Necessary and Sufficient Conditions on Measurements of Quantum Channels
John Burniston, Michael Grabowecky, Carlo Maria Scandolo, Giulio Chiribella, Gilad Gour
NNecessary and Sufficient Conditions on Measurements of Quantum Channels
John Burniston,
1, 2
Michael Grabowecky,
3, 1, 2
Carlo Maria Scandolo,
1, 2, ∗ Giulio Chiribella,
4, 5 and Gilad Gour
1, 2, † Department of Mathematics & Statistics, University of Calgary, Calgary, AB, Canada Institute for Quantum Science and Technology, University of Calgary, Calgary, AB, Canada Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada Department of Computer Science, The University of Hong Kong, Hong Kong, China Department of Computer Science, University of Oxford, Oxford, UK
Quantum supermaps are a higher-order generalization of quantum maps, taking quantum mapsto quantum maps. It is known that any completely positive, trace non-increasing (CPTNI) mapcan be performed as part of a quantum measurement. By providing an explicit counterexamplewe show that, instead, not every quantum supermap sending a quantum channel to a CPTNI mapcan be realized in a measurement on quantum channels. We find that the supermaps that can beimplemented in this way are exactly those transforming quantum channels into CPTNI maps evenwhen tensored with the identity supermap.
One of the most puzzling aspects of quantum mech-anics has always been the need to consider probabil-istic processes to describe the observation of physicalsystems. The development of quantum information the-ory has turned this puzzling feature into a resource formany protocols. Think, for instance, of the implement-ation of quantum computation through measurements(measurement-based quantum computation) [1, 2], ofquantum cryptographic protocols [3–6], or of the gen-eration of random numbers [7].Specializing our attention to finite-dimensionalquantum systems, the most general quantum measuringdevice can be described by a set of linear maps that arecompletely positive and trace non-increasing (CPTNI).The maps in this set must sum to a quantum channel,namely to a completely positive and trace-preserving(CPTP) linear map [8–10]. This situation is describedby a quantum instrument [10–13], a quantum channelthat takes a quantum system as input, and outputs aa classical-quantum system, where the classical systemrepresents the “meter” read by the experimenter. Fromthe classical outcome read on the meter, one can inferwhich CPTNI map occurred during the experiment. Thischaracterization of quantum experiments, in conjunctionwith the fact that quantum channels with trivial (i.e.1-dimensional) input represent states [14], singles outchannels as the fundamental objects of quantum theory,encapsulating all the other processes.For this reason, it is important to understand how tomanipulate quantum channels. The study of such manip-ulations, initiated in [14–16], has both practical [14, 17–28] and foundational consequences [16, 17, 25, 29–31],and has led to the development of new research areas,such as resource theories of quantum processes [28, 32–54]. The manipulation of quantum channels is imple-mented by supermaps [14–16, 38, 55, 56], which are lin-ear transformations sending linear maps to linear maps.In this setting, superchannels [14, 38] represent the waya channel can evolve deterministically, in the same wayas channels represent the deterministic evolution of a quantum state. Superchannels are the supermaps thattake quantum channels to quantum channels even whentensored with the identity supermap [16]. Measurementson channels are then described by a set of supermapsthat sum to a superchannel, giving rise to the notion ofa quantum super-instrument .In this letter we focus on measurements performed onquantum channels, and we show that a naive applicationof a condition similar to CPTNI in quantum theory is not enough to single out physical supermaps, viz. thosethat can arise in an experiment performed on quantumchannels.We will adopt the following notation.
Notation. B ( H ) denotes the set of bounded linear oper-ators on the finite-dimensional Hilbert space H , B h ( H ) the set of bounded hermitian operators on H , and D ( H ) the set of density matrices on H . Every letter without asubscript denotes a pair of systems A := A A , where A is usually regarded as an input system, and A as an out-put system. Thus E A := E A → A denotes a linear mapwith input A and output A , and L A := L A → A is theset of such linear maps, from B (cid:0) H A (cid:1) to B (cid:0) H A (cid:1) . | A | denotes the dimension of H A . A supermap Θ A → B takeselements of L A to elements of L B , and its action on a lin-ear map E A is denoted with square brackets: Θ A → B (cid:2) E A (cid:3) .Finally, a tilde over a system, like in A (cid:101) A , indicates thatwe are considering two identical copies of a system (in thiscase A ). We adopt the following convention concern-ing partial traces: if M AB is a matrix on A A B B , M AB denotes the partial trace on the missing system B : M AB := Tr B (cid:2) M AB (cid:3) . In summary, when a super-script is missing, we have taken the partial trace over themissing system of the original matrix.The first condition one must require of physical super-maps is that they be completely CP-preserving (CPP):they should send CP maps to CP maps even whentensored with the identity supermap. In formula, asupermap Θ A → B is CPP if for all bipartite CP maps a r X i v : . [ qu a n t - ph ] N ov E RA ∈ L RA , we have that (cid:0) R ⊗ Θ A → B (cid:1) (cid:2) E RA (cid:3) , (1)where R := R → R is the identity supermap, is still a CPmap. This is analogous to the CP condition for quantummaps.The second condition, analogous to being TNI forquantum maps, is that a physical supermap should sendCPTNI maps to CPTNI maps. If a supermap is CPP,demanding this is equivalent to requiring that it shouldtake CPTP maps to CPTNI maps (see Appendix A).More precisely, a supermap Θ A → B is CPTNI-preserving if it is CPP and Tr (cid:2) Θ A → B (cid:2) N A (cid:3) (cid:0) ρ B (cid:1)(cid:3) ≤ , (2)for any CPTP map N A ∈ L A and any ρ B ∈ D (cid:0) H B (cid:1) .A measurement on quantum channels (called a super-measurement ) is described by a set of CPTNI-preservingsupermaps (cid:8) Θ A → Bx (cid:9) x ∈ X , indexed by the outcome x ofthe measurement, such that (cid:80) x ∈ X Θ A → Bx is a superchan-nel. This gives rise to the super-instrument Υ A → X B (cid:2) E A (cid:3) = (cid:88) x ∈ X | x (cid:105) (cid:104) x | X ⊗ Θ A → Bx (cid:2) E A (cid:3) , (3)for every CP map E A , where system X represents theclassical meter and (cid:110) | x (cid:105) X (cid:111) is an orthonormal basis of X .Our main result is that, surprisingly, not all CPTNI-preserving supermaps can arise in a quantum super-measurement, therefore not all
CPTNI-preserving super-maps are physical. An example is the supermap Θ A → B whose action on a generic CP map E A is Θ A → B (cid:2) E A (cid:3) (cid:0) ρ B (cid:1) = Tr (cid:104) E A → B (cid:0) u A (cid:1) Y B (cid:0) ρ B (cid:1) T Y B (cid:105) u B (4)where all systems are qubits, u is the maximally mixedstate, and Y is the Pauli Y matrix ( ρ B is a generic dens-ity matrix, used to define the action of the CPTNI map Θ A → B (cid:2) E A (cid:3) on its input). Note that, if E A is CPTP, onehas indeed Tr (cid:2) Θ A → B (cid:2) E A (cid:3) (cid:0) ρ B (cid:1)(cid:3) ≤ , because ρ B is adensity matrix. This means that the supermap Θ A → B isCPTNI-preserving (cf. Eq. (2)). Full details are presen-ted in Appendix C.We find that the right condition to ensure that aCPTNI-preserving supermap Θ A → B is physical is thatit be completely CPTNI-preserving. This means that itis CPTNI-preserving even when tensored with the iden-tity supermap: Tr (cid:2)(cid:0) R ⊗ Θ A → B (cid:1) (cid:2) N RA (cid:3) (cid:0) ρ R B (cid:1)(cid:3) ≤ , (5)where N RA is a CPTP map, and ρ R B ∈ D (cid:0) H R B (cid:1) .The example in Eq. (4) highlights that, in general, not all CPTNI-preserving supermaps are completely CPTNI-preserving.For superchannels the situation is different: it is suf-ficient to demand that they be CPP and TP-preserving(TPP), without requiring that they be TPP in a complete sense [38]. The situation of generic supermaps is also dif-ferent from linear maps acting on quantum states. In thelatter case, to have a physical CP map, it is enough to re-quire that it be TNI, without demanding it in a completesense. The ultimate reason for these different behaviorsis related to causality and no-signaling [57], and it is fullyexamined in Appendix F.Following [15, 38], we work in the Choi picture forquantum maps and supermaps. A summary of usefulfacts is presented in Appendix B 1. Let J AB Θ be the Choimatrix of a supermap Θ A → B , and J A E the Choi matrixof a linear map E A ∈ L A . Then Θ A → B is a CPTNI-preserving supermap if and only if J AB Θ ≥ (since it isCPP), and it satisfies the additional condition derivingfrom Eq. (2): Tr (cid:104) J AB Θ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) ≤ , (6)for every CPTP map N A ∈ L A , and every ρ B ∈ D (cid:0) H B (cid:1) (see Appendix B 1). In a similar spirit, we canexpress the requirement of complete CPTNI preservationin Eq. (5) in the Choi picture as J AB Θ ≥ plus the re-markably simple additional constraint Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) ≤ , (7)for every positive semi-definite matrix M AB with mar-ginal M A B = I A ⊗ ρ B , for some ρ B ∈ D (cid:0) H B (cid:1) . Thetechnical details are provided in Appendix B 2.It is not hard to check that all the matrices of the form J A N ⊗ ρ B , with N A CPTP, are a strict subset of thematrices M AB , confirming that complete CPTNI pre-servation is at least as strict a condition as CPTNI pre-servation. In fact, it is stricter, as our counterexamplein Eq. (4) shows: the supermap in Eq. (4) is CPTNI-preserving but not completely CPTNI-preserving. Con-sequently, the set of completely CPTNI-preserving super-maps is strictly contained in the set of CPTNI-preservingsupermaps. The situation is illustrated in Fig. 1.To obtain our main result, namely the characteriza-tion of which CPTNI-preserving supermaps are physical,we consider a semi-definite program (SDP) inspired byEq. (7): Find α = max M Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) Subject to: M AB ≥ M A B = I A ⊗ ρ B . (8) c-CPTNICPTNICPP Figure 1. Inclusions between sets of supermaps. Here c-CPTNI denotes completely CPTNI-preserving supermaps.
If we consider the dual of the SDP (8)Find β = | A | min r Subject to: r | A | J A B Φ ⊗ u A ≥ J AB Θ J A B Φ ≥ J A B Φ = I A B r ≥ r ∈ R Φ superchannel , we convert Eq. (8) from an SDP having a constraint on M AB into one having an explicit condition on J AB Θ .This condition is exactly what we need to derive our mainresult. Theorem 1.
A CPTNI-preserving supermap can becompleted to a superchannel if and only if it is completely
CPTNI-preserving.
The full proof is presented in Appendix D.Using the Choi picture, we can re-obtain a result of[14, theorem 2], namely that every completely CPTNI-preserving supermap can be expressed in terms of aCPTP pre-processing map and a CPTNI post-processingmap, as depicted in Fig. 2. In formula, if Θ A → Bx is a com-pletely CPTNI-preserving supermap, associated with theoutcome x of a quantum super-instrument, its Choi mat-rix J AB Θ x can be expressed in the following form: J AB Θ x = (cid:16) I AB ⊗ Γ (cid:101) A E → B post x (cid:17) ◦ (cid:16) I A (cid:101) A B ⊗ Γ (cid:101) B → A E pre (cid:17) (cid:16) φ B (cid:101) B + ⊗ φ A (cid:101) A + (cid:17) (9)where I AB is the identity channel on AB and φ B (cid:101) B + = (cid:80) j,k | jj (cid:105) (cid:104) kk | B (cid:101) B is the unnormalized maximally en-tangled state of B (cid:101) B . Here Γ (cid:101) B → A E pre is the CPTPpre-processing map, and Γ (cid:101) A E → B post x is the CPTNI post-processing map. Being that Θ A → Bx is part of a quantumsuper-instrument, we also have that (cid:80) x Γ (cid:101) A E → B post x is a Figure 2. Representation of a completely CPTNI-preservingsupermap. Here, a bipartite input map E RA is insertedbetween a CPTP pre-processing map and a CPTNI post-processing map. The output is a bipartite CPTNI map.Note the presence of the ancillary system E , which actsas a “memory” between the pre-processing and the post-processing. This realization of a supermap is called a quantum1-comb [19]. CPTP map. The proof of this result is reported in Ap-pendix E. Note that the pre-processing Γ (cid:101) B → A E pre is in-dependent of x , therefore it can be chosen to be the samefor all the supermaps in the same super-instrument.To summarize, for the first time we exactly pinneddown the necessary and sufficient conditions determ-ining which supermaps can appear in quantum super-instruments. Specifically, we found that only com-pletely CPTNI-preserving supermaps can be imple-mented in a quantum super-instrument. Addition-ally, we showed an explicit example of a supermapthat is CPTNI-preserving, but not completely CPTNI-preserving (Eq. (4)). Viewing CPTNI preservation asa higher-order generalization of the CPTNI conditionfor quantum maps, we cannot fail to note the differ-ence between the theory of quantum supermaps—whereCPTNI maps are regarded as states—and quantum the-ory. Indeed, in quantum theory, all CPTNI maps E A arealso completely CPTNI, the latter meaning Tr (cid:2)(cid:0) I R ⊗ E A (cid:1) (cid:0) ρ R A (cid:1)(cid:3) ≤ , (10)for every ρ R A ∈ D (cid:0) H R A (cid:1) . The ultimate reason forthis difference is that the theory of quantum supermapsdoes not satisfy the fundamental property of causality[57]. Axiom 2 (Causality) . The probability of a transform-ation occurring in an experiment is independent of thechoice of experiments performed on its output.
Loosely speaking, causality means that informationcannot “come back from the future”. One of its con-sequences is that all bipartite states are non-signaling.The existence of signaling bipartite channels [58, 59] is aclear signature that causality does not hold in the theoryof quantum supermaps. Moreover, we can get an intuit-ive grasp of the failure of causality from the realizationof physical supermaps expressed in Eq. (9) and in Fig. 2.Consider a super-measurement (cid:8) Θ A → Bx (cid:9) performed on aCPTNI map E A , which means that the measurement oc-curs after E A is prepared in a laboratory or in a quantumcircuit. The presence of a pre-processing in the realiz-ation of every Θ A → Bx implies that Θ A → Bx acts on the input of E A too, meaning, in some sense, that part of Θ A → Bx also acts before E A . Somehow in this situationthere is not a well-defined notion of what comes “before”and “after”, so causality cannot hold; for it would se-lect a clear “arrow of time” in information processing. Arigorous proof of this, and of the implications of the fail-ure of causality for the theory of quantum supermaps, ispresented in Appendix F.The results we obtained in this letter improve our un-derstanding of the operational viewpoint in quantum the-ory, and more generally in physics. In particular, weshowed that the correct conditions to impose on a lin-ear transformation to guarantee its physicality, be it aquantum map or a quantum supermap, must always beformulated in a complete sense . This means that theymust always involve the tensor product with the iden-tity transformation. Thus, for quantum supermaps wehave the CPP condition and the complete CPTNI pre-servation condition. For quantum maps we have the CPcondition and the complete TNI condition of Eq. (10).Since quantum theory satisfies causality, Eq. (10) be-comes equivalent to the TNI condition we impose ordin-arily on quantum maps (see Appendix F 2). However,the fundamental requirement is still the one expressed byEq. (10). In other words, the existence of signalling bi-partite states in the theory of quantum supermaps is thereason for the gap between CPTNI preservation and com-plete CPTNI preservation; in the very same way as theexistence of entangled states in quantum theory createsa gap between positive and completely positive maps,which are instead the same notion in classical physics.The fact that conditions expressed in a complete senseare the right thing to demand is apparent if one ad-opts the framework of operational probabilistic theories[57, 60–62], presented in Appendix F. This is an opera-tional approach to physical theories based on the notionof circuits, and of composition of physical transforma-tions occurring in experiments. Our results confirm andstrengthen the validity of this approach to the study ofthe fundamental operational properties of physical the-ories. Contributions
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Quantum Theory: Informational Foundationsand Foils , edited by G. Chiribella and R. W. Spekkens(Springer Netherlands, Dordrecht, 2016) pp. 309–366.[75] A. Kitaev, D. Mayers, and J. Preskill, Phys. Rev. A ,052326 (2004).[76] G. M. D’Ariano, F. Manessi, P. Perinotti, and A. Tosini,Int. J. Mod. Phys. A , 1430025 (2014).[77] G. Chiribella and C. M. Scandolo, “Entanglement as anaxiomatic foundation for statistical mechanics,” (2016),arXiv:1608.04459 [quant-ph].[78] G. Chiribella and C. M. Scandolo, New J. Phys. ,123043 (2017).[79] C. M. Scandolo, Information-theoretic foundations ofthermodynamics in general probabilistic theories , Ph.D.thesis, University of Oxford (2018).[80] G. Chiribella and C. M. Scandolo, New J. Phys. ,103027 (2015). Appendix A: General facts about quantum maps and supermaps
Quantum maps describe the evolution of quantum systems, in both the deterministic and the probabilistic case(e.g. when a measurement is performed). To be consistent with the interpretation of mixed states as probabilisticensembles, a quantum map E A must be linear, E A ∈ L A . This is not enough, because a quantum map must sendquantum states to quantum states even when applied only to half of a bipartite state. For this reason we first demandthat it be completely positive (CP): for every ρ R A ∈ D (cid:0) H R A (cid:1) it must be (cid:0) I R ⊗ E A (cid:1) (cid:0) ρ R A (cid:1) ≥ , where I R is the identity map on system R . This means that E A sends positive semi-definite operators to positivesemi-definite operators even when tensored with the identity. We also require that a map E A be trace non-increasing (TNI): Tr (cid:2) E A (cid:0) ρ A (cid:1)(cid:3) ≤ , for every ρ A ∈ D (cid:0) H A (cid:1) . In particular, if the trace is preserved, that is Tr (cid:2) E A (cid:0) ρ A (cid:1)(cid:3) = 1 , for every ρ A ∈ D (cid:0) H A (cid:1) ,we say that the map is trace-preserving (TP). The allowed quantum maps are those that are both CP and TNI(CPTNI). CPTP maps are also called quantum channels and represent the most general deterministic evolutions aquantum system can undergo. CPTNI maps that are not CPTP represent non-deterministic transformations. This iswhat happens in a quantum measurement, which can be seen as a collection of CPTNI maps (cid:8) E Ax (cid:9) , indexed by theoutcomes x of that measurement, such that (cid:80) x E Ax is a CPTP map. If we know the outcome x of the measurement,then we know that the system evolved under the CPTNI map E Ax . We can therefore construct a quantum instrument E A → X A = (cid:88) x | x (cid:105) (cid:104) x | X ⊗ E Ax , (A1)where (cid:110) | x (cid:105) X (cid:111) is an orthonormal basis of system X . E A → X A is a quantum channel with classical-quantum output.Here X is the classical system, recording the measurement outcome. As such it represents the meter read by theexperimenter performing the quantum measurement (cid:8) E Ax (cid:9) .These notions can be easily generalized to quantum supermaps [14, 15, 55], namely to transformations sendingquantum maps to quantum maps. Again these are linear maps, and an easy translation of the requirements of CP andTNI leads to the requirement of CPP (Eq. (1)) [14, 38] and TNI preservation. Specifically, a map is CPTNI-preservingif it is CPP, and sends CPTNI maps to CPTNI maps: Tr (cid:2) Θ A → B (cid:2) E A (cid:3) (cid:0) ρ B (cid:1)(cid:3) ≤ , (A2)for any CPTNI map E A and any ρ B ∈ D (cid:0) H B (cid:1) . In fact, if Θ A → B is CPP, it is enough to require that inequality (A2)be satisfied by quantum channels E A , namely by CPTP maps. Figure 3. Realization of a superchannel. Here an input quantum channel is inserted between a CPTP pre-processing map anda CPTP post-processing map. The output is another quantum channel. The ancillary system E plays the role of a quantummemory between the pre- and the post-processing. To see it, let E A be a CPTNI map. We can always find another CPTNI map E (cid:48) A such that E A + E (cid:48) A is CPTP. Nowassume that Θ A → B is CPP, and sends CPTP maps to CPTNI maps. Then, for every ρ B ∈ D (cid:0) H B (cid:1) , ≥ Tr (cid:2) Θ A → B (cid:2) E A + E (cid:48) A (cid:3) (cid:0) ρ B (cid:1)(cid:3) = Tr (cid:2) Θ A → B (cid:2) E A (cid:3) (cid:0) ρ B (cid:1)(cid:3) + Tr (cid:2) Θ A → B (cid:2) E (cid:48) A (cid:3) (cid:0) ρ B (cid:1)(cid:3) . Since Θ A → B is CPP, then Tr (cid:2) Θ A → B (cid:2) E A (cid:3) (cid:0) ρ B (cid:1)(cid:3) ≥ and Tr (cid:2) Θ A → B (cid:2) E (cid:48) A (cid:3) (cid:0) ρ B (cid:1)(cid:3) ≥ , therefore we conclude that itmust be Tr (cid:2) Θ A → B (cid:2) E A (cid:3) (cid:0) ρ B (cid:1)(cid:3) ≤ , which means that Θ A → B satisfies Eq. (A2).A CPTNI-preserving supermap Θ A → B is called superchannel if it sends CPTP maps to CPTP maps [38]. Theoriginal definition in [16] required that it should send quantum channels to quantum channels in a complete sense,i.e. even when tensored with the identity supermap. In other words, Tr (cid:2)(cid:0) R ⊗ Θ A → B (cid:1) (cid:2) N RA (cid:3) (cid:0) ρ R B (cid:1)(cid:3) = 1 . (A3)for any CPTP map N RA and any ρ R B ∈ D (cid:0) H R B (cid:1) , where R is the identity supermap on R . Actually, in [38,theorem 1], using the Choi picture, it was proved that we need not consider this requirement in a complete sense: aCPTNI-preserving supermap Θ A → B is a superchannel if and only if Tr (cid:2) Θ A → B (cid:2) N A (cid:3) (cid:0) ρ B (cid:1)(cid:3) = 1 , for any CPTP map N A and any ρ B ∈ D (cid:0) H B (cid:1) . In Appendix F 2 we will prove this result in an alternative way,without using the Choi isomorphism.Superchannels are intimately related to channels: it was proved that all superchannels can be represented in termsof a pre- and a post-processing CPTP map [14, 38], as depicted in Fig. 3. Such a representation is called a quantum1-comb [14].Superchannels play an important role because they represent all physical ways a quantum channel can evolve in anopen system, and provide a framework for measurements on quantum operations, called super-measurements . Theseare described by a set (cid:8) Θ A → Bx (cid:9) of CPTNI-preserving supermaps such that (cid:80) x Θ A → Bx is a superchannel. Then wecan construct a quantum super-instrument as the generalization of the quantum notion (see Eq. (3)): Υ A → X B (cid:2) E A (cid:3) = (cid:88) x | x (cid:105) (cid:104) x | X ⊗ Θ A → Bx (cid:2) E A (cid:3) , where system X again represents the classical meter, like in Eq. (A1), and E A is any CP map. The main resultof this letter is that, unlike CPTNI quantum maps, not all CPTNI-preserving supermaps can be part of a quantumsuper-instrument. In Appendix F we link this fact to the failure of causality [57] in the theory of quantum supermaps.
Appendix B: The Choi picture for quantum maps and supermaps
In this Appendix we collect some results about the Choi representation of quantum maps and supermaps. Specific-ally, Appendix B 1 provides the general background information and some basic results about the Choi isomorphism.Appendix B 2 instead focuses on the derivations related to complete CPTNI preservation in the Choi form, and inparticular on obtaining Eq. (7).
1. Choi matrices of quantum maps and supermaps
The first ingredient to define the Choi isomorphism is to consider the super-normalized maximally entangled state | φ + (cid:105) A (cid:101) A = (cid:80) | A | j =1 | j (cid:105) A | j (cid:105) (cid:101) A , where (cid:110) | j (cid:105) A (cid:111) is a fixed orthonormal basis of H A (and therefore of H (cid:101) A too, since (cid:101) A is just another copy of A ). The Choi matrix of a linear map E A ∈ L A is defined as J A E := (cid:16) I A ⊗ E (cid:101) A → A (cid:17) (cid:16) φ A (cid:101) A + (cid:17) , where φ A (cid:101) A + := | φ + (cid:105) (cid:104) φ + | A (cid:101) A , and I A is the identity channel. Again, since (cid:101) A is just another copy of A , the linearmap E (cid:101) A → A is well defined.In particular, E A is CP if and only if J A E ≥ . E A is CPTP if and only if in addition one has J A E = I A . Instead, E A is CPTNI if and only if, besides J A E ≥ , one has J A E ≤ I A . The Choi matrix J A E encodes all the informationabout E A because one can reconstruct the action of E A on quantum states from its Choi matrix: E A (cid:0) ρ A (cid:1) = Tr A (cid:104) J A E (cid:16)(cid:0) ρ A (cid:1) T ⊗ I A (cid:17)(cid:105) , (B1)for every ρ A ∈ D (cid:0) H A (cid:1) .To define the Choi matrix of a supermap Θ A → B , we follow the approach presented in [38]. Let us consider thefollowing basis of the space L A : E Ajklm (cid:0) ρ A (cid:1) = (cid:104) j | ρ | k (cid:105) A | l (cid:105) (cid:104) m | A , for j, k ∈ { , . . . , | A |} and l, m ∈ { , . . . , | A |} . The Choi matrix of the supermap Θ A → B can be defined as J AB Θ := (cid:88) j,k,l,m J A E jklm ⊗ J B Θ[ E jklm ] . Again, J AB Θ encodes all the information about Θ A → B . For instance, Θ A → B is CPP if and only if J AB Θ ≥ . Moreover,we can express the action of a supermap on a quantum map E A using their Choi matrices: if F B = Θ A → B (cid:2) E A (cid:3) , wehave [38] J B F = Tr A (cid:104) J AB Θ (cid:16)(cid:0) J A E (cid:1) T ⊗ I B (cid:17)(cid:105) . (B2)A full characterization of superchannels from their Choi matrices was given in [38]: Θ A → B is a superchannel if andonly if J AB Θ ≥ , and one has J AB Θ = J A B Θ ⊗ u A and J A B Θ = I A B . Here u A = | A | I A is the maximally mixedstate. Combining Eqs. (B1) and (B2) for a CPTP map N A , we have Θ A → B (cid:2) N A (cid:3) (cid:0) ρ B (cid:1) = Tr AB (cid:104) J AB Θ (cid:16)(cid:0) J A N ⊗ ρ B (cid:1) T ⊗ I B (cid:17)(cid:105) ; therefore, Tr (cid:2) Θ A → B (cid:2) N A (cid:3) (cid:0) ρ B (cid:1)(cid:3) = Tr B (cid:110) Tr AB (cid:104) J AB Θ (cid:16)(cid:0) J A N ⊗ ρ B (cid:1) T ⊗ I B (cid:17)(cid:105)(cid:111) = Tr (cid:104) J AB Θ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) (B3)Hence, by Eq. (A3) Θ A → B is a superchannel if and only if Tr (cid:104) J AB Θ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) = 1 . Similarly, we can characterize CPTNI-preserving supermaps in the Choi picture. By Eq. (2), a supermap is CPTNI-preserving if Tr (cid:2) Θ A → B (cid:2) N A (cid:3) (cid:0) ρ B (cid:1)(cid:3) ≤ , for every CPTP map N A and every density matrix ρ B . By Eq. (B3) wecan rewrite this condition in the Choi picture as Tr (cid:104) J AB Θ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) ≤ . This proves Eq. (6).
2. Some technical derivations about completely CPTNI-preserving supermaps
Now we will focus on expressing the complete CPTNI preservation condition in the Choi picture. Looking at Eq. (5)tells us that we need to find an expression for Tr (cid:2)(cid:0) R ⊗ Θ A → B (cid:1) (cid:2) N RA (cid:3) (cid:0) ρ R B (cid:1)(cid:3) in the Choi picture. Note that theidentity supermap does not change the systems it acts on. Therefore, to express Eq. (5) in the Choi form, we onlyconsider how N RA is acted on by the supermap Θ A → B , representing the action of the identity superchannel with theidentity matrix I R . Therefore, combining Eqs. (B1) and (B2) this time yields Tr (cid:2)(cid:0) R ⊗ Θ A → B (cid:1) (cid:2) N RA (cid:3) (cid:0) ρ R B (cid:1)(cid:3) = Tr (cid:104)(cid:0) I R ⊗ J AB Θ (cid:1) (cid:16)(cid:0) J RA N (cid:1) T A ⊗ I B (cid:17) (cid:16)(cid:0) ρ R B (cid:1) T ⊗ I R AB (cid:17)(cid:105) = Tr R AB (cid:110) Tr R B (cid:104)(cid:0) I R ⊗ J AB Θ (cid:1) (cid:16)(cid:0) J RA N (cid:1) T A ⊗ I B (cid:17) (cid:16)(cid:0) ρ R B (cid:1) T ⊗ I R AB (cid:17)(cid:105)(cid:111) = Tr R AB (cid:20)(cid:16) I R ⊗ J AB Θ (cid:17) (cid:18)(cid:16) J R A N (cid:17) T A ⊗ I B (cid:19) (cid:16)(cid:0) ρ R B (cid:1) T ⊗ I A (cid:17)(cid:21) . (B4)Now let us define M AB := Tr R (cid:20)(cid:0) ρ R B ⊗ I A (cid:1) (cid:18)(cid:16) J R A N (cid:17) T R ⊗ I B (cid:19)(cid:21) , (B5)and let us calculate Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) = Tr R AB (cid:34)(cid:16) I R ⊗ J AB Θ (cid:17) (cid:18)(cid:16) J R A N (cid:17) T R ⊗ I B (cid:19) T (cid:0) ρ R B ⊗ I A (cid:1) T (cid:35) = Tr (cid:20)(cid:16) I R ⊗ J AB Θ (cid:17) (cid:18)(cid:16) J R A N (cid:17) T A ⊗ I B (cid:19) (cid:16)(cid:0) ρ R B (cid:1) T ⊗ I A (cid:17)(cid:21) . As we can see, this coincides with Eq. (B4). Therefore Tr (cid:2)(cid:0) R ⊗ Θ A → B (cid:1) (cid:2) N RA (cid:3) (cid:0) ρ R B (cid:1)(cid:3) = Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) ,where M AB is defined in Eq. (B5). Now the complete CPTNI preservation condition of Eq. (5) becomes Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) ≤ (B6)for every M AB of the form (B5). Note that Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) ≥ for every CPP supermap Θ A → B , whence M AB is positive semi-definite. Furthermore, M A B = Tr R A (cid:20)(cid:0) ρ R B ⊗ I A (cid:1) (cid:18)(cid:16) J R A N (cid:17) T R ⊗ I B (cid:19)(cid:21) = Tr R (cid:2) ρ R B (cid:3) ⊗ I A = I A ⊗ ρ B , where we have used the fact that Tr R A (cid:20)(cid:16) J R A N (cid:17) T R (cid:21) = Tr R A (cid:104) J R A N (cid:105) , and that Tr A (cid:104) J R A N (cid:105) = Tr R A (cid:2) J RA N (cid:3) = I R ⊗ I A because N is CPTP (cf. Appendix B 1). So M AB has marginal M A B = I A ⊗ ρ B .Now we prove a key result, namely that every positive semi-definite matrix M AB with marginal M A B = I A ⊗ ρ B ,where ρ B is any density matrix, can be written as in Eq. (B5). In this way, instead of stating complete CPTNIpreservation as in Eq. (B6) for M AB of the form (B5), we will state it in a remarkably simpler way: Θ A → B iscompletely CPTNI-preserving if and only if Eq. (B6) is satisfied for any positive semi-definite M AB with marginal M A B = I A ⊗ ρ B . This technical result will be crucial for the main finding of this letter, namely the characterizationof physical supermaps (see Appendix D). Lemma 3.
Let M AB ≥ such that M A B = I A ⊗ ρ B , for some ρ B ∈ D (cid:0) H B (cid:1) . Then M BC = Tr R (cid:20)(cid:0) ρ R B ⊗ I A (cid:1) (cid:18)(cid:16) J R A N (cid:17) T R ⊗ I B (cid:19)(cid:21) , where N AB is some CPTP map and ρ R B ∈ D (cid:0) H R B (cid:1) .Proof. Let φ A (cid:101) A + ⊗ ϕ E B be a purification of M A B = I A ⊗ ρ B , where ϕ E B ∈ D (cid:0) H B E (cid:1) is a purification of ρ B .Now let τ AB F be a purification of M AB , so τ AB F is also a purification of M A B . Thus, these two purificationscan be related by an isometry channel V (cid:101) A E → A F such that [63] τ AB F = (cid:16) I A B ⊗ V (cid:101) A E → A F (cid:17) (cid:16) φ A (cid:101) A + ⊗ ϕ E B (cid:17) . F yields M AB = (cid:16) I A B ⊗ Γ (cid:101) A E → A (cid:17) (cid:16) φ A (cid:101) A + ⊗ ϕ E B (cid:17) (B7)where Γ (cid:101) A E → A := Tr F ◦ V (cid:101) A E → A F is a CPTP map. The action of Γ (cid:101) A E → A on a generic state χ (cid:101) A E ∈ D (cid:16) H (cid:101) A E (cid:17) can be written in terms of its Choi matrix as Γ (cid:101) A E → A (cid:16) χ (cid:101) A E (cid:17) = Tr (cid:101) A E (cid:20) J (cid:101) A E A Γ (cid:18)(cid:16) χ (cid:101) A E (cid:17) T ⊗ I A (cid:19)(cid:21) . (B8)Let us substitute Eq. (B8) into Eq. (B7). Note that the identity map does not change the systems it acts on.Therefore, to express Eq. (B7) in the Choi form, we only consider how φ A (cid:101) A + ⊗ ϕ E B is acted on by the map Γ (cid:101) A E → A , representing the action of the identity channel with the identity matrix I A B . Thus Eq. (B7) becomes M AB = Tr (cid:101) A E (cid:20)(cid:16) I A B ⊗ J (cid:101) A E A Γ (cid:17) (cid:18)(cid:16) φ A (cid:101) A + (cid:17) T (cid:101) A ⊗ (cid:0) ϕ E B (cid:1) T E ⊗ I A (cid:19)(cid:21) . Expanding φ A (cid:101) A + , and using the cyclic property of the trace, we get Tr (cid:101) A E (cid:20)(cid:16) I A B ⊗ J (cid:101) A E A Γ (cid:17) (cid:18)(cid:16) φ A (cid:101) A + (cid:17) T (cid:101) A ⊗ (cid:0) ϕ E B (cid:1) T E ⊗ I A (cid:19)(cid:21) == (cid:88) x,y | x (cid:105) (cid:104) y | A ⊗ Tr E (cid:34)(cid:32) I B ⊗ (cid:68) x (cid:12)(cid:12)(cid:12) J (cid:101) A E A Γ (cid:12)(cid:12)(cid:12) y (cid:69) (cid:101) A (cid:33) (cid:16)(cid:0) ϕ E B (cid:1) T E ⊗ I A (cid:17)(cid:35) . Since (cid:101) A is a copy of A , (cid:88) x,y | x (cid:105) (cid:104) y | A (cid:68) x (cid:12)(cid:12)(cid:12) J (cid:101) A E A Γ (cid:12)(cid:12)(cid:12) y (cid:69) (cid:101) A =: J E A Γ where we have replaced system (cid:101) A with system A , and we have set A := A A as usual. Now Γ is regarded as achannel from A E to A . With this in mind, we can write M AB = Tr E (cid:104)(cid:16) J E A Γ ⊗ I B (cid:17) (cid:16)(cid:0) ϕ E B (cid:1) T E ⊗ I A (cid:17)(cid:105) . Taking the transpose on E , this expression can be rewritten as M AB = Tr E (cid:20)(cid:0) ϕ E B ⊗ I A (cid:1) (cid:18)(cid:16) J E A Γ (cid:17) T E ⊗ I B (cid:19)(cid:21) . Now rename E as R , and define J R A N := J R A Γ , and ρ R B := ϕ R B . We find that M AB can be written in theform of Eq. (B5).This means that once we require M AB to be positive semi-definite with marginal M A B = I A ⊗ ρ B , for somedensity matrix ρ B , this automatically implies that M AB has the special form of Eq. (B5). Consequently, we canexpress the requirement of complete CPTNI preservation in the Choi form as follows: Θ A → B is complete CPTNI-preserving if and only if J AB Θ ≥ and Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) ≤ for every positive semi-definite M AB with marginal M A B = I A ⊗ ρ B , where ρ B ∈ D (cid:0) H B (cid:1) . This is Eq. (7). In particular, Θ A → B is a superchannel, which is acompletely CPTP-preserving supermap if and only if Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) = 1 for every M AB as above.Note that, among these M AB ’s we can find matrices of the form J A N ⊗ ρ B , where N A is a CPTP map. Thesematrices are those used to check the CPTNI preservation condition (cf. Eq. (6)). Indeed, J A N ⊗ ρ B ≥ , and themarginal is Tr A (cid:2) J A N ⊗ ρ B (cid:3) = Tr A (cid:2) J A N (cid:3) ⊗ ρ B = I A ⊗ ρ B because J A N is the Choi matrix of a CPTP map (see Appendix A). Therefore, as it must be, we recover in the Choipicture that CPTNI preservation is not stronger than complete CPTNI preservation. In fact, it is strictly weaker, asshown in Appendix C.1 Appendix C: A supermap that is CPTNI-preserving, but not completely CPTNI-preserving
In this Appendix we present the concrete counterexample of a supermap Θ A → B that is CPTNI-preserving, but not completely CPTNI-preserving. In this construction we take | A | = | A | = | B | = 2 . Consider a supermap Θ A → B that has a Choi matrix with marginal J AB Θ = I A ⊗ ψ A B − , where ψ A B − = | ψ − (cid:105) (cid:104) ψ − | A B , and | ψ − (cid:105) A B = √ (cid:16) | (cid:105) A B − | (cid:105) A B (cid:17) is the singlet state. Given this marginal, a possible Choi matrix of the supermap Θ A → B is J AB Θ = I A ⊗ ψ A B − ⊗ u B , where u B is the maximally mixed state of B . Now we will prove that this supermap isCPTNI-preserving, but not completely CPTNI-preserving.To this end, we first show that J AB Θ satisfies Eq. (6). If N A is a CPTP map and ρ B is a density matrix, we have Tr (cid:104) J AB Θ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) = Tr (cid:104)(cid:16) I A ⊗ ψ A B − (cid:17) (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) . Now we express ψ A B − in terms of the super-normalized maximally entangled state φ A B + : ψ A B − = 12 (cid:0) I A ⊗ Y B (cid:1) φ A B + (cid:0) I A ⊗ Y B (cid:1) , (C1)where Y B is the Pauli Y matrix. Then Tr (cid:104) J AB Θ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) = 12 Tr (cid:104)(cid:16) I A ⊗ (cid:0) I A ⊗ Y B (cid:1) φ A B + (cid:0) I A ⊗ Y B (cid:1)(cid:17) (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) = 12 Tr A B (cid:110) Tr A (cid:104)(cid:16) I A ⊗ (cid:0) I A ⊗ Y B (cid:1) φ A B + (cid:0) I A ⊗ Y B (cid:1)(cid:17) (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105)(cid:111) = 12 Tr A B (cid:20) φ A B + (cid:18)(cid:16) J A N (cid:17) T ⊗ Y B (cid:0) ρ B (cid:1) T Y B (cid:19)(cid:21) , using the cyclic property of the trace. Now let us expand φ A B + .
12 Tr (cid:20) φ A B + (cid:18)(cid:16) J A N (cid:17) T ⊗ Y B (cid:0) ρ B (cid:1) T Y B (cid:19)(cid:21) = 12 Tr (cid:20)(cid:80) x,y =1 | xx (cid:105) (cid:104) yy | A B (cid:18)(cid:16) J A N (cid:17) T ⊗ Y B (cid:0) ρ B (cid:1) T Y B (cid:19)(cid:21) = 12 Tr (cid:34)(cid:80) x,y =1 (cid:28) y (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) J A N (cid:17) T (cid:12)(cid:12)(cid:12)(cid:12) x (cid:29) A | x (cid:105) (cid:104) y | B Y B (cid:0) ρ B (cid:1) T Y B (cid:35) = 12 Tr (cid:20)(cid:80) x,y =1 (cid:68) x (cid:12)(cid:12)(cid:12) J A N (cid:12)(cid:12)(cid:12) y (cid:69) A | x (cid:105) (cid:104) y | B Y B (cid:0) ρ B (cid:1) T Y B (cid:21) . (C2)Here the expression (cid:80) x,y =1 (cid:68) x (cid:12)(cid:12)(cid:12) J A N (cid:12)(cid:12)(cid:12) y (cid:69) A | x (cid:105) (cid:104) y | B means considering N A with its output system transformed from A to B . With this simplification, Eq. (C2) reads Tr (cid:104) J AB Θ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) = 12 Tr (cid:104) J B N Y B (cid:0) ρ B (cid:1) T Y B (cid:105) . Now, both J B N and Y B (cid:0) ρ B (cid:1) T Y B are density operators, therefore Tr (cid:104) J AB Θ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) = Tr (cid:20)(cid:18) J B N (cid:19) (cid:16) Y B (cid:0) ρ B (cid:1) T Y B (cid:17)(cid:21) ≤ . Hence J AB Θ satisfies Eq. (6); therefore Θ A → B is a CPTNI-preserving supermap.Now we show that Θ A → B violates Eq. (7). To this end, let us take M AB = (cid:16) J AB Θ (cid:17) T . This choice of M AB complies with the two requests on M AB in Eq. (7). Since J AB Θ = I A ⊗ ψ A B − , (cid:16) J AB Θ (cid:17) T is positive semi-definite;and its marginal M A B = Tr A (cid:20)(cid:16) J AB Θ (cid:17) T (cid:21) = (cid:16) Tr A (cid:104) J AB Θ (cid:105)(cid:17) T = (cid:0) I A ⊗ u B (cid:1) T = I A ⊗ u B I A ⊗ ρ B , with ρ B density matrix. Then Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) = Tr (cid:104)(cid:16) I A ⊗ ψ A B − (cid:17) (cid:16) I A ⊗ ψ A B − (cid:17)(cid:105) = Tr (cid:20) I A ⊗ (cid:16) ψ A B − (cid:17) (cid:21) = Tr A B (cid:110) Tr A (cid:104) I A ⊗ ψ A B − (cid:105)(cid:111) = 2Tr (cid:104) ψ A B − (cid:105) = 2 > . This is in contrast with Eq. (7), therefore the supermap Θ A → B is not a completely CPTNI-preserving supermap,despite being CPTNI-preserving.We conclude this Appendix by reconstructing Θ A → B from its Choi matrix J AB Θ = I A ⊗ ψ A B − ⊗ u B . By Eqs. (B1)and (B2), we have, if E A is a generic CP map, Θ A → B (cid:2) E A (cid:3) (cid:0) ρ B (cid:1) = Tr AB (cid:104) J AB Θ (cid:0) J A E ⊗ ρ B ⊗ I B (cid:1) T (cid:105) = Tr AB (cid:104)(cid:16) I A ⊗ ψ A B − (cid:17) (cid:0) J A E ⊗ ρ B (cid:1) T (cid:105) u B = Tr A B (cid:110) Tr A (cid:104)(cid:16) I A ⊗ ψ A B − (cid:17) (cid:0) J A E ⊗ ρ B (cid:1) T (cid:105)(cid:111) u B = Tr (cid:20) ψ A B − (cid:16) J A E ⊗ ρ B (cid:17) T (cid:21) u B . Recalling Eq. (C1), we get Θ A → B (cid:2) E A (cid:3) (cid:0) ρ B (cid:1) = 12 Tr (cid:20) φ A B + (cid:18)(cid:16) J A E (cid:17) T ⊗ Y B (cid:0) ρ B (cid:1) T Y B (cid:19)(cid:21) u B , and using an argument similar to the one in Eq. (C2), we finally obtain Θ A → B (cid:2) E A (cid:3) (cid:0) ρ B (cid:1) = 12 Tr (cid:104) J B E Y B (cid:0) ρ B (cid:1) T Y B (cid:105) u B . (C3)By Eq. (B1), E A → B (cid:0) u A (cid:1) = Tr A (cid:104) J A B E I A B (cid:105) = J B E . An equivalent form of Eq. (C3) is therefore Θ A → B (cid:2) E A (cid:3) (cid:0) ρ B (cid:1) = Tr (cid:104) E A → B (cid:0) u A (cid:1) Y B (cid:0) ρ B (cid:1) T Y B (cid:105) u B . This is exactly Eq. (4).
Appendix D: The main result
In this Appendix we prove the main result of this letter, namely that a supermap can be part of a super-instrumentif and only if it is completely CPTNI-preserving. To this end, it is useful to consider the SDP (8), reported here forthe reader’s convenience. Find α = max M Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) Subject to: M AB ≥ M A B = I A ⊗ ρ B . Theorem 4.
Suppose Θ A → B is CPTNI-preserving supermap. Then there exists another CPTNI-preserving supermap Θ (cid:48) A → B such that Θ A → B + Θ (cid:48) A → B is a superchannel if and only if Θ A → B is completely CPTNI-preserving.Proof.
First we will show sufficiency, namely that any completely CPTNI-preserving supermap Θ A → B can be com-pleted to a superchannel. Following [64], let us write the SDP (8) in a different form. To do so, consider the linearmap L : B h (cid:0) H AB (cid:1) → R ⊕ B h (cid:0) H A B (cid:1) , defined as L (cid:0) M AB (cid:1) := (cid:0) Tr (cid:2) M AB (cid:3) , M A B − u A ⊗ M B (cid:1) M AB . We are working with with positive semi-definite matrices M AB with marginal M A B = I A ⊗ ρ B , where ρ B ∈ D (cid:0) H B (cid:1) , whence Tr (cid:2) M AB (cid:3) = Tr A B (cid:8) Tr A (cid:2) M AB (cid:3)(cid:9) = Tr (cid:2) M A B (cid:3) = Tr (cid:2) I A ⊗ ρ B (cid:3) = | A | . In addition, M B = Tr A (cid:2) M AB (cid:3) = Tr A (cid:2) M A B (cid:3) = Tr A (cid:2) I A ⊗ ρ B (cid:3) = | A | ρ B . Using L , we can replace the condition M A B = I A ⊗ ρ B with L (cid:0) M AB (cid:1) − (cid:0) | A | , A B (cid:1) = (cid:0) , A B (cid:1) . Rewritingthe SDP (8) in terms of L one obtainsFind α = max M Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) Subject to: L (cid:0) M AB (cid:1) − (cid:0) | A | , A B (cid:1) = (cid:0) , A B (cid:1) M AB ≥ . We can now construct the associated dual problem as follows. The dual map of L is L ∗ : R ⊕ B h (cid:0) H A B (cid:1) → B h (cid:0) H AB (cid:1) such that L ∗ (cid:0) r, σ A B (cid:1) = (cid:0) rI A B + σ A B − u A ⊗ σ B (cid:1) ⊗ I A , where (cid:0) r, σ A B (cid:1) ∈ R ⊕ B h (cid:0) H A B (cid:1) . The dual problem is thenFind β = min (cid:10)(cid:0) r, σ A B (cid:1) , ( | A | , (cid:11) Subject to: (cid:0) rI A B + σ A B − u A ⊗ σ B (cid:1) ⊗ I A − J AB Θ ≥ r ∈ R σ A B ∈ B h (cid:0) H A B (cid:1) , where the inner product (cid:10)(cid:0) r, σ A B (cid:1) , (cid:0) s, τ A B (cid:1)(cid:11) is defined as (cid:10)(cid:0) r, σ A B (cid:1) , (cid:0) s, τ A B (cid:1)(cid:11) = rs + Tr (cid:2) σ A B τ A B (cid:3) . With this in mind, the dual problem simplifies toFind β = | A | min r Subject to: (cid:0) rI A B + σ A B − u A ⊗ σ B (cid:1) ⊗ I A ≥ J AB Θ (D1) r ∈ R σ A B ∈ B h (cid:0) H A B (cid:1) . Notice that the matrix rI A B + σ A B − u A ⊗ σ B must be positive semi-definite, otherwise the first constraint couldnot be satisfied. In particular this implies r ≥ . Indeed, if r < , for some σ A B the matrix rI A B + σ A B − u A ⊗ σ B would have negative eigenvalues. Factoring r | A | out of the first term of the constraint in Eq. (D1), we get (cid:0) rI A B + σ A B − u A ⊗ σ B (cid:1) ⊗ I A = r | A | (cid:0) u A ⊗ I B + σ (cid:48) A B − u A ⊗ σ (cid:48) B (cid:1) ⊗ I A , where σ (cid:48) A B := r | A | σ A B if r (cid:54) = 0 . Note that this does not alter the constraint on the dual SDP, so we can forgetthe primes, and rewrite Eq. (D1) asFind β = | A | min r Subject to: r | A | (cid:0) u A ⊗ I B + σ A B − u A ⊗ σ B (cid:1) ⊗ I A ≥ J AB Θ r ≥ σ A B ∈ B h (cid:0) H A B (cid:1) . In particular this implies that u A ⊗ I B + σ A B − u A ⊗ σ B ≥ . Now let us define J AB Φ := (cid:0) u A ⊗ I B + σ A B − u A ⊗ σ B (cid:1) ⊗ I A . (D2)4Note that J AB Φ = J A B Φ ⊗ u A because J A B Φ = Tr A (cid:104) J AB Φ (cid:105) = | A | (cid:0) u A ⊗ I B + σ A B − u A ⊗ σ B (cid:1) . Moreover, J A B Φ = Tr A (cid:104) J AB Φ (cid:105) = (cid:0) I B + σ B − σ B (cid:1) ⊗ I A = I A B . Since J A B Φ ≥ , by Appendix B 1 J AB Φ is the marginal Choi matrix of a superchannel Φ A → B . Eq. (D2) can be takenas the definition of the marginal J AB Φ of the Choi matrix of any superchannel. This is because any such marginal J AB Φ can be written as in Eq. (D2) for some hermitian matrix σ A B : it is enough to take σ A B to be | A | J A B Φ .Indeed, substituting σ A B = | A | J A B Φ in the right-hand side of Eq. (D2) yields | A | (cid:18) u A ⊗ I B + 1 | A | J A B Φ − | A | u A ⊗ J B Φ (cid:19) ⊗ u A = (cid:16) | A | u A ⊗ I B + J A B Φ − u A ⊗ J B Φ (cid:17) ⊗ u A = (cid:16) | A | u A ⊗ I B + J A B Φ − u A ⊗ Tr AB (cid:2) J AB Φ (cid:3)(cid:17) ⊗ u A = (cid:16) | A | u A ⊗ I B + J A B Φ − u A ⊗ Tr A (cid:8) Tr A B (cid:2) J AB Φ (cid:3)(cid:9)(cid:17) ⊗ u A = (cid:16) | A | u A ⊗ I B + J A B Φ − u A ⊗ Tr A (cid:104) J A B Φ (cid:105)(cid:17) ⊗ u A = (cid:16) | A | u A ⊗ I B + J A B Φ − u A ⊗ Tr A (cid:2) I A B (cid:3)(cid:17) ⊗ u A = (cid:16) | A | u A ⊗ I B + J A B Φ − | A | u A ⊗ I B (cid:17) ⊗ u A = J A B Φ ⊗ u A . Therefore, in the light of these remarks, the dual SDP can be equivalently formulated in the following terms:Find β = | A | min r Subject to: r | A | J A B Φ ⊗ u A ≥ J AB Θ J A B Φ ≥ J A B Φ = I A B r ≥ . Strong duality states that the primal and dual problem have the same optimal solution, therefore α = β . Since Θ A → B is completely CPTNI-preserving, α = max M Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) ≤ . Hence β ≤ . Clearly taking r | A | = β satisfies the constraint r | A | J A B Φ ⊗ u A ≥ J AB Θ , and we have J A B Φ ⊗ u A ≥ β J A B Φ ⊗ u A ≥ J AB Θ because β ≤ . Now define Θ (cid:48) A → B to be a new supermap such that J AB Θ (cid:48) := J A B Φ ⊗ u A − J AB Θ . By construction J AB Θ (cid:48) ≥ ; and by substituting J AB Θ (cid:48) into the left-hand side of Eq. (6) one obtains Tr (cid:104) J AB Θ (cid:48) (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) = Tr (cid:104)(cid:16) J AB Φ − J AB Θ (cid:17) (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) = Tr (cid:104) J AB Φ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) − Tr (cid:104) J AB Θ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) = 1 − Tr (cid:104) J AB Θ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) , where we have used the fact that Φ A → B is a superchannel (see Appendix B 1). Now, Tr (cid:104) J AB Θ (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) ≥ because Θ A → B is CPP. Therefore Tr (cid:104) J AB Θ (cid:48) (cid:0) J A N ⊗ ρ B (cid:1) T (cid:105) ≤ for every J A N ⊗ ρ B , thus Θ (cid:48) A → B is CPTNI-preserving.To conclude the proof, let us prove necessity. Assume that Θ A → B is a CPTNI-preserving supermap such that Φ A → B = Θ A → B + Θ (cid:48) A → B is a superchannel, where Θ (cid:48) A → B is another CPTNI-preserving supermap. We will provethat Θ A → B must be completely CPTNI-preserving. In the Choi picture we have J AB Θ + J AB Θ (cid:48) = J AB Φ . (D3)5Let us multiply both sides of Eq. (D3) by the transpose of any matrix M AB ≥ with marginal M A B = I A ⊗ ρ B , ρ B ∈ D (cid:0) H B (cid:1) , and then take the trace. Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) + Tr (cid:104) J AB Θ (cid:48) (cid:0) M AB (cid:1) T (cid:105) = Tr (cid:104) J BC Φ (cid:0) M AB (cid:1) T (cid:105) (D4)By the results in Appendix B 2, the right-hand side is 1 because Φ A → B is a superchannel. Thus Eq. (D4) becomes Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) + Tr (cid:104) J AB Θ (cid:48) (cid:0) M AB (cid:1) T (cid:105) = 1 , which implies Tr (cid:104) J AB Θ (cid:0) M AB (cid:1) T (cid:105) ≤ for all M AB because Θ A → B is CPP. Therefore Θ A → B satisfies Eq. (7), whichmeans that it is completely CPTNI-preserving. This concludes the proof.Applying the statement of this theorem to Θ (cid:48) A → B , we get that Θ (cid:48) A → B is completely CPTNI-preserving too. Appendix E: Quantum super-instruments
In this section we re-derive one of the results of [14], but in a different way. This new proof is based on our mainresult: every completely CPTNI-preserving supermap can be completed to a superchannel. Specifically, we show thateach completely CPTNI-preserving supermap Θ A → Bx in a super-measurement (cid:8) Θ A → Bx (cid:9) can be expressed in terms ofa CPTP pre-processing channel, independent of x , and a CPTNI post-processing map, as depicted in Fig. 3. Proposition 5.
The Choi matrix J AB Θ x of each completely CPTNI-preserving supermap Θ A → Bx in a super-measurement (cid:8) Θ A → Bx (cid:9) x ∈ X can be written in terms of a common CPTP pre-processing map Γ (cid:101) B → A E pre , and a CPTNI post-processingmap Γ (cid:101) A E → B post x as J AB Θ x = (cid:16) I AB ⊗ Γ (cid:101) A E → B post x (cid:17) ◦ (cid:16) I A (cid:101) A B ⊗ Γ (cid:101) B → A E pre (cid:17) (cid:16) φ B (cid:101) B + ⊗ φ A (cid:101) A + (cid:17) . Proof.
Define Θ A → B := (cid:80) x ∈ X Θ A → Bx , which we know to be a superchannel. In the Choi picture this is translatedinto J AB Θ = (cid:80) x J AB Θ x . In [38, theorem 1] one of the authors showed that the Choi matrix of a superchannel can bewritten in terms of its pre-processing Γ B → A E pre and post-processing Γ A E → B post as J AB Θ = (cid:16) I AB ⊗ Γ (cid:101) A E → B post (cid:17) (cid:16) ψ A B E ⊗ φ A (cid:101) A + (cid:17) , where ψ A B E := (cid:16) I AB ⊗ Γ (cid:101) B → A E pre (cid:17) (cid:16) φ B (cid:101) B + (cid:17) . ψ A B E can be shown to be a purification of | A | J A B Θ [38]. Now,summing over all outcomes x ∈ X , let us construct the matrix (cid:80) x ∈ X | x (cid:105) (cid:104) x | X ⊗ J AB Θ x , where (cid:110) | x (cid:105) X (cid:111) | X | x =1 is anorthonormal basis of H X . Let ϕ X ABF be a purification of (cid:80) x | x (cid:105) (cid:104) x | X ⊗ J AB Θ x . Note that ϕ X ABF is a purificationof J AB Θ too, because Tr X F (cid:2) ϕ X ABF (cid:3) = Tr X (cid:8) Tr F (cid:2) ϕ X ABF (cid:3)(cid:9) = Tr X (cid:34)(cid:88) x | x (cid:105) (cid:104) x | X ⊗ J AB Θ x (cid:35) = (cid:88) x J AB Θ x = J AB Θ . If we take the isometry V (cid:101) A E → B G to be a Stinespring dilation of Γ (cid:101) A E → B post , namely Γ (cid:101) A E → B post = Tr G ◦V (cid:101) A E → B G ,then χ ABG := (cid:16) I AB ⊗ V (cid:101) A E → B G (cid:17) (cid:16) ψ A B E ⊗ φ A (cid:101) A + (cid:17) is another purification of J AB Θ . Indeed, Tr G (cid:2) χ ABG (cid:3) = I AB ⊗ (cid:16) Tr G ◦ V (cid:101) A E → B G (cid:17) (cid:16) ψ A B E ⊗ φ A (cid:101) A + (cid:17) = (cid:16) I AB ⊗ Γ (cid:101) A E → B post (cid:17) (cid:16) ψ A B E ⊗ φ A (cid:101) A + (cid:17) = J AB Θ . ϕ X ABF and χ ABG are purifications of J AB Θ , they are related by an isometry channel U G → X F suchthat ϕ X ABF = (cid:0) I AB ⊗ U G → X F (cid:1) (cid:0) χ ABG (cid:1) = (cid:0) I AB ⊗ U G → X F (cid:1) (cid:16) I AB ⊗ V (cid:101) A E → B G (cid:17) (cid:16) ψ A B E ⊗ φ A (cid:101) A + (cid:17) =: (cid:16) I AB ⊗ W (cid:101) A E → B X F (cid:17) (cid:16) ψ A B E ⊗ φ A (cid:101) A + (cid:17) , where we have defined W (cid:101) A E → B X F := U G → X F ◦ V (cid:101) A E → B G , which is another isometry channel, and anotherStinespring dilation of Γ (cid:101) A E → B post . Now let us trace out system F , recalling that Tr F (cid:2) ϕ X ABF (cid:3) = (cid:80) y | y (cid:105) (cid:104) y | X ⊗ J AB Θ y ,where we have changed the index from x to y for convenience. We get (cid:88) y | y (cid:105) (cid:104) y | X ⊗ J AB Θ y = (cid:16) I AB ⊗ (cid:101) Γ (cid:101) A E → B X (cid:17) (cid:16) ψ A B E ⊗ φ A (cid:101) A + (cid:17) , (E1)where (cid:101) Γ (cid:101) A E → B X := Tr F ◦ W (cid:101) A E → B X F is a CPTP map. To get J AB Θ x we apply the projector | x (cid:105) (cid:104) x | X to bothsides of Eq. (E1), tracing over X . J AB Θ x = Tr X (cid:104) | x (cid:105) (cid:104) x | X (cid:16) I AB ⊗ (cid:101) Γ (cid:101) A E → B X (cid:17) (cid:16) ψ A B E ⊗ φ A (cid:101) A + (cid:17)(cid:105) = (cid:104) I AB ⊗ (cid:16) Tr X | x (cid:105) (cid:104) x | X ◦ (cid:101) Γ (cid:101) A E → B X (cid:17)(cid:105) (cid:16) ψ A B E ⊗ φ A (cid:101) A + (cid:17) . (E2)Now let us define Γ (cid:101) A E → B post x := Tr X | x (cid:105) (cid:104) x | X ◦ (cid:101) Γ (cid:101) A E → B X , which is a CPTNI map whose action on a density matrix ρ (cid:101) A E is Γ (cid:101) A E → B post x (cid:16) ρ (cid:101) A E (cid:17) = (cid:68) x (cid:12)(cid:12)(cid:12)(cid:101) Γ (cid:101) A E → B X (cid:16) ρ (cid:101) A E (cid:17)(cid:12)(cid:12)(cid:12) x (cid:69) X . Therefore Eq. (E2) becomes J AB Θ x = (cid:16) I AB ⊗ Γ (cid:101) A E → B post x (cid:17) (cid:16) ψ A B E ⊗ φ A (cid:101) A + (cid:17) . Recalling ψ A B E = (cid:16) I AB ⊗ Γ (cid:101) B → A E pre (cid:17) (cid:16) φ B (cid:101) B + (cid:17) , where Γ (cid:101) B → A E pre is the pre-processing of the superchannel Θ A → B ,we get the thesis J AB Θ x = (cid:16) I AB ⊗ Γ (cid:101) A E → B post x (cid:17) ◦ (cid:16) I A (cid:101) A B ⊗ Γ (cid:101) B → A E pre (cid:17) (cid:16) φ B (cid:101) B + ⊗ φ A (cid:101) A + (cid:17) . Therefore we can realize every completely CPTNI-preserving supermap Θ A → Bx that is part of a quantum super-instrument as a quantum 1-comb, like in Fig. 3. More precisely we have Θ A → Bx = B Γ pre A A Γ post x B E , (E3)where Γ B → A E pre is the CPTP pre-processing of the superchannel Θ A → B = (cid:80) x Θ A → Bx . The pre-processing of acompletely CPTNI-preserving supermap is therefore independent of x and common to all the supermaps in thesame quantum super-instrument. In fact, even the post-processing is almost shared by all supermaps in the samesuper-instrument: it is given by Γ A E → B post x = Tr X | x (cid:105) (cid:104) x | X ◦ (cid:101) Γ A E → B X , namely by a reading performed on theclassical output X of (cid:101) Γ A E → B X . (cid:101) Γ A E → B X depends only on the superchannel Θ A → B , so it is common to allthe supermaps in the same super-instrument. Eq. (E3) then becomes Θ A → Bx = B Γ pre A A (cid:101) Γ B E X x . From the proof of proposition 5 we have Tr X ◦ (cid:101) Γ A E → B X = Tr X F ◦ W A E → B X F = Γ A E → B post (E4)7because W A E → B X F is a Stinespring dilation of Γ A E → B post . Therefore if we forget the outcome x of the super-measurement, Eq. (E4) yields (cid:88) x Γ A E → B post x = Tr X ◦ (cid:101) Γ A E → B X = Γ A E → B post , and we recover the post-processing channel Γ A E → B post of Θ A → B . Appendix F: OPT interpretation of the result
The theory of quantum supermaps, where generic evolutions of quantum maps are described by supermaps, can beanalyzed using the framework of operational-probabilistic theories (OPTs) [57, 60–62, 65–67], which is a formalismto describe arbitrary physical theories admitting probabilistic processes. OPTs differ from the convex set approachto general probabilistic theories [68–70] in that they take the composition of physical processes and systems as aprimitive. Mathematically, this is based on the graphical language of circuits [71–74] and probability theory.
1. The general framework
OPTs describe the experiments that can be performed on a given set of systems by a given set of physical processes.The framework is based on a primitive notion of composition, whereby every pair of physical systems A and B canbe combined into a composite system AB . Physical processes can be connected in sequence or in parallel to buildcircuits, in the very same way as the corresponding devices are connected in a laboratory to build an experiment. Forinstance, ρ A A A (cid:48) A (cid:48) A (cid:48)(cid:48) a B B B (cid:48) b . (F1)In this example, A , A (cid:48) , A (cid:48)(cid:48) , B , and B (cid:48) are systems , ρ is a bipartite state , A , A (cid:48) and B are transformations , a and b are effects . Note that inputs are on the left and outputs on the right.For generic systems A and B , we denote by • St (A) the set of states of system A , • Eff (A) the set of effects on A , • Transf (A , B) the set of transformations from A to B , • B ◦ A (or BA , for short) the sequential composition of two transformations A and B , with the input of B matching the output of A , • I A the identity transformation on system A , represented by the plain wire A , • A ⊗ B the parallel composition (or tensor product) of the transformations A and B .Among the list of valid physical systems, every OPT includes the trivial system I , corresponding to the degrees offreedom ignored by theory and to the lack of input (or output) system. States (resp. effects) are transformations withthe trivial system as input (resp. output).A circuit with no external wires, like in Eq. (F1), is identified with a real number in the interval [0 , , interpreted asthe probability of the joint occurrence of all the transformations present in the circuit. We will often use the notation ( a | ρ ) to denote the circuit ( a | ρ ) := ρ A a , and the notation ( b |C| ρ ) to mean the circuit ρ A C B b . Let us clarify these concepts in quantum theory.8
Example 6.
In quantum theory we associate a Hilbert space H A with every system A . States are positive semi-definite operators ρ with Tr [ ρ ] ≤ . The reason why we also consider states with trace less than 1 will be explainedin example 8. An effect is, instead, represented by a positive semi-definite operator E , with E ≤ I , where I is theidentity operator. The pairing between states and effects is given by the trace: ( E | ρ ) = Tr [ Eρ ] .The fact that some circuits represent real numbers induces a notion of sum for transformations, so that the sets St (A) , Transf (A , B) , and Eff (A) become spanning sets of real vector spaces. We will denote the vector space of statesas St R (A) and the vector space of transformations as Transf R (A , B) . Effects become linear functionals on St R (A) ,and transformations in Transf (A , B) are linear transformations from St R (A) to St R (B) .If we restrict ourselves to linear combinations of states with non-negative coefficients (conical combinations), weobtain a proper convex cone [57], called the cone of states St + (A) . Note that effects take non-negative values on thecone of states. Indeed if ξ ∈ St + (A) , then ξ is a conical combination of some states ρ i : ξ = (cid:80) i λ i ρ i , where λ i ≥ forevery i . Therefore when an effect a acts on ξ , we have ( a | ξ ) = (cid:88) i λ i ( a | ρ i ) ≥ as λ i ≥ , and ≤ ( a | ρ i ) ≤ , because an effect yields a probability when applied to a state. Example 7.
In quantum theory, St R (A) is the vector space of hermitian matrices on H A , and St + (A) is the cone ofpositive semi-definite matrices.In general, an experiment in a laboratory can be non-deterministic, i.e. it can result into a set of alternativetransformations applied to the input system, heralded by different outcomes, which can (at least in principle) beaccessed by an experimenter. General non-deterministic processes are described by tests : a test from A to B is acollection of transformations {C x } x ∈ X from A to B , where X is the set of outcomes. If A (resp. B ) is the trivial system,the test is called a preparation-test (resp. observation-test ). If the set of outcomes X contains a single element, wesay that the test is deterministic , because only one transformation can occur, and we can predict the outcome of theexperiment. We refer to deterministic transformations as channels . If we sum over all the transformations in a test weget a deterministic transformation, viz. a channel: C := (cid:80) x ∈ X C x . This is because the sum of all the transformationsarising in a test can be viewed as the full coarse-graining over all outcomes [57], resulting in a new, deterministic,test. Example 8.
In quantum theory, a channel from A to A is a CPTP map from B (cid:0) H A (cid:1) to B (cid:0) H A (cid:1) . A test from A to A is a collection of CPTNI maps from B (cid:0) H A (cid:1) to B (cid:0) H A (cid:1) summing to a CPTP map. Note that this isconsistent with the fact that the sum over all the transformations in a test yields a channel.Deterministic states are positive semi-definite operators ρ with Tr [ ρ ] = 1 . A non-deterministic preparation-test is acollection of positive semi-definite operators ρ i with Tr [ ρ i ] < (non-deterministic states) that sum to a deterministicstate ρ . This is essentially a random preparation: a state ρ i is prepared with a probability given by Tr [ ρ i ] . This iswhy we consider all positive semi-definite operators ρ with Tr [ ρ ] ≤ as states.Observation-tests are POVMs. In quantum theory there is only one deterministic effect: the identity I (moreprecisely it is the functional Tr [ I • ] ). This is not a coincidence, but it follows from the fact that quantum theory is acausal theory (see definition 9).Among all theories, causal theories [57] are particularly important: in these theories, loosely speaking, informationcannot come back from the future. They are particularly simple in their structure, and generally speaking they arewell understood. Causality can also be shown to imply no-signaling in space-like separated systems [57]. The formalstatement of the property of causality is as follows. Axiom 9 (Causality [57]) . For every state ρ , take two observation-tests { a x } x ∈ X and { b y } y ∈ Y . One has (cid:88) x ∈ X ( a x | ρ ) = (cid:88) y ∈ Y ( b y | ρ ) . Causality can be equivalently characterized in terms of deterministic effects: an OPT is causal if and only if, forevery system A , there is a unique deterministic effect u A [57]. This characterization is very practical to work with. Example 10.
In quantum theory there is only one deterministic effect, the identity operator (or the trace functional).Hence quantum theory is causal.9Causal theories enjoy an important property: the unique deterministic effect for a composite system AB alwaysfactorizes as the parallel composition of the deterministic effects on A and on B . In symbols, u AB = u A ⊗ u B . Thisis because if u A and u B are the deterministic effects of A and B , then u A ⊗ u B is a deterministic effect on AB . Sincethe theory is causal, there is a unique deterministic effect on AB , so u A ⊗ u B is the deterministic effect of AB .Moreover in causal theories there is a nice characterization of channels: a transformation C ∈
Transf (A , B) is achannel if and only if [57] u B C = u A . (F2)In quantum theory, since u is the trace, this condition amounts to saying that channels are trace-preserving.Let us conclude this section showing how the theory of quantum supermaps fits into the OPT formalism. Example 11.
In the theory of quantum supermaps, every system A is a pair of input and output quantum systems A = ( A , A ) ; deterministic states are CPTP maps, and non-deterministic ones are CPTNI maps. The cone of statesis given by all CP maps. Transformations in this theory are supermaps [14, 16, 38, 55, 56]. As our results show, it isnot immediate to pin down the mathematical properties that make a generic linear supermap from A to B physical.We will analyze this issue from the OPT perspective in the next subsection.Now let us show that the theory of quantum supermaps is not causal. Suppose we want to construct a deterministiceffect in this theory. According to [14, 15, 19], to this end it is enough to consider a 1-comb made of deterministicquantum operations, which means a circuit fragment of the form B A A A S , where both A and A are deterministic quantum operations. Since this comb must output a probability, its pre-processing A must be a deterministic bipartite quantum state ρ ∈ St (A S) , and its post-processing A must be adeterministic bipartite quantum effect u ∈ Eff (A S) , for some system S : ρ A A u S . Now recall that in causal theories the deterministic effect of a bipartite system A S factorizes as u A ⊗ u S , and that u is nothing but the trace (cf. example 8). Then ρ A A u S = ρ A A Tr S Tr = ρ (cid:48) A A Tr , where ρ (cid:48) = Tr S [ ρ ] . In this way, for any choice of ρ ∈ D (cid:0) H A S (cid:1) we obtain all quantum states ρ (cid:48) ∈ D (cid:0) H A (cid:1) . Thereforethe generic deterministic effect on system A = ( A , A ) of the theory of quantum supermaps is of the form u ρ = ρ A A Tr , for any quantum state ρ ∈ D (cid:0) H A (cid:1) . This means that there is a whole family of deterministic effects, labelled byquantum states. Therefore the theory of quantum supermaps is not causal, a fact that is confirmed by the presenceof signalling bipartite quantum channels [58]. The failure of causality implies here that there are some deterministiceffects for a bipartite system AB = ( A , A ) ( B , B ) that do not factorize. Indeed, if we take a non-product bipartitequantum state ρ ∈ D (cid:0) H A B (cid:1) , the associated deterministic effect is u ρ = ρ A A Tr B B Tr , (F3)which does not factorize. This fact will play an important role in Appendix F 2, and it is ultimately the reason whywe need the CPTNI preservation condition in a complete sense.
2. Necessary conditions for physical transformations
In the OPT approach, however we construct a diagram, this represents a physical object: a valid state, a validtransformation, a valid effect. Specializing our analysis to transformations from a system A to a system B , a linear0map A from St R (A) to St R (B) is a valid physical transformation if and only if ρ A A BS (F4)is a valid state of system BS , for every choice of ρ and S . Here we will derive some necessary conditions to guaranteethis. In particular if (F4) is a valid state, for every bipartite effect E ∈ Eff (BS) we have ≤ ρ A A B E S ≤ , (F5)because this is the probability of E occurring on ( A ⊗ I S ) ρ . Remark . Condition (F5) is only necessary , but in general not sufficient to guarantee that (F4) represents a validphysical state. Indeed, the theory may have additional restrictions on the allowed states, as it happens in the presenceof superselection rules [75–79]. If the theory is completely unrestricted, like quantum theory or the theory of quantumsupermaps, condition (F5) is sufficient as well.Let us analyze the two inequalities in (F5) separately. If ( A ⊗ I S ) ρ is in the cone of states of BS , then we immediatelyhave ρ A A B E S ≥ , for every effect E ∈ Eff (BS) . Definition 13.
We say that a transformation A in Transf R (A , B) is completely positive if, for every system S andevery element ξ ∈ St + (AS) , we have ( A ⊗ I S ) ξ ∈ St + (BS) .In words, a completely positive transformation is a linear transformation that maps elements in the input cone ofstates to elements in the output cone of states in a complete sense, i.e. even when there is an ancillary system S . Thisis clearly a necessary condition for a transformation to be physical.Note that it is equivalent to define complete positivity just on states in St (AS) , instead of on generic elementsof ξ ∈ St + (AS) : A is completely positive if and only if, for every system S and every state ρ ∈ St (AS) , we have ( A ⊗ I S ) ρ ∈ St + (BS) . To see the non-trivial implication, recall that if ξ is a generic element of St + (AS) , it can bewritten as a conical combination of states ρ i of AS : ξ = (cid:80) i λ i ρ i , with λ i ≥ for every i . Then, if we know that ( A ⊗ I S ) ρ ∈ St + (BS) for every ρ ∈ St (AS) , we have ( A ⊗ I S ) ξ = (cid:88) i λ i ( A ⊗ I S ) ρ i ∈ St + (BS) , because St + (BS) is closed under conical combinations. Example 14.
In quantum theory, the cone of states is the cone of positive semi-definite operators; therefore completelypositive transformations in the sense of definition 13 are exactly CP maps.In the theory of quantum supermaps, the cone of states is the cone of CP maps. In this case, completely positivetransformations are CPP supermaps [14, 38].Now let us analyze the second inequality in (F5), namely ρ A A B E S ≤ , (F6)for every effect E ∈ Eff (BS) . Assume A is completely positive. Then, demanding the validity of inequality (F6) forevery state ρ ∈ St (AS) and every effect E ∈ Eff (BS) is equivalent to demanding its validity when ρ is any deterministic state and E any deterministic effect. To see the non-trivial implication, recall that if ρ is non-deterministic, it arises1in a preparation-test { ρ, ρ (cid:48) } . Similarly, if E is non-deterministic, it arises in an observation-test { E, E (cid:48) } . Clearly (cid:101) ρ = ρ + ρ (cid:48) is a deterministic state, and (cid:101) E = E + E (cid:48) is a deterministic effect. Then ≥ (cid:101) ρ A A B (cid:101) E S = ρ A A B E S + ρ A A B E (cid:48) S + ρ (cid:48) A A B E S + ρ (cid:48) A A B E (cid:48) S . Now, each term in the right-hand side is non-negative because A is completely positive. It follows that each term isalso less than or equal to 1, and specifically ρ A A B E S ≤ . We summarize these necessary requirements in the following theorem.
Theorem 15.
Let
A ∈
Transf R (A , B) . Then A is a physical transformation only if both these conditions are satisfied:1. ( A ⊗ I S ) ρ ∈ St + (BS) for every system S and every state ρ ∈ St (AS) ;2. ρ A A B u S ≤ , for every system S , every deterministic state ρ ∈ St (AS) , and every deterministic effect u ∈ Eff (BS) . Note that in particular, condition 2 implies that ρ A A B u ≤ , (F7)for it is enough to take S to be the trivial system I . However, in general, this condition is weaker than condition 2,such as in the theory of quantum supermaps. Let us analyze the role of conditions 1, 2, and (F7) in this theory. Example 16.
First of all, since the theory of quantum supermaps has no restrictions, the conditions in theorem 15become sufficient as well. We have already examined condition 1. Let us focus on condition 2, and unfold its meaning.In this case, ρ is actually a bipartite channel N , and A acts as a supermap Θ on half of N . Recalling Eq. (F3),condition 2 becomes Tr B S (cid:2)(cid:0) Θ A → B ⊗ S (cid:1) (cid:2) N AS (cid:3) (cid:0) ρ B S (cid:1)(cid:3) ≤ . This is nothing but requiring that Θ be completelyCPTNI-preserving (cf. Eq. (5)).In conclusion, the two conditions of theorem 15 are exactly the two conditions we found in this letter. Note thatcondition (F7), expressing CPTNI preservation (but not in a complete sense), is weaker than condition 2, as there isno way to recover condition 2 from condition (F7). This is essentially because not all bipartite deterministic effectscan be reduced to single-system deterministic effects (cf. Eq. (F3)). Thus condition (F7) cannot be used to assesswhether a candidate supermap is physical or not, and CPTNI preservation is not enough.If theorem 15 is valid in all physical theories, why do we not need to impose the trace non-increasing condition ina complete sense in quantum theory? This is because the theory is causal. Indeed in all causal theories, condition 2becomes equivalent to condition (F7). Proposition 17.
In a causal theory with deterministic effect u , one has ρ A A B u S ≤ , for every system S and every deterministic state ρ ∈ St (AS) , if and only if ρ A A B u ≤ . for every deterministic state ρ ∈ St (A) . Proof.
We have already seen one implication (necessity), now let us focus on the other. Assume condition (F7) holds.Take an arbitrary system S and an arbitrary deterministic state Σ ∈ St (AS) . Then Σ A A B u S = Σ A A B u S u = ρ A A B u ≤ , where we have used the fact that the deterministic effect of a composite system factorizes, and that Σ AS u =: ρ A is a deterministic state.In other words, for causal theories condition 2 can be formulated only for single systems, without the need of anancillary system S . Recall that in quantum theory u is the trace, so condition (F7) means that A is trace-non-increasing. Proposition 17 is the ultimate reason why in quantum theory it is enough to require that a CP map beTNI (on single system) rather than completely TNI. In conclusion, the ultimate origin of the unexpected behavior ofthe theory of quantum supermaps is the failure of causality.However, in [38] one of the authors showed that for a CPP map to be a superchannel, instead, it is not necessaryto demand that it be completely TPP, but it is enough that it be TPP. Why do we not need CPTP preservation in acomplete sense for superchannels? Let us understand it using the OPT formalism.Clearly, a superchannel Θ A → B must send channels to channels in a complete sense: for any bipartite quantumchannel N AB , R ⊗ Θ A → B (cid:2) N RA (cid:3) = M RB , where M RB is still a quantum channel. By Eq. (F2), this is true if andonly if (Tr R ⊗ Tr B ) ◦ (cid:0) R ⊗ Θ A → B (cid:2) N RA (cid:3)(cid:1) = Tr R ⊗ Tr B , (F8)where we have denoted the deterministic effect u explicitly as the trace. Now let us try to prove Eq. (F8) knowingthat Θ A → B is just TPP. Now consider the following channel A N (cid:48) A := ρ N R Tr A A , (F9)where ρ is some density matrix on R . Since Θ A → B is TPP, we have that M (cid:48) B := Θ A → B (cid:2) N (cid:48) A (cid:3) is still a quantumchannel. In other words Tr B ◦ Θ A → B (cid:2) N (cid:48) A (cid:3) = Tr B . Then if we take a density matrix σ ∈ D (cid:0) H B (cid:1) , we have Tr B ◦ Θ A → B (cid:2) N (cid:48) A (cid:3) (cid:16) σ B (cid:17) = Tr B (cid:104) σ B (cid:105) = 1 . Now, recalling the definition of N (cid:48) A in Eq. (F9), we have Tr R Tr B (cid:0) R ⊗ Θ A → B (cid:2) N RA (cid:3)(cid:1) (cid:16) ρ R ⊗ σ B (cid:17) = 1 , (F10)for any ρ ∈ D (cid:0) H R (cid:1) and any σ ∈ D (cid:0) H B (cid:1) . If we manage to prove that Tr R Tr B (cid:0) R ⊗ Θ A → B (cid:2) N RA (cid:3)(cid:1) (cid:0) τ R B (cid:1) = 1 for every bipartite state τ R B , then the validity of Eq. (F8) is shown. Now, recall that in quantum theory everybipartite state can be written as an affine combination of product states. Therefore τ R B = (cid:80) j λ j ρ R j ⊗ σ B j , with (cid:80) j λ j = 1 . Therefore Tr R Tr B (cid:0) R ⊗ Θ A → B (cid:2) N RA (cid:3)(cid:1) (cid:0) τ R B (cid:1) = Tr R Tr B (cid:0) R ⊗ Θ A → B (cid:2) N RA (cid:3)(cid:1) (cid:88) j λ j ρ R j ⊗ σ B j = (cid:88) j λ j Tr R Tr B (cid:0) R ⊗ Θ A → B (cid:2) N RA (cid:3)(cid:1) (cid:16) ρ R j ⊗ σ B j (cid:17) = (cid:88) j λ j = 1 , Tr R Tr B (cid:0) R ⊗ Θ A → B (cid:2) N RA (cid:3)(cid:1) (cid:16) ρ R ⊗ σ B (cid:17) ≤ , we cannot conclude that Tr R Tr B (cid:0) R ⊗ Θ A → B (cid:2) N RA (cid:3)(cid:1) (cid:0) τ R B (cid:1) ≤ for every bipartite state τ R B . The reason is that we are only dealingwith an affine combination, possibly even containing negative terms. This does not allow us to conclude anythingabout (cid:80) j λ j Tr R Tr B (cid:0) R ⊗ Θ A → B (cid:2) N RA (cid:3)(cid:1) (cid:16) ρ R j ⊗ σ B j (cid:17) .We conclude this Appendix with an interesting remark: sometimes, even with a non-causal theory, the weakercondition (F7) is enough to characterize which completely positive transformations are physical, in that it becomesequivalent to the stronger condition 2 in theorem 15. This happens when the only deterministic states of the theoryare separable [57, 80]: i.e. they can be written as a convex combination of product deterministic states. In this case,suppose we know that condition (F7) holds. Let us assess ( u |A ⊗ I S | Σ) , where S is an arbitrary system, Σ ∈ St (AS) is an arbitrary deterministic state, and u ∈ Eff (BS) is an arbitrary deterministic effect. We have Σ A A B u S = (cid:88) j p j α j A A B uσ j S =: (cid:88) j p j α j A A B u j , where { p j } is a probability distribution, α j and β j are deterministic states, and u j is the deterministic effect definedas u j := u BS ( I B ⊗ σ j, S ) . Now, each term ( u j |A| ρ j ) ≤ by condition (F7), so any convex combination of them willyield a number less than or equal to 1. In this case we were able to prove that condition (F7) implies condition 2 oftheorem 15.We can follow the same argument when, dually, all deterministic effects are separable. Again, let us assumecondition (F7) holds, and let us assess ( u |A ⊗ I S | Σ) , where S is an arbitrary system, Σ ∈ St (AS) is an arbitrarydeterministic state, and u ∈ Eff (BS) is an arbitrary deterministic effect, as above. One has Σ A A B u S = (cid:88) j p j Σ A A B u j, BS u j, S =: (cid:88) j p j σ j A A B u j, B ≤ , where { p j } is a probability distribution, u j, B and u j, S are deterministic effects, and σ j is a deterministic state, definedas σ j := ( I A ⊗ u j, S ) Σ) Σ