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Remarks on Conjugate Channels

Let () =

k FkFk be a quantum channel. The Stinespring/Lindblad

representation of is equivalent to the following. Let

F() =jk

|ejek|FjFk (1)

be the block matrix with blocks FjFk . Then () = Tr1 F(). One can

define the reverse or conjugate channel as

R() = Tr2 F() = jk |

ej

ek|

Tr FjFk . (2)

This is equivalent to the definition given by Matsumoto, and allows one toprove several useful results.

Kraus operators: First, suppose that Fk =

m vkmGm with V V = I. (We

do not require V to be unitary, but allow the possibility of a partial isometryto include Kraus representations with different numbers of operators.) Then() =

m GmG

m is another Kraus representation, and

Tr2 F() = V[Tr2 G()]V. (3)

This gives

RF() = V[RG()]V

=

V RG

() (4)

where V() = V V. Thus, the effect of using a different Kraus represen-

tation is simply composition with conjugation by a partial isometry.

EBT maps: Recall that an EBT map can be written using Kraus operatorsFk = |xkwk| with rank one so that

() =k

|xkxk|wk, wk, (5)

and

k xk|wkwk| = I. Then

F() =jk

|ejek| |xjxk| wj, wk (6)

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Then

R() =jk

|ejek| xj , xk wj, wk = X W (7)

where denotes the Hadamard product, X is the matrix with elementsxj, xk and W the matrix with elements wj , wk. When the wk forman orthonormal basis, is an extreme CQ channel and W is the usual ma-trix representative of in the O.N. basis wk so that

R() = X . King[] has shown that this implies that R has simultaneously diagonal Krausoperators. Thus, the conjugate of an extreme CQ channel is a diagonalchannel. When the xk are O.N. so that

xj, xk

= jk , the channel is QC and

R() =

j |ejej | wj , wj is also a QC channel.The conjugate of a general EBT channel can always be written as a

Hadamard product as in (7), but with a non-standard representative W.An equivalent statement is that R() can be written with simultaneouslydiagonal Kraus in a pair of bases of which only one need be O.N. Moreprecisely, the Kraus operators of the conjugate of an EBT map can be writtenin the form

Fj =m

cmj|fmvm| (8)

with fm O.N. and vm a set of vectors which do not depend upon j. To seethat (7) has Kraus operators of the form (8), it suffices to swtich j m andchoose cjm such that CC

= xj, xk.Conversely, suppose that the Kraus operators of a map have the form

(8). Then

F() =jk

mn

|ejek| cmjcnk|fmfn| vm, vn (9)

and

R() =jk

m

|ejek|cmjcmk vm, vm (10)

= |mm|vm, vm (11)

where |m =

j cmj|ej. This gives an EBT map.

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Thus R() is the Hadamard product of and a positive semi-definite block

matrix. I dont believe further reduction is possible since, in general, a non-extreme CQ map could require d2 Kraus operators which is precisely whatwe have here. However, one could also write this in the form

R() = ( J) (18)

where has elements fn

Rk

Rjfm and J denotes the matrix with allentries equal to 1 (and isthe identity with respect to the Hadamard product).Note that up to a (? missing ?) factor of d, J = |11| which has thesame spectrum as .

For QC maps

Unital qubit maps: Now consider a unital qubit channel. WLOG we canassume that () =

3

k=0 akkk with the convention that 0 = I. Onecan write

= 12

[I + w ] = 12

3k=0

wkk. (19)

Then one can choose Fk = ak k and

R() = A

w0 w1 w2 w3w1 w0 iw3 iw2w2 iw3 w0 iw1w3 iw2 iw1 w0

= 4A NR() (20)

where A is the matrix with elements

ajak and NR() is the conjugate of

the completely noisy map for which all ak =1

4.

Moreover, this result extends to d-dimensions for the class of channels

() =d21m=0

amTmTm (21)

and Tm is some ordering of the generalized Pauli matrices XjZk, j , k =

0 . . . d 1 with T0 = I. The Tk form an orthonormal basis for Md since

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TmTn = d mn, and one can write =1

d

d21m=0

amTm. Everything goes through

as above.

Note that the map NR() always takes a pure state to a multiple of ad-dimensional projection. Thus, although the iso-spectral property holds,the map is not trivial. The multiplicativity of the maximal p-norm of suchmaps was recently proved by Wolf and Eisert [?] (extending the Alicki-Fannesargument for the WH channel). So the conjugate channel 4A NR() is acomposition of maps whose multiplicativity we know how to prove.

The special case for which all am =1

d2

1

(1

a0) for all m

1 is a

depolarizing channel.

Convexity issues: The set of CPT maps : Md Md is convex and Itis natural to ask if the set of conjugate maps is also convex and if extremepoints maps to extreme points. The answer to both is negative.

First, recall that a set of Kraus operators can be obtained from the Choimatrix

jk |ejek| (|ejek|); this is done by stacking the elements

of the eigenvectors associated with non-zero eigenvalues. Moreover, Choishowed that () = k Gk G

k is an extreme point of the set of CPT maps

if and only if {GjGk} is a linear independent set in Md. This implies thatan extreme point can not require more than d Kraus operators. I believethat these results should generalize to maps : Mm Mn with m == n,though we need to check carefully. (The key issue for extreme points is thatwhen GkGk = I , then G

jGk is linear independent. The set of dual maps

wrt the Hilbert-Schmidt inner product is convex with a 1-1 correspondencebetween extreme points. For CPT maps, we can apply this condition whenm n. For CP maps which are unital, but not necessarily TP, it can beapplied when m n.)

For the extreme point question, consider the unital qubit map

: 1+a2

+ 1a2

xx. (22)

One finds that the conjugate map is

R : 12

I + w 1

2

I +

1 a2 w1 1 + a 3

(23)

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The map (22) is a convex combination of unitary conjugations and, hence,

not a true extreme point of the CPT maps on qubit states. But (22) is atrue extreme point (one which is also CQ) as shown in [2].

However, Ruskai, Szarek and Werner [2] found it useful to generalize theclass of extreme points to one which contains all qubit maps which can bewritten using at most two Kraus operators, and (22) belongs to this class.One can similarly consider the class of maps : Md Md which can bewritten using at most d Kraus operators. It follows easily from Matsumotosapproach that the conjugate of any such map can also be written using atmost d Kraus operators. So in this generalized sense, extreme points map toextreme points.

For CPT maps with > d Kraus operators, the conjugate R : Md M.The set of these conjugate maps is a subset of the set of the convex set ofall CPT maps : Md M. Since d < , the extension of Chois resultsapply so that the extreme points require at most d Kraus operators. Thus,the conjugate maps correspond to extreme points (in the generalizedsense ofRSW) of the convex of the CPT maps from Md to M rather than a convexsubset. The generic situation is = d2 and all others can be embedded intothis by formally adding Kraus operators equal to zero.

Another approach can be illustrated by observing that any qubit CPT

map can be written as the convex combination of two (generalized) extrememaps, each of which has (at most) two Kraus operators. Thus we can write = x1 + (1 x)2 so that

() = x()1 + (1x)2() (24)= xG1 G

1 + xG2 G

2 + (1x)G3 G3 + (1x)G4 G4

If 1() = Tr2 U1 1U1 and 2() = Tr2 U2 2U2 with k = |kk|and k C2 it does not follow that () = Tr2U U for some (possiblymixed) state M2 and U M4 unitary. Even in the exceptional cases

where this holds, the isospectral property of the conjugate channel breaksdown for mixed states.

Another approach would be to embed each of the representations on C2C2 into C2 C4 by adding Kraus operators which are zero. This seems to

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work, and would lead to a block diagonal representation of the form

() = Tr2 U

x1 (1x)2

U (25)

= Tr2

U1 00 U2

x 1 0

0 (1x) 2

U1 00 U2

(26)

with U = U1 U2. But we again have a mixed state. A better alternativewould be

=

x1

1x2

x1

1x2

=

x|11|

x(1x)|12|

x(1x)|22| (1x)|22|

() = Tr2

U1 00 U2

U1 0

0 U2

(27)

=

x U1 |11|U1

x(1x) U1 |12|U2

x(1x) U1 |22|U1 (1x) U2 |22|U2

By writing everything in terms of Kraus operators, one sees that the choice1| = 2| = (11) reduces to (3). Thus, we do not seem to gain much fromthis convex decomposition, except a partition of the Kraus operators.

To extend this to : Md Md, we would need to know if every CPT mapcan be written as a convex combination of (at most) d generalized extremepoints. Then we would have a similar partition into d subsets of Krausoperators (each of size at most d.) This seems like an interesting questionwhich as far as I know is open. Otherwise, we know only that we canwrite every CPT map as a convex combination of m extreme points, each ofwhich can be written using d Kraus operators. However, for m > d the totalnumber of Kraus operators needed for the kind of partition used above couldthen be dm > d2 which is necessarily redundant.

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A qutrit example: The following example describes a case in which the

canonical Kraus operators arising as eigenvectors of the Choi matrix do notform a minimal set, but do allow a partition as above.

Let |0, |1, |2 be an orthonormal basis for C3 and define the followingfour vectors as in [1].

|v0 = 13

+ |0 + |1 + |2

|v1 = 13

+ |0 |1 |2

|v2 =1

3 |0 + |1 |2|v3 = 1

3

|0 |1 + |2

The map with Kraus operators Fk = |vkvk|,

() =3

4

3i=0

|vivi| Tr |vivi| (28)

is EBT; in fact, it was shown in [1] that it is an extreme point of the the set

of qutrit EBT maps that is neither CQ nor an extreme point of the set of allquitrit CPT maps.

A straightforward calculation yields the following set of seven (7) Krausoperators from the Choi matrix

13

I, 13|01|, 1

3|02|, 1

3|10|, 1

3|12|, 1

3|20|, 1

3|21|. (29)

Moreover, () = 13

1() +1

32() +

1

33() with

1() = I

2() = 1, 1|00| + 2, 2|11| + 0, 0|22| |01|, |12|, |20|3() = 2, 2|00| + 0, 0|11| + 1, 1|22| |02|, |10|, |21|

the decomposition into extreme points corresponding to a partition of theKraus operators. Although this choice gives unital CPT maps, there are

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other decompositions into extreme points which do not give unital maps.

For example, the choice

1() = I

2() =1, 1 + 2, 2|00| + 0, 0|22| |01|, |02|, |20|

3() =0, 0 + 2, 2|11| + 1, 1|22| |12|, |10|, |21|

gives extreme points which are CPT but not unital.

The conjugate channel for this map is

R() =

3j,k=0

|ejek| (jk 14) vj vk (30)

References

[1] M.Horodecki, P. Shor, and M. B. Ruskai Entanglement Breaking Chan-nels Rev. Math. Phys 15, 629641 (2003). (quant-ph/030203)

[2] M. B. Ruskai, S. Szarek, E. Werner, An analysis of completely positivetrace-preserving maps M2 Lin. Alg. Appl. 347, 159 (2002).

[3] M.M. Wolf, J. Eisert Classical information capacity of a class of quantumchannels quant-ph/0412133

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