Transforming quantum operations: quantum supermaps

Transforming quantum operations: quantum supermaps

Giulio Chiribella QUIT Group, Dipartimento di Fisica “A. Volta” and INFM, via Bassi 6, 27100 Pavia, Italy http://www.qubit.it    Giacomo Mauro D’Ariano QUIT Group, Dipartimento di Fisica “A. Volta” and INFM, via Bassi 6, 27100 Pavia, Italy http://www.qubit.it    Paolo Perinotti QUIT Group, Dipartimento di Fisica “A. Volta” and INFM, via Bassi 6, 27100 Pavia, Italy http://www.qubit.it
July 14, 2019
Abstract

We introduce the concept of quantum supermap, describing the most general transformation that maps an input quantum operation into an output quantum operation. Since quantum operations include as special cases quantum states, effects, and measurements, quantum supermaps describe all possible transformations between elementary quantum objects (quantum systems as well as quantum devices). After giving the axiomatic definition of supermap, we prove a realization theorem, which shows that any supermap can be physically implemented as a simple quantum circuit. Applications to quantum programming, cloning, discrimination, estimation, information-disturbance trade-off, and tomography of channels are outlined.

pacs:
03.65.Ta, 03.67.-a

The input-output description of any quantum device is provided by the quantum operation of Kraus kraus (), which yields the most general probabilistic evolution of a quantum state. Precisely, the output state is given by the quantum operation applied to the input state as follows

 (1)

where is the probability of occurring on state , when is one of a set of alternative transformations, such as in a quantum measurement. Owing to its physical meaning, a quantum operation must be a linear, trace non-increasing, completely positive (CP) map (see, e.g. Nielsen2000 ()). The most general form of such a map is known as Kraus form

 E(ρ)=∑jEjρE†j, (2)

where the operators satisfy the bound so that . Trace-preserving maps, i.e. those achieving the bound, are a particular kind of quantum operations: they occur deterministically and are referred to as quantum channels.

In general it is convenient to consider two different input and output Hilbert spaces and , respectively. In this way, the concept of quantum operation can be used to treat also quantum states, effects, and measurements, which describe the properties of elementary quantum objects such as quantum systems and measuring devices. Indeed, states can be described as quantum operations with one-dimensional , i.e. with Kraus operators given by ket-vectors , thus yielding the output state . A quantum effect ludwig () corresponds instead to a quantum operation with one-dimensional , i.e. with Kraus operators given by bra-vectors , yielding the probability with . More generally, any quantum measurement can be viewed as a particular quantum operation, namely as a quantum-to-classical channel qc ().

Channels, states, effects, and measurements are all special cases of quantum operations. What about then considering maps between quantum operations themselves? They would describe the most general kind of transformations between elementary quantum objects. For example a programmable channel NielsenProg () would be a map of this type, with a quantum state at the input and a channel at the output. Or else, a device that optimally clones a set of unknown unitary gates would be a map from channels to channels. We will call such a general class of quantum maps quantum supermaps, as they transform CP maps (sometimes referred to as superoperators) into CP maps.

In this paper we develop the basic tools to deal with quantum supermaps. The concept of quantum supermap is first introduced axiomatically, by fixing the minimal requirements that a map between quantum operations must fulfill. We then prove a realization theorem that provides any supermap with a physical implementation in terms of a simple quantum circuit with two open ports in which the input operation can be plugged. This result allows one to simplify the description of complex quantum circuits and to prove general theorems in quantum information theory. Moreover, the generality of the concept of supermap makes it fit for application in many different contexts, among which quantum programming, calibration, cloning, and estimation of devices.

To start with, we define the deterministic supermaps as those sending channels to channels. Conversely, a probabilistic supermap will send channels to arbitrary trace-non-increasing quantum operations. The minimal requirements that a deterministic supermap must satisfy in order to be physical are the following: it must be i) linear and ii) completely positive. Linearity is required to be consistent with the probabilistic interpretation. Indeed, if the input is a random choice of quantum operations , the output must be given by the same random choice of the transformed operations , and, if the input is the quantum operation with probability , the output must be the with probability , implying . Clearly, these two conditions imply that is a linear map on the linear space generated by quantum operations. Complete positivity is needed to ensure that the output of is a legitimate quantum operation even when is applied locally to a bipartite joint quantum operation, i.e. a quantum operation with bipartite input space and bipartite output space . If is a supermap transforming quantum operations with input (output) space (), complete positivity corresponds to require that is a CP map for any bipartite quantum operation , denoting the identity supermap on the spaces labeled by .

In order to deal with complete positivity it is convenient to use the Choi representation choi () of a CP map in terms of the positive operator on

 E:=E⊗I(|I⟩⟨I|), (3)

where is the maximally entangled vector , an orthonormal basis, and is the identity operation. The correspondence is one-to-one, the inverse relation of Eq. (3) being

 E(ρ):=TrHin[(I⊗ρ⊺)E], (4)

where denotes transposition in the basis . In terms of the Choi operator, the probability of occurrence of is given by , where is the effect . To have unit probability on any state, a quantum channel must have , i.e. its Choi operator must satisfy the normalization

 TrHout[E]=IHin . (5)

A supermap maps quantum operations into quantum operations as . In the Choi representation, the supermap induces a linear map on Choi operators, as . Using Eq. (4), we can get back from as follows

 E′(ρ)=~S(E)(ρ)=TrKout[(I⊗ρ⊺)S(E)]. (6)

Of course complete positivity of implies that the map is positive. On the other hand, it is easily seen that the bipartite structure of a joint operation over a composite system induces a bipartite structure of the Choi operator . The local application of the supermap —given by —then corresponds to the local application of —given by —whence is CP if and only if is CP.

Since the correspondence is one-to-one, in the following we will focus our attention on . The supermap sends positive operators on to positive operators on generally different Hilbert spaces . Complete positivity of is equivalent to the existence of a Kraus form

 S(E)=∑iSiES†i, (7)

where are operators from to .

The following Lemmas provide the characterization of deterministic supermaps:

Lemma 1

Any linear operator on such that for all Choi operators of channels has the form , with on satifying . For one has .

Proof. Consider a Choi operator with effect . Upon defining for some state on we have that is the Choi operator of a channel, normalized as in Eq. (5). Since by hypothesis , we have that

 Tr[CE]= 1−Tr[CD]=1−Tr[C(σ⊗I)]+Tr[C(σ⊗P)] = Tr[C(σ⊗P)], (8)

since is the Choi operator of a channel. Therefore,

 Tr[CE]=Tr[C(σ⊗P)]=Tr[PTrHout[C(σ⊗I)]]=Tr[TrHout[E]TrHout[C(σ⊗I)]]=Tr[ρP], (9)

where . Since does not depend on , the last equality can be rewritten as for all positive , whence , and in order to have for all , we must have . Clearly implies .

Lemma 2

The supermap is deterministic iff there exists a channel from states on to states on such that, for any state on , one has

 S∗(IKout⊗ρ)=IHout⊗N∗(ρ) , (10)

where is the dual map of defined in terms of the Kraus form in Eq. (7) by

 S∗(O):=∑iS†iOSi . (11)

Proof. One has . Consider a positive operator on , where is a state on . We have that

 1=Tr[CS(E)]=Tr[S∗(C)E], (12)

for all Choi operators of channels. According to Lemma 7, this implies , where is a state for any state . Since the maps , and are all CP, we have , where is a CP trace preserving map from states on to states on .

Remarkably, the same mathematical structure of Lemma 2 characterizes semi-causal quantum operations ESW (), i.e. operations on bipartite systems that allow signaling from system to system but not viceversa. In our case, this structure originates from the causality of input-output relations. An equivalent condition for a supermap to be deterministic is given by the following:

Lemma 3

The supermap is deterministic iff there exists an identity preserving completely positive map such that, for any operator on , one has

 TrKout[S(E)]=N(TrHout[E]). (13)

Proof. This lemma follows from the previous one by considering that

 Tr[ρTrKout[S(E)]]=Tr[(I⊗ρ)S(E)]=Tr[S∗(I⊗ρ)E]=Tr[(I⊗N∗(ρ))E]=Tr[(I⊗ρ)(I⊗N)(E)]=Tr[ρN(TrHout[E])], (14)

for all states on . The map is identity preserving because it represents in the Schrödinger picture.

Eq. (13) shows that the effect depends only on the effect , e. g. not on . Basically, this reflects the fact that, in the input/output bipartition of the Choi operator, the output must not influence the transformation of the input effect.

Now we show that deterministic supermaps, so far introduced on a purely axiomatic level, can be physically realized with simple quantum circuits. Upon writing a canonical Kraus form for the completely positive map as follows

 N(P)=∑lN†lPNl, (15)

and substituting the Kraus forms (7) and (15) into Eq. (13), one obtains

 ∑n(⟨kn|⊗I)SiES†i(I⊗|kn⟩)=∑m(⟨hm|⊗N†j)E(Nj⊗|hm⟩), (16)

where and are orthonormal basis for and , respectively, and identity operators must be considered as acting on the appropriate Hilbert spaces— on the top and on the bottom part of Eq. (16). Eq. (16) gives two equivalent Kraus forms for the same CP map, of which the second one is canonical (since is canonical and are orthogonal). Therefore, there exists an isometry connecting the two sets of Kraus operators as follows

 (⟨kn|⊗I)Si=∑mjWni,mj(⟨hm|⊗N†j), (17)

with . Explicitly

 Wni,mj:=(⟨kn|⊗⟨ai|)W(|hm⟩⊗|bj⟩), (18)

where and are orthonormal basis for two ancillary systems with Hilbert spaces and . From Eq. (17) we then obtain

 Si=(I⊗⟨ai|)W(I⊗Z), (19)

where

 Z=∑j|bj⟩⊗N†j . (20)

Using Eq. (7) we can now evaluate the output Choi operator as follows

 S(E)=TrA[W(I⊗Z)E(I⊗Z†)W†]. (21)

Finally, using Eq. (6) we get

 E′(ρ)=TrKin[(I⊗ρ⊺)S(E)]=TrKin⊗A[(IKout⊗A⊗ρ⊺)W(I⊗Z)E(I⊗Z†)W†]=TrA[W(E⊗IB)(VρV†)W†], (22)

where is the partial transposed of (see Eq. (20)) on the second space. Since the map is identity preserving, is an isometry, namely . We have then proved the following realization theorem

Theorem 1

Every deterministic supermap can be realized by a four-port quantum circuit where the input operation is inserted between two isometries and and a final ancilla is discarded as in Fig. Transforming quantum operations: quantum supermaps. The output operation is given by

 ~S(E)(ρ)=TrA[W(E⊗IB)(VρV†)W†]. (23)

Since any isometry can be realized as a unitary interaction with an ancilla initialized in some reset state, the above Theorem entails a realization of supermaps in terms of unitary gates. However, we preferred stating it in terms of isometries in order to avoid the arbitrariness in the choice of the initial ancilla state.

You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters