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Characterisation of the input-output inversions of bistochastic channels

The foundation of our framework is the characterisation of the bidirectional quantum devices. The logic of our argument is the following: first, we observe that the input-output inversion must be linear in its argument (Supplementary Note 5). Hence, the action of the map Θ on the unitary channels uniquely determines the action of the map Θ on every channel in the linear space generated by the unitary channels. This linear space is characterised by the following theorem from²⁸, for which we provide a new, constructive proof in Supplementary Note 6.

Theorem 1

The linear span of the set of unitary channels coincides with the linear span of the set of bistochastic channels.

Theorem 1 implies that the action of the map Θ on of bistochastic channels is uniquely determined by the action of the map Θ on unitary channels. For unitary channels, we have seen that there are only two possible choices: either the action of Θ on the bistochastic channels is unitarily equivalent to the adjoint, or it is unitarily equivalent to the transpose. In either case, \({{\Theta }}({{{{{{{\mathcal{C}}}}}}}})\) is a valid quantum channel (CPTP map) for every bistochastic channel \({{{{{{{\mathcal{C}}}}}}}}\). Hence, all bistochastic channels are bidirectional.

Characterisation of the bidirectional channels

We now show that the set of bidirectional channels coincides with the set of bistochastic channels. The key of the argument is the following result:

Theorem 2

If a channel \({{{{{{{\mathcal{C}}}}}}}}\) admits an input-output inversion satisfying Requirements 1, 2, and 4, then its input-output inversion \({{\Theta }}({{{{{{{\mathcal{C}}}}}}}})\) is a bistochastic channel.

The proof is provided in Supplementary Note 7. Theorem 2, combined with Requirement 3 (the input-output inversion maps distinct channels into distinct channels), implies that only bistochastic channels can admit an input-output inverse. Indeed, if a non-bistochastic channel had an input-output inversion, then the input-output inverse would coincide with the input-output inversion of a bistochastic channel, in contradiction with Requirement 3.

In Supplementary Note 8 we show that, even if Requirement 3 is dropped, defining a non-trivial input-output inversion satisfying requirements 1, 2, and 4 is impossible for every system of dimension d > 2. For d = 2, instead, a map Θ satisfying conditions (1), (2), and (4) can be defined on all channels, but it maps all channels into bistochastic channels, in agreement with Theorem 2.

Summarising, the set of bidirectional channels is the set of bistochastic channels, and the input-output inversion is either equivalent to the adjoint or to the transpose. The adjoint and the transpose exhibit a fundamental difference when applied locally to bipartite processes. Suppose that a composite system S ⊗ E undergoes a joint evolution with the property that the reduced evolution of system S is bistochastic for every initial state of system E. Then, one may want to apply the input-output inversion only on the S-part of the evolution, while leaving the E-part unchanged. In Supplementary Note 9 we show that, when the dimension of system S is larger than two, the local application of the input-output inversion generates valid quantum evolutions (CPTP maps) if and only if the input-output inversion is described by the transpose.

Characterisation of the operations on bistochastic channels

A basic way to interact with a bidirectional quantum device is described by a particular type of quantum supermap²² that transforms bistochastic channels into ordinary channels (CPTP maps).

Hereafter, we will denote by \(L({{{{{{{\mathcal{H}}}}}}}},{{{{{{{\mathcal{K}}}}}}}})\) the set of linear operators on a generic Hilbert space \({{{{{{{\mathcal{H}}}}}}}}\) to another generic Hilbert space \({{{{{{{\mathcal{K}}}}}}}}\), and we will use the shorthand notation \(L({{{{{{{\mathcal{H}}}}}}}}):=L({{{{{{{\mathcal{H}}}}}}}},{{{{{{{\mathcal{H}}}}}}}})\). Also, we will denote by Map(S_i, S_o) the set of linear maps from \(L({{{{{{{{\mathcal{H}}}}}}}}}_{{S}_{{{{{{{{\rm{i}}}}}}}}}})\) to \(L({{{{{{{{\mathcal{H}}}}}}}}}_{{S}_{{{{{{{{\rm{o}}}}}}}}}})\), by Chan(S_i, S_o) ⊂ Map(S_i, S_o) the set of all quantum channels (CPTP maps), and by BiChan(S_i, S_o) the subset of all bistochastic channels. The set of density matrices of system S will be denoted as St(S).

A quantum supermap transforming bistochastic channels in BiChan(A_i, A_o) into channels Chan(B_i, B_o) is a linear map \({{{{{{{\mathcal{S}}}}}}}}:{\mathsf{Map}}({A}_{{{{{{{{\rm{i}}}}}}}}}\,,\,{A}_{{{{{{{{\rm{o}}}}}}}}})\to {\mathsf{Map}}({B}_{{{{{{{{\rm{i}}}}}}}}},{B}_{{{{{{{{\rm{o}}}}}}}}})\). The map \({{{{{{{\mathcal{S}}}}}}}}\) is required to produce valid channels even when acting locally on part of bipartite processes. Explicitly, the requirement is that \(({{{{{{{\mathcal{S}}}}}}}}\otimes {{{{{{{{\mathcal{I}}}}}}}}}_{{E}_{{{{{{{{\rm{i}}}}}}}}}{E}_{{{{{{{{\rm{o}}}}}}}}}})\,({{{{{{{\mathcal{C}}}}}}}})\) must be a valid quantum channel in Chan(B_iE_i ,  B_oE_o) for every \({{{{{{{\mathcal{C}}}}}}}}\in {\mathsf{Chan}}({A}_{{{{{{{{\rm{i}}}}}}}}}{E}_{{{{{{{{\rm{i}}}}}}}}}\,,\,{A}_{{{{{{{{\rm{o}}}}}}}}}{E}_{{{{{{{{\rm{o}}}}}}}}})\) satisfying the condition that the reduced channel \({{{{{{{{\mathcal{C}}}}}}}}}_{\sigma }:\rho \mapsto {{{{{{{{\rm{Tr}}}}}}}}}_{{E}_{{{{{{{{\rm{o}}}}}}}}}}[{{{{{{{\mathcal{C}}}}}}}}(\rho \otimes \sigma )]\) is in BiChan(A_i, A_o) for every density matrix σ ∈ St(E_i)²².

A convenient way to represent quantum supermaps is to use the Choi representation⁵⁴. A generic linear map \({{{{{{{\mathcal{M}}}}}}}}:L({{{{{{{{\mathcal{H}}}}}}}}}_{{S}_{{{{{{{{\rm{i}}}}}}}}}})\to L({{{{{{{{\mathcal{H}}}}}}}}}_{{S}_{{{{{{{{\rm{o}}}}}}}}}})\) is in one-to-one correspondence with its Choi operator \({{{{{{{\rm{Choi}}}}}}}}({{{{{{{\mathcal{M}}}}}}}})\in L({{{{{{{{\mathcal{H}}}}}}}}}_{{S}_{{{{{{{{\rm{o}}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{S}_{{{{{{{{\rm{i}}}}}}}}}})\), defined by \({{{{{{{\rm{Choi}}}}}}}}({{{{{{{\mathcal{M}}}}}}}}):={\sum }_{m,n}{{{{{{{\mathcal{M}}}}}}}}(\left|m\right\rangle \left\langle n\right|)\otimes \left|m\right\rangle \left\langle n\right|\), where \(\{\left|n\right\rangle \}\) is a fixed orthonormal basis. For a bistochastic channel \({{{{{{{\mathcal{C}}}}}}}}\in {\mathsf{BiChan}}({A}_{{{{{{{{\rm{i}}}}}}}}},{A}_{{{{{{{{\rm{o}}}}}}}}})\) with A_i ≃ A_o, the Choi operator C satisfies the conditions

Equivalently, the operator C can be decomposed as

where d is the dimension of systems A_i and A_o, and T is an operator such that

and ∥T∥≤1/d.

Now, every supermap \({{{{{{{\mathcal{S}}}}}}}}:{\mathsf{Map}}({A}_{{{{{{{{\rm{i}}}}}}}}}\,,\,{A}_{{{{{{{{\rm{o}}}}}}}}})\to {\mathsf{Map}}({B}_{{{{{{{{\rm{i}}}}}}}}},{B}_{{{{{{{{\rm{o}}}}}}}}})\) is itself a linear map, and, as such, it can be represented by Choi operator \(S\in L({{{{{{{{\mathcal{H}}}}}}}}}_{{B}_{{{{{{{{\rm{o}}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{B}_{{{{{{{{\rm{i}}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{A}_{{{{{{{{\rm{o}}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{A}_{{{{{{{{\rm{i}}}}}}}}}})\). The operator S is completely specified by the relation

where \({{{{{{{\mathcal{M}}}}}}}}\) is an arbitrary map in Map(A_i ,  A_o), and T denotes the transpose with respect to the basis \(\{\left|n\right\rangle \}\). This relation can be used, for example, to compute the Choi operator of the quantum time flip. In the case of the quantum time flip, the systems A_i and A_o have the same dimension, and the systems B_i and B_o are of the bipartite form B_i = B_itB_ic and A_i = A_itA_ic, where B_it (B_ot) is a target system, of the same dimension as A_i and A_o, and B_ic (B_oc) is a two-dimensional control system. Using this notation, we can express the Choi operator of the quantum time flip as

with

and \(\left.\left|I\right\rangle \right\rangle :={\sum }_{m}\,\left|m\right\rangle \otimes \left|m\right\rangle\). (Here the vector \(\left|V\right\rangle\) belongs to the Hilbert space \({{{{{{{{\mathcal{H}}}}}}}}}_{{B}_{{{{{{{{\rm{ot}}}}}}}}}{B}_{{{{{{{{\rm{oc}}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{B}_{{{{{{{{\rm{it}}}}}}}}}{B}_{{{{{{{{\rm{ic}}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{A}_{{{{{{{{\rm{o}}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{A}_{{{{{{{{\rm{i}}}}}}}}}}\), and it is understood that the Hilbert spaces in the r.h.s. have to be reordered consistently according to the systems’ labels).

In the Choi representation, the requirement that \({{{{{{{\mathcal{S}}}}}}}}\) be applicable locally on part of a larger process is equivalent to the requirement that the operator S be positive semidefinite^22,29. The requirement that \({{{{{{{\mathcal{S}}}}}}}}\) transforms any bistochastic channel into a CPTP map is equivalent to the condition

where C is an arbitrary Choi operators of a bistochastic channel \({{{{{{{\mathcal{C}}}}}}}}\in {\mathsf{Bi}}{\mathsf{Chan}}({A}_{{{{{{{{\rm{i}}}}}}}}},{A}_{{{{{{{{\rm{o}}}}}}}}})\).

The normalisation condition (9) can be put in a more explicit form by decomposing the operator S into orthogonal components, in a similar way as it was done in²¹ for the characterisation of the operations with definite time direction.

Choosing T = 0 in Eq. (4) and inserting the operator \(C={I}_{{A}_{{{{{{{{\rm{o}}}}}}}}}}\otimes {I}_{{A}_{{{{{{{{\rm{i}}}}}}}}}}/d\) into Eq. (9), we obtain

Choosing an arbitrary T, instead, we obtain

The combination of conditions (10) and (11) is equivalent to the original condition (9).

We will now cast condition (11) in a more explicit form. Condition (11) is equivalent to the requirement that S be orthogonal (with respect to the Hilbert-Schmidt product) to all operators of the form \({I}_{{B}_{{{{{{{{\rm{o}}}}}}}}}}\otimes {J}_{{B}_{{{{{{{{\rm{i}}}}}}}}}}\otimes {T}_{{A}_{{{{{{{{\rm{o}}}}}}}}}{A}_{{{{{{{{\rm{i}}}}}}}}}}\), where \({J}_{{B}_{{{{{{{{\rm{i}}}}}}}}}}\) is an arbitrary operator on \({{{{{{{{\mathcal{H}}}}}}}}}_{{B}_{{{{{{{{\rm{i}}}}}}}}}}\) and \({T}_{{A}_{{{{{{{{\rm{i}}}}}}}}}{A}_{{{{{{{{\rm{o}}}}}}}}}}\) is an arbitrary operator satisfying Eq. (5). This condition implies that S can be decomposed into the sum of four mutually orthogonal operators, namely

where \({G}_{{B}_{{{{{{{{\rm{o}}}}}}}}}{B}_{{{{{{{{\rm{i}}}}}}}}}}\) is an arbitrary operator on \({{{{{{{{\mathcal{H}}}}}}}}}_{{B}_{{{{{{{{\rm{o}}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{B}_{{{{{{{{\rm{i}}}}}}}}}}\), and the remaining operators on the right hand side satisfy the relations

(The operator \({L}_{{B}_{{{{{{{{\rm{o}}}}}}}}}{B}_{{{{{{{{\rm{i}}}}}}}}}{A}_{{{{{{{{\rm{i}}}}}}}}}}\otimes {I}_{{A}_{{{{{{{{\rm{o}}}}}}}}}}\) in Eq. (12) is understood as acting on \({{{{{{{{\mathcal{H}}}}}}}}}_{{B}_{{{{{{{{\rm{o}}}}}}}}}}\otimes {{{{{{{{\mathcal{B}}}}}}}}}_{{{{{{{{\rm{i}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{A}_{{{{{{{{\rm{o}}}}}}}}}}\otimes {{{{{{{{\mathcal{H}}}}}}}}}_{{A}_{{{{{{{{\rm{o}}}}}}}}}}\), with an implicit reordering of the Hilbert spaces according to the systems’ labels. In the following, this implicit reordering will be used).

We now express the first three operators in the right-hand side of Eq. (12) in terms of the partial traces of S. Explicitly, we have

Inserting the above relations into Eq. (12), we obtain

or equivalently,

In other words, the left-hand side of the equation should be an operator that satisfies the last three conditions of Eq. (13). The first two conditions are automatically guaranteed by the form of the right-hand side of Eq. (16), while the third condition reads

Summarising, we have shown that the normalisation of the supermap \({{{{{{{\mathcal{S}}}}}}}}\) is expressed by the two conditions (10) and (17).

As an example, one can easily verify that the Choi operator of the quantum time flip, provided in Eq. (7) satisfies conditions (10) and (17). In fact, the quantum time flip satisfies these conditions even when the roles of B_i and B_o are exchanged. This additional property expresses the fact that the quantum time flip supermap transforms bistochastic channels into bistochastic channels.

Multipartite quantum operations on bistochastic channels

Multipartite operations with indefinite time direction can also be described as quantum supermaps on the set of N-partite no-signalling bistochastic channels, that is, the set of N-partite quantum channels of the form

where each c_j is a real coefficient, each \({{{{{{{{\mathcal{A}}}}}}}}}_{i,j}\) is a bistochastic channel. We denote the set of channels of this form as BiNoSig(A_1i, A_1o ∣ A_2i, A_2o ∣  ⋯  ∣A_Ni, A_No) where A_ni (A_no) is the input (output) of channel \({{{{{{{{\mathcal{A}}}}}}}}}_{n,j}\), for every possible n and every possible j.

A quantum supermap on no-signalling bistochastic channels is then defined as a linear map \({{{{{{{\mathcal{S}}}}}}}}:{\mathsf{Map}}({A}_{1{{{{{{{\rm{i}}}}}}}}}{A}_{2{{{{{{{\rm{i}}}}}}}}}\cdots {A}_{N{{{{{{{\rm{i}}}}}}}}}\,,\,{A}_{1{{{{{{{\rm{o}}}}}}}}}{A}_{2{{{{{{{\rm{o}}}}}}}}}\cdots {A}_{N{{{{{{{\rm{o}}}}}}}}})\to {\mathsf{Map}}({B}_{{{{{{{{\rm{i}}}}}}}}},{B}_{{{{{{{{\rm{o}}}}}}}}})\), where B_i (B_o) is the input (output) of the channel produced by \({{{{{{{\mathcal{S}}}}}}}}\). The map \({{{{{{{\mathcal{S}}}}}}}}\) is required to transform no-signalling bistochastic channels into CPTP maps even when acting locally on part of a composite process. Explicitly, this means that the map \(({{{{{{{\mathcal{S}}}}}}}}\otimes {{{{{{{{\mathcal{I}}}}}}}}}_{{E}_{{{{{{{{\rm{i}}}}}}}}}{E}_{{{{{{{{\rm{o}}}}}}}}}})\,({{{{{{{\mathcal{N}}}}}}}})\) must be a valid quantum channel in Chan(B_i, B_o) for every \({{{{{{{\mathcal{N}}}}}}}}\in {\mathsf{Chan}}({A}_{1{{{{{{{\rm{i}}}}}}}}} \,{A}_{2{{{{{{{\rm{i}}}}}}}}} \,\cdots \, {A}_{N{{{{{{{\rm{i}}}}}}}}} \, {E}_{{{{{{{{\rm{i}}}}}}}}} , \, {A}_{1{{{{{{{\rm{o}}}}}}}}}\, {A}_{2{{{{{{{\rm{o}}}}}}}}}\, \cdots \, {A}_{N{{{{{{{\rm{o}}}}}}}}} {E}_{{{{{{{{\rm{o}}}}}}}}} )\) satisfying the condition that the reduced channel \({{{{{{{{\mathcal{N}}}}}}}}}_{\sigma }:\rho \,\mapsto\, {{{{{{{{\rm{Tr}}}}}}}}}_{{E}_{{{{{{{{\rm{o}}}}}}}}}}[{{{{{{{\mathcal{N}}}}}}}}(\rho \otimes \sigma )]\) belongs to BiNoSig(A_1i, A_1o ∣ A_2i, A_2o ∣  ⋯  ∣A_Ni, A_No) for every density matrix σ ∈ St(E_i).

Quantum supermaps on bistochastic no-signalling channels describe the most general way in which N bidirectional quantum processes can be combined into a single channel. In general, this combination can be incompatible with a definite direction of time, and, at the same time, incompatible with a definite ordering of the N channels.

Here we provide three examples of bipartite supermaps. To specify each supermap, we specify its action on the set of product channels, which by definition are a spanning set of the set of bipartite bistochastic no-signalling channels. The first supermap, \({{{{{{{{\mathcal{S}}}}}}}}}_{1}\), is defined as

where {A_1m} and {A_2,n} are Kraus operators of channels \({{{{{{{{\mathcal{A}}}}}}}}}_{1}\) and \({{{{{{{{\mathcal{A}}}}}}}}}_{2}\), respectively. This supermap can be generated by applying two independent quantum time flips to channels \({{{{{{{{\mathcal{A}}}}}}}}}_{1}\) and \({{{{{{{{\mathcal{A}}}}}}}}}_{2}\), respectively: indeed, one has \({{{{{{{{\mathcal{S}}}}}}}}}_{1}({{{{{{{{\mathcal{A}}}}}}}}}_{1}\otimes {{{{{{{{\mathcal{A}}}}}}}}}_{2})={{{{{{{\mathcal{F}}}}}}}}({{{{{{{{\mathcal{A}}}}}}}}}_{1})\circ {{{{{{{\mathcal{F}}}}}}}}^{\prime} ({{{{{{{{\mathcal{A}}}}}}}}}_{2})\), where \({{{{{{{\mathcal{F}}}}}}}}^{\prime}\) is the variant of the quantum time flip in which the roles of the control states \(\left|0\right\rangle\) and \(\left|1\right\rangle\) are exchanged.

The supermap \({{{{{{{{\mathcal{S}}}}}}}}}_{1}\) describes the winning strategy in Eq. (2). This strategy cannot be realized by using the two channels \({{{{{{{{\mathcal{A}}}}}}}}}_{1}\) and \({{{{{{{{\mathcal{A}}}}}}}}}_{2}\) in a definite time direction, but is compatible with a definite causal order: the channel \({{{{{{{{\mathcal{A}}}}}}}}}_{1}\) (\({{{{{{{{\mathcal{A}}}}}}}}}_{1}^{T}\)) always acts after channel \({{{{{{{{\mathcal{A}}}}}}}}}_{2}^{T}\) (\({{{{{{{{\mathcal{A}}}}}}}}}_{2}\)).

The second supermap, \({{{{{{{{\mathcal{S}}}}}}}}}_{2}\), is the quantum SWITCH ^20,22, defined as

Note that here the quantum SWITCH is restricted to act on the set of bistochastic no-signalling channels. Interestingly, however, this definition determines the action of the quantum SWITCH on arbitrary channels (and on arbitrary linear maps as well): the reason is that the set of bistochastic no-signalling channels includes the set of all products of unitary channels, and it is known that the quantum SWITCH is uniquely determined by its action on such channels⁵⁵. In the quantum SWITCH, the order of the channels \({{{{{{{{\mathcal{A}}}}}}}}}_{1}\) and \({{{{{{{{\mathcal{A}}}}}}}}}_{2}\) is indefinite, but each channel is used in the forward time direction.

Finally, our third example is a supermap \({{{{{{{{\mathcal{S}}}}}}}}}_{3}\) arising from the combination of the quantum time flip with the quantum SWITCH. It is defined as follows:

This supermap describes a coherent superposition of the process \({{{{{{{{\mathcal{A}}}}}}}}}_{1}\circ {{{{{{{{\mathcal{A}}}}}}}}}_{2}\) and its input-output inverse \({{\Theta }}({{{{{{{{\mathcal{A}}}}}}}}}_{1}\circ {{{{{{{{\mathcal{A}}}}}}}}}_{2})={{{{{{{{\mathcal{A}}}}}}}}}_{2}^{T}\circ {{{{{{{{\mathcal{A}}}}}}}}}_{1}^{T}\). Such supermap is incompatible with both a definite time direction and with a definite causal order.

Choi representation of multipartite operations

An equivalent way to represent quantum supermaps on bistochastic no-signalling channels is to use the Choi representation. When this is done, one obtains a generalisation of the notion of process matrix²¹, originally used for supermaps that combine processes in an indefinite order while using each process in a definite time direction.

Since \({{{{{{{\mathcal{S}}}}}}}}\) is a linear map, it has a Choi operator S ∈ L(B_oB_iA_1oA_1iA_2oA_2i   ⋯  A_NoA_Ni), uniquely determined by the relation

where \({{{{{{{\mathcal{M}}}}}}}}\) is an arbitrary map in Map(A_1iA_2i ⋯ A_Ni ,  A_1oA_2o ⋯ A_No).

As in the N = 1 case, the requirement that \({{{{{{{\mathcal{S}}}}}}}}\) be applicable locally on part of a larger process is equivalent to the requirement that the operator S be positive semidefinite. The requirement that \({{{{{{{\mathcal{S}}}}}}}}\) transforms any bistochastic no-signalling channel into a CPTP map is equivalent to the condition

where N is the Choi operator of an arbitrary bistochastic no-signalling channel in BiNoSig(A_1i, A_1o ∣  A_2i, A_2o ∣  ⋯  ∣A_Ni, A_No). A more explicit characterization can be obtained using an orthogonal decomposition of the operator S as illustrated earlier in the N = 1 case.