Quantum Channels, Kraus Operators, POVMs

1 Qitd412 Quantum Channels, Kraus Operators, POVMsRobert B. GriffithsVersion of 22 March 2012 Contents1 Introduction12 Kraus Operators23 Quantum Introduction .. Model Quantum channel .. Single qubit channel .. Geometrical interpretation .. Quantitative measures of noise .. Types of Quantum information .. 84 Quantum Operations and Introduction .. Quantum operations .. One qubit .. Kraus representation of Quantum operations .. Transition operator and dynamical operator .. 155 Definition.

2 Example of a POVM .. Naimark extension .. 18 References:CQT =Consistent Quantum Theoryby R. B. Griffiths (Cambridge, 2002)QCQI = Quantum computation and Quantum Informationby M. A. Nielsen and I. L. Chuang(Cambridge, 2000).Peres = Quantum Theory: Concepts and Methodsby A. Peres (Kluwer, 1995).1 Introduction The Quantum circuit in the following figure will be central toour discussion. In Fig. 1(a) we have systemsaande, with Hilbert spacesHa,He, of dimensiondaandde,initially in states| iand| ei, respectively, that interact, and the time development fromt0tot1isdescribed by a unitary operator or gate T, resulting in a state| i, see (3) below.

3 We shall think of| ias a state on the combined systembandf, Hilbert spaceHb Hf, ata later time. Simplest situation:bas the same asa, andfis the same | i| eiaebfT(a)| iabfJ(b)Figure 1: Quantum channel diagram as (a) unitary mapT; (b) isometryJ. So why not use the same names? Because we want a formalism thatallows the dimensiondbofHbto be different from the dimensiondaofHa. This is consistent with a unitaryTprovideddbdf=dade. Ifdb=daanddf=deone can identifyHbwithHaandHfwithHe. The verysimplest situation of interest is the one in whichTmaps two qubits to two qubits.

4 We suppose thateis always in the same initial normalized state| ei, whereas there are variouspossibilities for the initial normalized state| iofa; think of| eias a constant and| ias a variable. Because| eiis held fixed we can take the alternative perspective indicated in Fig. 1(b) anddefine the operatorJ| i=T | i | ei (1)as a map fromHatoHb Hf. This map is anisometryin that if| i=J| iand| i=J| i,thenh | i=h | i, , theJmap preserves inner products, and therefore also the norm and themetricbased upon the norm. (Isometry = preserves the metric.)

5 2 Exercise. Check thatJas defined in (1) preserves norms, providedTis unitary and| eiisnormalized. The isometryJmaps the whole Hilbert spaceHaonto a subspace of dimensiondaof thetensor productHb Hf. Consequently, we must haveda dbdf.(2) While relatingJto the unitaryTis important and useful for various physical applications,it is worth noting that much of what interests us depends onlyon the fact thatJis an isometryfromHatoHb Kraus Operators Choose some orthonormal basis (orbasis){|fki}forHf, and expand| i=J| i=T | i | ei =Xk| ki |fki(3)in this basis, assuming| iis normalized, where the expansion coefficients, the kets| ki, are ingeneral neither normalized nor orthogonal.

6 One can think of carrying out a simple measurement onfin the{|fki}basis, and if theoutcome (position of the pointer of the measuring apparatus) isk, then just before the measurementfwas in the state|fki. This measurement outcome, or the corresponding state itself just beforethe measurement, occurs with a probabilitypk=k kk2=h k| ki,(4)2and when it occurs we know that we should assign toHbthe normalized pre-probability| ki=| ki/k kk(5)if we want to calculate probabilities of properties ofb. It is useful to write the relationship between| kiand| i, assuming the isometryJis heldfixed (which will be true if| eiandTremain the same), and the basis{|fki}is also held fixed, inthe form| ki=Kk| i(6)where theKrausoperatorKkis a linear map fromHatoHb(hence fromHato itself in the casein whichbis the same asa).

7 Then (3) takes the formJ| i=T | i | ei =Xk Kk| i |fki.(7) The operatorsEkused in QCQI Sec. and later (pp. 360ff) are what we call Krausoperators; QCQI never uses the term Kraus . Also, QCQI focuses on the situationHb=Ha, butit is useful to also allow cases in whichdbis less than or larger thanda. Likewise, the operatorsMmintroduced in QCQI Sec. , which they call measurementoperators, are Kraus operators. That there is a linear relationship (6) between| kiand| imay not be immediately follows from (3), and the fact that the expansion coefficient| kiis uniquely determined by| iif everything else is held fixed.

8 See the following Show that the relationship between| iand| kifor a fixedkgiven by (3) is linearby looking at what happens when (i)| iis multiplied by a complex numberc, (ii)| iis replacedby a sum| i+| i. The fact thatJis an isometry (implied by| einormalized andTunitary) means thath | i=h | i=Xkh k| ki=Xkh |K kKk| i=h | XkK kKk | i,(8)will be true for every| i. This will be the case if and only if theclosure conditionXkK kKk=Ia(9)is satisfied, whereIais the identity onHa. See the Show that if for a fixed operatorAit is the case that foreverynormalized| iinHathe quantityh |A| iis 1, thenA=Ia.

9 [Hint. Introduce an orbasis{|aji}, expandA|ajiinthis basis, and show thatA|aji=|aji.] Note that sinceKkmapsHatoHb, its adjointK kmapsHbtoHa, and consequently theproductK kKkmapsHato itself, which is whyIaappears on the right side of (9), and notIb. Thisdistinction is important whenHbis different fromHa. WhenHb=Ha(as in QCQI) one does notneed the subscript onI. It is obvious from (7) that the Kraus operatorsdepend on the choice of orbasis{|fki} not misinterpret this as meaning that different types of measurements onf, corresponding todifferent choices of orbasis, will somehow influence Hb.

10 Instead, they reflect different frameworks3for relating properties ofbto those off. Remember that the{| ki=Kk| i}are conditionalpre-probabilities(up to normalization) used to calculate correlations; theyare not physical propertiescreated by some mysterious action-at-a-distance. Put in other words, what Fred, who measuresf, can learn about the systembin Bob spossession depends on what he learns aboutf, which in turn depends on the type of measurementhe carries out onf, saySxas againstSzin the case of a qubit. Fred s measurement has absolutelyno physical effect onb; to suppose otherwise is to fall prey to the nonlocality Quantum Introduction A basic issue in both Quantum computation and Quantum cryptography is that one needs toget information from one point to another in a reliable way.