ ] _ are _ completely positive . a. s. holevo ,  on the capacity of quantum communication channel " , _ probl . peredachi inform . _ , * 15 * , no . 4 , 3 - 11 ( 1979 ) ( english translation : _ problems of information transm . _ , * 15 * , no . 4 , 247 - 253 ( 1979 ) ) .                      * * figure 1 . * the four eigenvalues of the product state of minimal entropy , with their direction of movement shown as @xmath599 increases away from @xmath6 . * * figure 2 . * the entropy difference @xmath600 $ ] for the two channels @xmath601 \otimes \phi[\mu , \mu , \mu]$ ] ( upper curve ) and @xmath601 \otimes \phi[\mu , -\mu , \mu]$ ] ( lower dashed curve ) , in the range @xmath339 . note that the curves coincide over most the interval . * * figure 3 . * the entropy difference @xmath600 $ ] for the three channels @xmath601 \otimes \phi[\mu , \mu , \mu]$ ] ( dots ) , @xmath601 \otimes \phi[\mu , 2\mu -1 , \mu]$ ] ( dashes ) , and @xmath602 \otimes \phi[\mu , 2\mu -1 , \mu]$ ] ( full ) in the range @xmath343 . * * figure 4 . * the entropy difference @xmath600 $ ] for the channel @xmath603 $ ] in the range @xmath604 . * * figure 5 . * the bloch sphere and its image ( the ellipsoid ) under a unital map with one large and two small singular values . the endpoints of the ellipsoid are shown separately these are the unique states of minimal entropy . * * figure 6 . * the bloch sphere and its image under a unital map with two large and one small singular values . the `` waistband '' of the ellipsoid is shown separately ; this entire circle consists of minimal entropy states . * * figure 7 . * the bloch sphere ( the circle of radius 1 ) and its image ( the translated ellipse ) under the fuchs map , together with the circle of radius @xmath605 . the endpoints of the ellipse are marked @xmath606 , and the points of minimal entropy are marked @xmath607 .
in this paper , we consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel , which is modeled by the action of a completely positive trace - preserving ( or stochastic ) map . 
 we provide strong support for the conjecture that this minimal entropy is additive , namely that the minimum entropy can be achieved when product states are transmitted . 
 explicitly , we prove that for a tensor product of two unital stochastic maps on qubit states , using an entanglement that involves only states which emerge with minimal entropy can not decrease the entropy below the minimum achievable using product states . 
 we give a separate argument , based on the geometry of the image of the set of density matrices under stochastic maps , which suggests that the minimal entropy conjecture holds for non - unital as well as for unital maps . 
 we also show that the maximal norm of the output states is multiplicative for most product maps on @xmath0-qubit states , including all those for which at least one map is unital .    for the class of _ unital _ channels on @xmath1 
 , we show that additivity of minimal entropy implies that the holevo capacity of the channel is _ additive _ over two inputs , achievable with orthogonal states , and equal to the shannon capacity . 
 this implies that superadditivity of the capacity is possible only for non - unital channels .    _ 
 key words : _ entangled state ; minimal entropy ; qubit ; stochastic map .