Quantum Channel Construction with Circuit Quantum Electrodynamics

PHYSICAL REVIEW B 95, 134501 (2017)

Quantum channel construction with circuit quantum electrodynamics

Chao Shen,* Kyungjoo Noh, Victor V. Albert, Stefan Krastanov, M. H. Devoret, R. J. Schoelkopf, S. M. Girvin, and Liang Jiang Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA and Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA (Received 18 November 2016; revised manuscript received 26 February 2017; published 4 April 2017)

Quantum channels can describe all transformations allowed by quantum mechanics. We adapt two existing works [S. Lloyd and L. Viola, Phys.Rev.A65, 010101 (2001) and E. Andersson and D. K. L. Oi, ibid. 77, 052104 (2008)] to superconducting circuits, featuring a single qubit ancilla with quantum nondemolition readout and adaptive control. This construction is efﬁcient in both ancilla dimension and circuit depth. We point out various applications of quantum channel construction, including system stabilization and quantum error correction, Markovian and exotic channel simulation, implementation of generalized quantum measurements, and more general quantum instruments. Efﬁcient construction of arbitrary quantum channels opens up exciting new possibilities for quantum control, quantum sensing, and information processing tasks.

DOI: 10.1103/PhysRevB.95.134501

I. INTRODUCTION quantum measurement, called positive-operator valued measure (POVM) [23]. As detailed in Ref. [25], an explicit Quantum channels or quantum operations, more formally binary tree construction has been provided to implement any known as completely positive and trace preserving (CPTP) given POVM. maps between density operators [1–3], give the most general In this paper we concretize the idea developed in [23,25] description of quantum dynamics. For closed quantum sys- and propose a superconducting circuit construction of arbitrary tems, unitary evolution is sufficient to describe the dynamics. CPTP maps, featuring minimal ancilla system—a single qubit For open quantum systems, however, the interaction between and low circuit depth (logarithmic with the system dimension). the system and environment leads to nonunitary evolution of We provide an explicit proposal to implement such a treelike the system (e.g., dissipation), which requires CPTP maps for series using a minimal and currently feasible set of operations full characterization. Besides describing open system dynam- from circuit quantum electrodynamics (cQED) [26–29], with ics, the system dissipation can further be engineered to protect the setup shown in Fig. 1. Furthermore, using concrete encoded quantum information from undesired decoherence examples, we argue that the ability to efficiently construct processes [4–9]. Hence, it is important to systematically extend arbitrary CPTP maps can lead to exciting new possibilities quantum control techniques from closed to open quantum in the field of quantum control and quantum information systems. processing in general. Theoretically, universal Lindbladian dynamics construc- The goal of this investigation is to expand the quantum tions have been investigated [10–12], which can be used control toolbox to efficiently implement all CPTP maps. In for stabilization of target quantum states [4], protection of contrast to investigations of analog/digital quantum simulators information encoded in subspaces [13], or even quantum of certain complex quantum dynamics [7,30–37], we focus on information processing [14–16]. Experimentally, dissipative the efficient implementation of CPTP maps for various quan- quantum control has been demonstrated using various physical tum control tasks, including state stabilization, information platforms [6–8,17–20]. Besides Lindbladian dynamics, CPTP processing, quantum error correction, etc. maps also include exotic indivisible channels that cannot This paper is organized as follows. First, we review the be expressed as Lindbladian channels [21]. Hence, use of basic notation of CPTP maps using the Kraus representation Lindbladian dynamics is insufficient to construct all CPTP in Sec. II. We then provide an explicit protocol that can imple- maps, which require more general techniques. ment arbitrary CPTP maps using an ancilla qubit with QND The textbook approach to construct all CPTP maps for a readout and adaptive control, and describe its implementation d-dimensional system (with d = 2m for a system consisting of with cQED in Sec. III. In Sec. IV we illustrate potential m qubits) requires a d2-dimensional ancilla and one round of applications of such constructed CPTP maps. In Sec. V we SU(d3) joint unitary operation (Stinespring dilation, see [1]). discuss further extensions and various imperfections. Finally, One recent work suggests that using a d-dimensional ancilla we conclude the paper in Sec. VI. and a probabilistic SU(d2) joint unitary operation might be sufficient for all CPTP maps, based on a mathematical II. KRAUS REPRESENTATION conjecture [22]. More interestingly, the ancilla dimension can be dramatically reduced to 2 for arbitrary system dimension d Mathematically we use the Kraus representation for CPTP [23], if we introduce adaptive control [24] based on quantum maps, N nondemolition (QND) readout of the ancilla which conditions † T (ρ) = K ρK , (1) a sequence of SU(2d) unitary operations. Besides CPTP i i = maps, the adaptive approach can be used for generalized i 1 which are trace preserving as ensured by the condition [38] N † K Ki = I. (2) * i [email protected] i=1

dynamics. Using that construction, to implement a CPTP map with Kraus rank N, we need a quantum circuit with L = N − 1 rounds of operations. Each round of operation consists of one joint unitary of system and ancilla and one QND measurement on the ancilla qubit. Andersson and Oi provided a scheme for a binary-tree construction to explicitly implement an arbitrary = POVM with L log2 N [25]. We extend the binary-tree FIG. 1. Schematic setup of a circuit QED system used for scheme to a more general protocol for arbitrary CPTP maps. constructing an arbitrary quantum channel. Later in Sec. IV E we will discuss the relations between CPTP maps, POVMs, and quantum instruments. The procedure to construct a CPTP map with Kraus rank N is associated with The Kraus operators Ki do not have to be unitary or Hermitian. = a binary tree of depth L log2 N , as shown in Fig. 2. With They can even be nonsquare matrices, if the input and output a qubit ancilla, the circuit depth of log2 N is the lowest Hilbert spaces have different dimensions. By padding with possible. This is what we mean by “efficient”. In the following, zeros, we can always make them square matrices that describe we first consider the simple case with L = 1, corresponding a dimension-preserving channel for a system with dimension to the CPTP maps with Kraus rank N 2. Then we provide d. The Kraus representation is not unique, because for any an explicit construction for general CPTP maps. After that we × = N N unitary matrix U, the set of new Kraus operators Fi outline how to physically implement the circuits using cQED j Uij Kj characterizes the same CPTP map [1]. as a promising physical platform. To efficiently construct a CPTP map, it is convenient to work with the Kraus representation with the minimum number of Kraus operators, called the Kraus rank of the CPTP map. A. Quantum channels with Kraus rank 2 2 Since there are at most d linearly independent operators for Given a single use of the ancilla qubit, we can construct any a Hilbert space of dimension d, the Kraus rank is no larger rank-2 CPTP map, characterized by Kraus operators {K0,K1}. 2 than d (for a rigorous treatment see [38]). There are efficient The procedure consists of the following: (1) initialize the procedures to convert different representations of a channel to ancilla qubit in |0, (2) perform a joint unitary operation the minimal Kraus representation [2,3,38]. For example, we U ∈ SU(2d), and (3) discard (“trace over”) the ancilla qubit. may convert the Kraus representation into the Choi matrix (a Since this procedure has only one round of operation, there is 2 2 d × d Hermitian matrix) and from there obtain the minimal no need for adaptive control and thus we can simply discard Kraus representation [38]. The second approach is to calculate the ancilla without any measurement. = † × the overlap matrix Cij Tr(Ki Kj ) and then diagonalize it, The 2d 2d matrix of unitary operation has the following = † ˜ = C V DV [1]. The new Kraus operators Ki j Vij Kj block matrix form [39]: will be the most economic representation with some of them | | ∗ being zero matrices if the original representation is redundant. = 0 U 0 U | | ∗, (3) For cases with the CPTP map provided in other representations 1 U 0 (e.g., superoperator matrix representation, Jamiolkowski-Choi where the d × d submatrices are 0|U|0=K , 1|U|0= matrix representation), we can also perform a well-defined 0 K , and “*” denotes irrelevant submatrices (as long as U is routine to bring them into the minimal Kraus representation 1 unitary). The 2d × d submatrix formed by 0|U|0 (as detailed in Appendix A). and 1|U|0 is an isometry, i.e., any matrix V that satisfies V †V = I. The trace preserving requirement III. UNIVERSAL CONSTRUCTION † + † = K0K0 K1 K1 I ensures that the isometry condition OF QUANTUM CHANNELS | | † | | = b=0,1 ( b U 0 ) b U 0 Id×d is fulfilled. After discarding As pointed out by Lloyd and Viola [23], repeated ap- the ancilla qubit, the procedure achieves the CPTP map, plication of Kraus rank-2 channels in an adaptive fashion T = † + † is in principle sufficient to construct arbitrary open-system U (ρ) K0ρK0 K1ρK1 .

FIG. 2. (a) Quantum circuit for arbitrary channel construction. The dimension of the system d can be arbitrary and the circuit depth depends = only on the Kraus rank of the target channel. (b) Binary tree representation with depth L 3. The Kraus operators Kb(L) are associated with the (L) ∈{ }L leaves of the binary tree, b 0,1 . The system-ancilla joint unitary to apply in lth round Ub(l) depends on the previous ancilla readout record (l) l b = (b1b2 ···bl ) ∈{0,1} associated with a node of the binary tree. For any given channel, all these unitaries can be explicitly constructed and efﬁciently implemented.

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2 Therefore, any channel with Kraus rank 2 can be simulated ≡ = (l) (l) (l) = where sgn(0) 0, so that Pb(l) Pb and Pb Db with a single use of the ancilla qubit [40]. = ⊥ = − Db(l) Pb(l) Db(l) . The orthogonal projection is Pb(l) I Pb(l) If we measure the ancilla qubit instead of discard- ⊥ † and we also define the related projection Q (l) ≡ V (l) P V . ing it, we can in principle obtain the “which trajectory” b b b(l) b(l) In addition, we define information. More specifically, the system state becomes † (0|U|0)ρ(0|U |0) (unnormalized) if we find the ancilla in −1 1/(Db(l) )j,k if (Db(l) ) = 0, † D = (8) |0, and it becomes (1|U|0)ρ(0|U |1) if we find the ancilla b(l) j,k 0 otherwise. in |1. We may use the “which trajectory” information to determine later operations, and thus construct more complicated and denote the Moore-Penrose pseudoinverse of Mb(l) as + = −1 † = = = −1 = CPTP maps with higher Kraus rank. Mb(l) Vb(l) Db(l) Vb(l) .Forl 0wefixVb(0) Db(0) Db(0) = ⊥ = Pb(0) I and Pb(0) 0. Finally, we have the explicit expression for the relevant B. Quantum channels with higher Kraus rank submatrices of the unitary matrix To implement a CPTP map with Kraus rank N, we need = + 1 a quantum circuit with L log N rounds of operations. + | (l) | = (l+1) + √ (l) 2 bl 1 Ub 0 Mb Mb(l) Qb , (9) Each round consists of (1) initialization of the ancilla qubit, 2 (2) joint unitary gate over the system and ancilla (conditional (l+1) (l) with b = (b ,bl+1)forl = 0,...,L− 2, and on the measurement outcomes from previous rounds), (3) + 1 † QND readout of the ancilla, and (4) storage of the classical | | = + + √ + bl+1 Ub(l) 0 Kb(l 1) M (l) Wb(l 1) V (l+1) Qb(l) (10) measurement outcome for later use. For a quantum circuit b 2 b consisting of L rounds of operations with adaptive control for l = L − 1. Since the isometric condition (based on binary outcomes), there are 2L − 1 possible interme- † = (bl+1|Ub(l) |0) bl+1|Ub(l) |0=Id×d is fulfilled diate unitary gates (associated with 2L − 1 nodes of a depth-L bl+1 0,1 L (as proven in Appendix B), we can complete the unitary binary tree) and 2 possible trajectories (associated with the | | L matrix Ub(l) with appropriate submatrices bl+1 Ub(l) 1 . 2 leaves of the binary tree). For L = 1weuseEq.(10)forl = 0 and obtain As illustrated in Fig. 2, we denote the lth round unitary (l) = gate as Ub(l) , associated with the node of the binary tree b = K1 for b1 0, l b1|Ub(0) |0=Kb(1) = (b1b2 ···bl) ∈{0,1} with l = 0,...,L− 1. [For L = 1 there K2 for b1 = 1, (0) is only one unitary gate for b = ∅, which is U (0)=∅ as given b which is consistent with the earlier construction for Kraus in Eq. (3).] Generally, the unitary gate U (l) has the following b rank-2 channels. block matrix form: With the above explicit construction of arbitrary CPTP 0|Ub(l) |0∗ maps, we will investigate the physical implementation with Ub(l) = , (4) 1|Ub(l) |0∗ cQED. where “*” again denote irrelevant submatrices (as long as U (l) b C. Physical implementation with cQED is unitary). Since the ancilla always starts in |0, it is sufficient to specify the d × d submatrices bl+1|Ub(l) |0 actingonthe The above channel construction scheme relies on three key system, with the projectively measured ancilla state |bl+1 for components: (1) ability to apply a certain class of unitary gates (L) bl+1 = 0,1. Associated with the leaves of the binary tree, b ∈ (recall that we engineer only the left half of the unitary) on {0,1}L are Kraus operators labeled in binary notation, the system and ancilla combined system; (2) QND readout of the ancilla qubit; and (3) adaptive control of all unitary Kb(L) = Ki , (5) gates based on earlier rounds of QND measurement outcomes. n − = ··· + = Although there are a total of (2 1) unitaries potentially to with i (b1b2 bL)2 1 and Ki>N 0. The singular be applied, they can all be precalculated and one only needs value decomposition of each Kraus operator is Kb(L) = † to decide which one to perform in real time based on the Wb(L) Db(L) Vb(L) . measurement record. In principle any quantum system that | | We now provide an explicit construction for bl+1 Ub(l) 0 . meets these three requirements can be used to implement our (l) = − First, for each node b with l 1,...,L 1, we may diago- scheme. In the following, we focus on a circuit QED system nalize the nonnegative Hermitian matrix (which is associated with a transmon qubit dispersively coupled to a microwave with the summation over all the leaves in the branch starting cavity with Hamiltonian [41] (l) from b ) † † Hˆ0 = ωcaˆ aˆ + ωq |ee|−χa a|ee|, † = 2 † ≡ 2 Kb(L) Kb(L) Vb(l) Db(l) Vb(l) Mb(l) , (6) where ωc and ωq are the cavity and qubit transition frequency, bl+1,...,bL respectively, aˆ is the the annihilation operator of a cavity | | with unitary matrix Vb(l) , diagonal matrix Db(l) consisting of excitation, χ is the dispersive shift parameter, and e e is nonnegative diagonal elements, and Hermitian matrix Mb(l) = the qubit excited state projection. This is a promising platform † to implement the channel construction scheme because the Vb(l) Db(l) V (l) . For notational convenience, we introduce Pb(l) as b dispersive shift χ can be three orders of magnitude larger than the support projection matrix of Db(l) , with elements the dissipation of the qubit and the cavity, allowing universal (Pb(l) )j,k = sgn[(Db(l) )j,k], (7) unitary control of the system [42,43].

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This above decomposition is known as the “cosine-sine decomposition” [44,45] and plays an important role in decomposing an arbitrary unitary/isometry into controlled-not gates and single qubit gates. This construction is sufﬁcient to perfectly match the relevant two submatrices of the desired unitary 0|U|0∗ U = , 1|U|0∗ FIG. 3. Level diagram for the dispersively coupled qubit-cavity system. It is straightforward to implement U for such a system by † † ent with 0|U|0=W0S0V and 1|U|0=W1S1V .Toimple- driving two level transitions that are spectrally separated. Here g/e ment the quantum circuit in Fig. 2(a), we may explicitly denote the ancilla qubit states (0/1 logical states) and n denotes the identify the W0/1, S0/1, and V matrices for unitary operations photon number state. at different rounds U = Ub(l) . To justify the above claim, we provide an explicit design

U U (l) The Fock states of a cavity mode can be used to encode of ent to perfectly match the left two submatrices of b in a d-dimensional system and the qubit can be used as the three steps. (1) We start with singular value decompositions | | = † | | = † ancilla. Universal unitary control on the d-level system has (SVD) 0 U 0 W0S0V0 and 1 U 0 W1S1V1 , where we been proposed in Ref. [43] and demonstrated experimentally have already set the W’s and S’s to their desired values. = = in Refs. [29,42]. The strong dispersive coupling of the cavity Now all that is left to do is to make sure that V0 V1 V . and qubit enables selective driving of transitions between To uniquely determine the decomposition, we require that |g,n and |e, n for different excitation numbers n, which the singular values in S0 are arranged in descending order can implement the following entangling unitary gate: (S0)j,j (S0)j+1,j+1, while the singular values in S1 are arranged in ascendingorder (S1)j,j (S1)j+1,j+1.(2)The | | † | | = Uent(θi ) isometric condition b=0,1 ( b U 0 ) b U 0 Id×d re- † 2 † 2 2 − quires that V V S V V = I × − S . Since both S and = S0 S1 0 1 1 1 0 d d 0 1 − 2 † S1 S0 Id×d S0 are diagonal with elements in ascending order, V1 V0 ⎛ ⎞ = = θ θ must be the identity, that is, V0 V1 V . Therefore, we have cos 1 − sin 1 | | = ⎜ 2 2 ⎟ obtained all the components of Uent, which fulfills 0 U 0 ⎜ . . ⎟ † | | = † ⎜ .. .. ⎟ W0S0V and 1 U 0 W1S1V . A similar property was ⎜ ⎟ ⎜ cos θd − sin θd ⎟ used in [46] to simplify the construction of generalized = ⎜ 2 2 ⎟ measurements of a qubit. In terms of circuits, we decomposed ⎜ θ1 θ1 ⎟ ⎜sin 2 cos 2 ⎟ the 2d-dimensional unitaries in Fig. 2 into a series of simpler ⎜ . . ⎟ ⎝ .. .. ⎠ operations, as shown in Fig. 4. sin θd cos θd 2 2 IV. APPLICATION EXAMPLES d−1 = exp(−iYnθn/2), (11) The concept of CPTP maps encompasses all physical n=0 operations ranging from cooling, quantum gates, measurements, to dissipative dynamics. The ability to construct an where Yn ≡−i|g,ne, n|+H.c. is the Pauli-Y operator for arbitrary CPTP map offers a unified approach to all aspects the two-dimensional subspace associated with n excitations of quantum technology. To illustrate the wide range of impact (see Fig. 3). This entangling gate gives a channel described by of quantum channel construction, we now investigate some † Kraus operators {S0,S1}. If we precede Uent with a unitary V interesting applications, including quantum system initial- acting on the system alone and perform an adaptive unitary on ization/stabilization, quantum error correction, Lindbladian the system after Uent depending on the ancilla measurement quantum dynamics, exotic quantum channels, and quantum W0 or W1, we end up with the unitary instruments. − † = W0 0 S0 S1 V 0 A. Initialization/stabilization Uent † 0 W1 S1 S0 0 V Almost all quantum information processing tasks require † W0S0V ∗ working with a well-defined (often pure) initial state. One = † . W1S1V ∗ common approach is to sympathetically cool the system to the

FIG. 4. For cQED systems, the 2d-dimensional unitary used to generate an arbitrary Kraus rank-2 channel can be eventually simpliﬁed to unitaries acting on the system alone and an entangling operation Uent [Eq. (11)], which is a series of independent two-level transitions between |g,n and |e, n,whereg/e denote the ancilla qubit states (0/1 logical states) and n denotes the photon number state.

134501-4 QUANTUM CHANNEL CONSTRUCTION WITH CIRCUIT . . . PHYSICAL REVIEW B 95, 134501 (2017) ground state by coupling to a cold bath, or optically pumping previous measurement outcomes b(l−1). Finally, we perform to a specific dark state, and then performing unitary operations the correction unitary operation Ub(Ns ) conditioned on the to bring the system to a desired initial state. This can be slow if syndrome b(Ns ). the system has a large relaxation time scale. Another approach Generally we may consider all QEC codes that fulfill is to actively cool the system by measurement and adaptive the quantum error-correction conditions associated with a control. Along the line of the second approach, the channel set of error operations [1,56]. For these QEC codes we can construction technique can be applied to discretely pump the explicitly obtain the Kraus representation of the QEC recovery system from an arbitrary state into the target state σ, which can map [1,56], which can be efficiently implemented with our be pure or mixed. The pumping time depends on the quantum construction of quantum channels. For example, let us consider gate and measurement speed, instead of the natural relaxation the binomial code [57], which uses the larger Hilbert space of rate. higher excitations to correct excitation loss errors in bosonic It is well known that the CPTP map systems. In order to correct up to two excitation losses, the ρ → E (ρ) = Tr(ρ)σ binomial code encodes the two logical basis states as Init √ stabilizes an arbitrary state σ [2,3]. If the target state has |0+ 3|6 |W↑≡ , diagonal representation σ = λ |ψ ψ |, where λ 0 μ μ μ μ μ √ 2 = and μλμ 1, one explicit form of Kraus operators is 3|3+|9 { μ = | |} | |W↓≡ . Ki λμ ψμ i , where i are a basis of the system 2 Hilbert space [47]. Contrary to the conventional approaches discussed in the previous paragraph, this dissipative map For small loss probability γ for each excitation, this encoding 2 bundles the cooling and state preparation steps and pumps scheme can correct errors up to O(γ ), which includes the ˆ an arbitrary state into state σ. In the case where σ is pure, following four relevant processes: identity evolution (I), losing 2 this channel reduces to the “measure and rotate” procedure. one excitation (aˆ), losing two excitations (aˆ ), and back-action Depending on the purity of σ, entropy can be extracted from induced dephasing (nˆ)[57]. Based on the Kraus representation or injected into the system by the ancilla qubit. If we run the of the QEC recovery (with Kraus rank 4), we can obtain the channel construction circuit repeatedly, state stabilization can following set of unitary operations Ub(l) for the construction of be achieved. This allows one to keep alive some nonclassical the QEC recovery channel with an adaptive quantum circuit: resource state in a noisy quantum memory. Pˆ3 Besides pure state initialization for quantum information U˜∅ = , Iˆ − Pˆ3 processing, preparation of carefully designed mixed states may find application in the study of foundational issues of quantum PˆW Pˆ1 U˜0 = , U˜1 = , mechanics such as quantum discord, quantum contextuality, Iˆ − PˆW Iˆ − Pˆ1 and quantum thermodynamics [48–53]. Iˆ U U˜ = , U˜ = nˆ , 00 0ˆ 01 0ˆ B. Quantum error correction U U 2 Besides unique steady states, there are CPTP maps that U˜ = aˆ , U˜ = aˆ , can stabilize multiple steady states or even a subspace of 10 0ˆ 11 0ˆ steady states, which may be used to encode useful classical or ˆ ≡ | + + | quantum information. A practically useful application of such where the projections are defined as Pi k 3k i 3k i CPTP maps with subspaces of steady states is quantum error and PˆW ≡|W↑W↑|+|W↓W↓|, and the unitary operators ˆ = 2 ˆ | correction (QEC). Typical QEC schemes encode quantum UOˆ (O a,ˆ aˆ , nˆ) transform the error states O Wσ back to information in some carefully chosen logical subspaces [1,54] |Wσ for σ =↑, ↓. Explicitly, (or subsystems [55]), and use syndrome measurement and | ˆ † conditional recovery operations to actively decouple the Wσ O ⊥ U ˆ = |Wσ + U , system from the environment. Despite the variety of QEC O ˆ † ˆ σ Wσ |O O|Wσ codes and recovery schemes, the operation of any QEC recovery can always be identified as a quantum channel. where U ⊥ is any isometry that takes the complement of For qubit-based stabilizer codes with Ns stabilizer gen- the syndrome subspace to the complement of the logical erators, the recovery is a CPTP map with Kraus rank 2Ns subspace. In the first two rounds, we perform the projective [1]. We may first use the ancilla to sequentially measure measurements to extract the error syndrome. In the last round, all Ns stabilizer generators to extract the syndrome, and we apply a correction unitary operation to restore the logical finally perform a correction unitary operation conditioned states. Specifically, if the measurement outcome b(2) = (0,0), on the syndrome pattern. Since the stabilizer generators there is no error and identify operation Iˆ is sufficient. If commute with each other, their ordering does not change b(2) = (0,1), there is back-action induced dephasing error, the syndrome. Moreover, the stabilizer measurement does which changes the coefficients of Fock states so we need (2) not require conditioning on previous measurement outcomes, to correct for that with Unˆ .Ifb = (1,1), there is a single because the unitary operation at the lth round is simply excitation loss, which can be fully corrected with Uaˆ .If (2) Ub(l) = Ul = P+ ⊗ Sˆl + P− ⊗ I with Sˆl for the lth stabilizer b = (1,0), there are two excitation losses, which can be fully 1 ± = | ±| |+ | 2 and P 2 ( g e )( g e ), which is independent of the corrected with Uaˆ . Repetitive application of the above QEC

134501-5 CHAO SHEN et al. PHYSICAL REVIEW B 95, 134501 (2017) recovery channel can stabilize the system in the code space where the jump operator J is spanned by |W↑ and |W↓. More interestingly, beyond exact QEC codes there are √ n approximate QEC codes [58–61], which can also efficiently J = κ (a − αi ). correct errors but only approximately fulfill the QEC criterion. i For approximate QEC codes, it is very challenging to ana- lytically obtain the optimal QEC recovery map, but one can The complex variables αi determine the coherent state compo- use semidefinite programming to numerically optimize the nents |αi that span the steady-state subspace. As proposed entanglement fidelity and obtain the optimal QEC recovery in [13] and demonstrated in [20], the dissipation can be map [62–65]. Alternatively one can use the transpose chan- engineered by coupling the system mode and another lossy † nel [59] or quadratic recovery channels [60,66,67] which are mode with Hamiltonian H = J b + H.c., where b is the known to be near optimal. All these recovery channels can annihilation operator for the lossy mode. Practically, it is be efficiently implemented with our general construction of challenging to generate desired engineered dissipation that is CPTP maps. much stronger than the undesired dissipations (e.g., dephasing, Kerr effect, etc). In addition, it is difficult to extract the Hamiltonian H associated with higher-order nonlinearity, in C. Markovian channels order to have a higher-dimensional steady state subspace Recently there has been growing interest in designing and with more coherent components. With our approach, however, engineering open system dynamics for quantum information the effective rate κ can be large and determined by the processing [5,10,11,14,68], which uses Markovian channels time scale to implement the circuits, which is limited by τ the duration of gates and measurements, and the delay of Lt dt ρ → EMC,t (ρ) = T[e 0 ]ρ, adaptive control. Moreover, the construction can easily extend to the case that simultaneously stabilizes many coherent L where T stands for time ordering, and t is the time-dependent components. Lindbladian operator that has general form With the channel construction presented here, we can now obtain Lindbladian dynamics EMC,t = exp(L t) for any given i →∞ Lt (ρ) =− [H,ρ] t. Sometimes we are interested in the channel for t h¯ (or equivalently the strong pumping limit κ →∞), E ∞, MC, + † − 1 † + † and it was recently shown that any more general (i.e., non- hn,m LnρLm (ρLmLn LmLnρ) , 2 Markovian) channel can be embedded in EMC,∞ [16]. For our n,m approach, sending t to ∞ does not cost us an infinite amount where Ln are jump operators. Markovian channels are a special of time, since the number of cycles in our construction circuit only scales logarithmically with the Kraus rank of E ∞. class of CPTP maps [21]. In contrast to the continuous time MC, τ Lt dt In numerical calculations, the Kraus rank is not a clear-cut evolution approach [10–12], we construct EMC,t = T [e 0 ] directly, which is advantageous in that it does not take more quantity even when we have obtained the most economic Kraus time to see results for larger τ because no Trotterization or representation. So we define and examine the “magnitudes” of ≡ † stroboscopic control is required. We consider the following the Kraus operators λi Tr(Ki Ki ) and remove Ki from the −10 cat-pumping example to manifest these points. description of the channel if λi < 10 . Note that λi /d is Using a specifically engineered dissipation for a cavity the probability for Ki to act on the system when the input = mode, one can stabilize a two-dimensional steady-state sub- state is the maximally mixed state ρ I/d.Theλi also turn space spanned by the so called cat code [13,20]. The required out to be the eigenvalues of the Choi matrix, see Appendix A dissipation can be described by the following time-independent for details. Numerically we found that E∞ has lower Kraus E Lindbladian: rank than MC,t with finite t, see Fig. 5 for two examples. In the infinite time limit, the Kraus rank scales linearly with the L = † − 1 † + † = + (ρ) JρJ 2 (J Jρ ρJ J ), dimension of the truncated Hilbert space d nc 1 (where

≡ † E = L FIG. 5. The magnitudes of the Kraus operators λi Tr(Ki Ki ), corresponding to t exp( t) for (a) two-legged cat pumping and (b) four-legged cat pumping. Here we set κ = 1. In the long time limit, both channels have Kraus rank approximately equal to the size of the −10 truncated Hilbert space d = nc + 1, where nc is the maximal photon number. We treat all λi smaller than 10 as 0. The figures show results with nc = 38 but we verified that our observation remains valid for any sufficiently large nc.

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FIG. 6. All possible trajectories for pumping a vacuum state |0 to the subspace spanned by |±α with α = 1.1. Depending on the probabilistic ancilla readout, the system evolves√ along different trajectories in each run of the circuit. However, since the steady state of the system is a pure state |ψf =(|α+|−α)/ 2, which cannot be decomposed as a probabilistic mixture of different states, the final state for each trajectory is always the same pure state |ψf . The two outcomes of the first round are only slightly different. Two of the four outcomes of the second round are also very similar to the others. nc is the photon number truncation), much smaller than the corner transpose” channel) for d-dimensional systems [21] largest possible value d2. Figures 6 and 7 (corresponding to n = 2 and n = 4 coherent components) show trajectories [69] of the system evolution ρTc + I Tr(ρ) T (ρ) = , under our constructed channel for a large t ∼ 103/κ. In each 1 + d run of the simulation, the ancilla measurement results that correspond to different trajectories are probabilistic. If the system starts in |0, |1,or|2, the correct steady state is where ρTc is the “corner transposed” density matrix (i.e., pure. So whichever trajectory the system follows, it ends up exchanging the matrix elements ρ1,d and ρd,1 while keeping in the same√ pure state. If the system starts in a state like all other elements unchanged). Following Ref. [21], the (|0+|2)/ 2, the steady state is a mixed state, in which partial corner transpose channel has diagonal representation case different trajectories lead to different final states. But in the generalized Gell-Mann basis, with identical eigenvalues the probabilistic mixture of all these final states make up the 1/(d + 1), except for two√ basis elements—the eigenvalue is 1 × − + expected steady state density matrix ρf = Et (ρinit). for basis element Id d / d, and the eigenvalue√ is 1/(d 1) Our approach of constructing CPTP maps thus provides for basis element (|d1|+|1d|)/ 2. Hence, the deter- 2 another promising pathway to efficiently pump the cavity minant detT =−(d + 1)1−d is negative. In contrast, the mode into the cat-code subspace using approximately log2(d) determinant for Markovian channels are always nonnegative. rounds of operations, each of which consists of adaptive Therefore, the partial corner transpose cannot be obtained from SU(2d) unitary gates, qubit QND measurement, and storing Markovian channels [70]. the measurement outcome. In the exact same fashion, we can We have obtained an explicit construction of {Ub(l) } for the construct CPTP maps that manipulate the logical states living partial corner transpose channel with d = 3, as detailed in in the code subspace, which can, e.g., implement a digital Appendix C. version of holonomic gates [15].

E. Quantum instrument and POVM D. Exotic channels The construction of CPTP maps can be further extended if Besides Markovian channels, there are also exotic CPTP the intermediate measurement outcomes are part of the output maps that cannot be obtained from time-dependent Lind- together with the state of the quantum system, which leads bladian master equations. Hence, these channels are not to an interesting class of quantum channel called a quantum accessible in previous proposals of open system evolution instrument (QI) [2,3,28]. QIs enable us to track both the under Lindbladian master equations [10–12]. For example, classical measurement outcome and the post-measurement we can deﬁne the following CPTP map (called the “partial state of the quantum system. Mathematically, the quantum

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√ FIG. 7. Example trajectories for four-component cat pumping starting with four different initial states |0, |2,(|0+|2)/ 2, and coherent state |α˜ = 2.3. Here the steady coherent components are |α, |iα, |−α,and|− iα with α = 2.5. The binary number on the arrow indicates the ancilla measurement outcome. For the first two cases, since the steady state is a pure state which cannot be decomposed as a probabilistic mixture of different states, the final state for each trajectory is always the same pure state |ψf . For the third case, the steady state is a mixed state ρf , so different trajectories give different pure states. Since the ancilla measurement results are discarded, the output state for the system is an ensemble of the different final states, which coincides with ρf . The fourth case starts near the steady state subspace and is slowly pulled into it. The trajectory shown is the dominant one which is taken with probability higher than 0.96. Dashed circles show the position of |α = 2.5.

134501-8 QUANTUM CHANNEL CONSTRUCTION WITH CIRCUIT . . . PHYSICAL REVIEW B 95, 134501 (2017) instrument has the following CPTP map: device

M M → E = | | ρ → EQI(ρ) = Eμ(ρ) ⊗|μμ|, (12) ρ POVM(ρ) Tr[ μρ] μ μ , = μ=1 μ 1 where |μμ| are orthogonal projections of the measurement which is characterized by a set of Hermitian positive semidef- E inite operators { }M that sum to the identify operator device with M classical outcomes, and μ are completely pos- μ μ=1 M E μ = I. For positive semidefinite μ, we can decompose itive trace nonincreasing maps, while μ=1 μ(ρ) preserves μ E = † the trace. Note that μ(ρ) gives the post-measurement state it as μ j Kμ,j Kμ,j with a set of Kraus operators associated with outcome μ. {K } . Therefore, the circuit for the quantum instru- μ,j j=1,...,Jμ As illustrated in Fig. 8, our channel construction can ment also implements the POVM if we remove the quantum implement the QI as follows. (1) Find the minimum Kraus system from the QI output, EPOVM(ρ) = Trsys[EQI(ρ)], which representation for Eμ (each with rank Jμ) with Kraus operators reduces to the binary tree construction scheme of a POVM as Kμ,j for j = 1, 2, ...,Jμ. (2) Introduce binary labeling of proposed by Andersson and Oi [25]. A POVM can be useful for these Kraus operators Kb(L) , where the binary label has length quantum state discrimination. It is known to be impossible for = + = (L1) L L1 L2 with the first L1 log2 M bits b to encode any detector to perfectly discriminate a set of nonorthogonal = μ μ and the remaining L2 log2 maxμ(J ) bits to encode j quantum states. An optimal detector can achieve the so-called (padding with zero operators to make a total of 2L Kraus Helstrom bound [71], by properly designing a POVM (in this operators). (3) Use the quantum circuit with L rounds of case a PVM—projection valued measure). For example, in adaptive evolution and ancilla measurement. (4) Output the optical communication, quadrature phase shift keying uses final state of the quantum system as well as b(L1) that encodes four coherent states with different phases |α, |iα, |−α, μ associated with the M possible classical outcomes. This and |−iα to send two classical bits of information. With our enables us to construct the arbitrary QI described in Eq. (12). scheme it is straightforward to implement the optimal POVM The QI is a very useful tool for implementation of complicated given in Ref. [72], which is a rank-4 POVM. conditional evolution of the system. It can be used for quantum As summarized in Fig. 8, we may classify three different information processing tasks that require measurement and situations for CPTP maps based on the output: (a) standard adaptive control. quantum channel with the quantum system as the output, (b) If we remove the quantum system from the QI output, POVM with the classical measurement outcomes as the output, we effectively implement a positive operator valued measure and (c) QI with both the quantum system and the classical (POVM), which is also referred to as a generalized quantum measurement outcomes for the output. In principle, all three measurement. A POVM is a CPTP map from the quantum situations can be reduced to the standard quantum channel state of the system to the classical state of the measurement with an expanded quantum system that includes an additional measurement device to keep track of the classical measurement outcomes. In practice, however, it is much more resource efficient to use a classical memory for classical measurement outcomes, so that we can avoid working with the expanded quantum system.

V. DISCUSSION So far we have assumed a two-level ancilla for our channel construction, which can be generalized to an ancilla with higher dimensions. If we use an s-dimensional ancilla, we can use an s-ary tree construction of the quantum channel with Kraus rank N, consisting of logs N rounds of adaptive evolution and ancilla measurement. We emphasize that the adaptive control is essential for arbitrary channel construction with a small (low-dimensional) ancilla. Without adaptive control, the constructed channel is a product of channels T =···T3T2T1, and it excludes FIG. 8. Three different types of CPTP maps. (a) To implement a indivisible channels which cannot be constructed with a single standard CPTP on the system qudit, all ancilla measurement records round of operation or decomposed into a product of nonuni- should be thrown away. (b) A generalized measurement does not tary channels [21]. Although the approach of Trotterization concern the system state after measurement, so only the ancilla and stroboscopic control can construct Markovian channels measurement record is kept. (c) A quantum instrument keeps the both without adaptive control, that approach has an overhead that the post-measurement state of the system and outcome μ, encoded increases with the duration of the Markovian evolution [12], by the ﬁrst L1 bits of the ancilla measurement record. The remaining while our construction has a bounded overhead that scales L2 bits of the measurement record are thrown away. In the ﬁgure, logarithmically with the relevant dimensions of the quantum L1 = 2andL2 = 1. system.

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Besides developing a control toolbox for quantum informa- superoperator T , such that tion processing, our channel construction protocol may also be useful for investigating open quantum systems, with the ρ˜ij = Tij,mnρmn potential advantages of reduced overhead in channel construc- m,n tion and the new ingredient of indivisible channels, which or are not accessible with conventional reservoir engineering of Markovian channels [4,9,34,73,74]. ρ˜ = T ·ρ, In experimental realizations, there will be imperfections in = T the unitary gates Ub(l) and ancilla measurements. Fortunately, whereρ ˜ (ρ). This matrix form is particularly useful the quantum circuit for channel construction only has n = when one considers the concatenation of channels. Applying T T log2 N 2log2 d rounds of gate and measurement. If the channel 1 first and then 2 results in the overall channel error per round is , then the overall error rate of the channel represented by the matrix T = T2 · T1, where “·” indicates ∼ construction is only n log2 d. More rigorously, we may matrix multiplication. The matrix form also allows one use the diamond norm distance to upper bound the error to characterize channels with the determinant det(T ). One associated with each round of operation [3], and n rigorously interesting property is that for Markovian channels or Kraus bounds the diamond norm distance of the constructed quantum rank-2 channels, the determinant is always positive [21]. The channel. downside of this representation is that it is not obvious whether agivenT qualifies as a CPTP map. We will need to convert it to the Jamiolkowski-Choi matrix representation or Kraus VI. CONCLUSION representation to verify that. Conversely, given a channel We have provided an explicit cQED proposal to construct in Kraus form, the superoperator matrix can be obtained arbitrary CPTP maps, assisted by an ancilla qubit with QND straightforwardly, readout and adaptive control. Our construction has vari- N ous applications, including system initialization/stabilization, ∗ T = K ⊗ K . quantum error correction, Markovian and exotic channel i i simulation, and generalized quantum measurement/quantum i instruments construction. Such a construction can also be implemented with other physical platforms such as cavity QED 2. Jamiolkowski-Choi matrix representation and motional modes of trapped ions. From the well known channel-state duality (Jamiolkowski- Note added. While finalizing the manuscript, the authors Choi isomorphism) [38,76] we know that each channel T for became aware of a related work on quantum channels [75], a system with d-dimensional Hilbert space H corresponds which studies a different way to construct a channel. In contrast (one-to-one) to a state (a density matrix) on H ⊗ H, to that work focusing on minimizing the number of C-NOT gates, here we consider adaptation to physical platforms, and τ = (T ⊗ I)(||), discuss various applications. where |= √1 |i⊗|i is the maximally entangled state d i of the two subsystems. A closely related matrix is the Choi ACKNOWLEDGMENTS matrix which is only a constant multiple of the Jamiolkowski = We thank Reinier Heeres, Phillip Reinhold, and Changling matrix M dτ, where d is the dimension of the Hilbert space. Zou for helpful discussions. We acknowledge support A convenient fact to note is that M and the superoperator from ARL-CDQI, ARO (W911NF-14-1-0011, W911NF-14- matrix T are related in a simple way, 1-0563), ARO MURI (W911NF-16-1-0349 ), NSF (DMR- T = M . 1609326, DGE-1122492), AFOSR MURI (FA9550-14-1- ij,mn im,jn 0052, FA9550-15-1-0015), the Alfred P. Sloan Foundation Being a density matrix, τ is Hermitian. Moreover, τ is (BR2013-0049), and the Packard Foundation (2013-39273). semipositive definite if and only if T is completely positive; τ is normalized if T is trace preserving. APPENDIX A: REPRESENTATIONS It is straightforward to convert the Choi matrix M to the OF QUANTUM CHANNELS Kraus representation. If M is diagonalized, In this Appendix we review some basics on alternative = † M λi vi vi , ways a CPTP map can be represented and how to convert i back and forth between different representations. Since our 2 scheme favors the Kraus representation as our “canonical where vi are d dimensional eigenvectors√ of τ, the Kraus representation”, it is important to understand how to convert a operators are obtained by rearranging λi vi as d × d matrices. target channel in other representations to the Kraus form. Clearly the number of nonzero eigenvalues λi is the Kraus rank of the corresponding channel. Later we will often check the eigenvalue spectrum of the Choi matrix of a channel to 1. Superoperator matrix representation determine its Kraus rank. For numerical calculation we usually Since CPTP maps are linear in the density matrix ρ,we make a truncation of the eigenvalues. For example, we may can treat ρ as a vector and write down the matrix form of the set all eigenvalues smaller than 10−10 to 0.

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APPENDIX B: PROOF OF QUANTUM which implies the same inequality for their support projections CHANNEL CONSTRUCTION † † + + Vb(l) Pb(l) Vb(l) Vb(l 1) Pb(l 1) Vb(l+1) . We now prove that our channel construction correctly † † ⊥ = = + = implements the target CPTP map. To justify the channel Moreover, since Vb(l) Vb(l) I Vb(l 1) Vb(l+1) and Pb(l) construction, we need to show that (a) the submatrices I − Pb(l) ,wehave bl+1|Ub(l) |0 fulﬁll the isometry condition ⊥ † ⊥ † + Vb(l) Pb(l) Vb(l) Vb(l 1) Pb(l+1) Vb(l+1) , (B4) † which demonstrates that the orthogonal support projection (bl+1|Ub(l) |0) bl+1|Ub(l) |0=Id×d (B1) grows with l. Using the fact that if projectors P1 P2 then bl+1=0,1 P1 = P1P2P1,wehave ⊥ † ⊥ † ⊥ † ⊥ † V (l) P V = V (l) P V V (l+1) P + V + V (l) P V , for all b(l) and l = 1,2,...,L− 1, and (b) the accumulated b b(l) b(l) b b(l) b(l) b b(l 1) b(l 1) b b(l) b(l) evolution along the binary tree indeed implements the corre- which is equivalent to sponding Kraus operator ⊥ ⊥ † ⊥ † ⊥ = + Pb(l) Pb(l) Vb(l) Vb(l 1) Pb(l+1) Vb(l+1) Vb(l) Pb(l) . (B5)

(bL|Ub(L−1) |0) ···(bl+1|Ub(l) |0) ···(b1|Ub(0) |0) = Kb(L) . Before we prove Eqs. (B1) and (B2), we ﬁrst note that (B2) + 1 | | = + + √ bl+1 Ub(l) 0 Mb(l 1) M (l) Qb(l) b 2 † −1 † First, we show that = + + Vb(l 1) Db(l 1) Vb(l+1) Vb(l) Db(l) Vb(l) ⎛ ⎞ + √1 ⊥ † Vb(l) Pb(l) Vb(l) 2 † = † = ⎝ † ⎠ 2 Vb(l) Db(l) Vb(l) Kb(L) Kb(L) Kb(L) Kb(L) ··· † −1 † bl+1,...,bL bl+1 bl+2, ,bL = V (l+1) D (l+1) V + V (l) D V b b b(l 1) b b(l) b(l) 2 † = V (l+1) D (l+1) V + . (B3) 1 ⊥ † ⊥ † b b b(l 1) + √ + Vb(l 1) P (l+1) V (l+1) Vb(l) P (l) V (l) = b b b b bl+1 0,1 2 † −1 = + + Since the right-hand side is a sum of two nonnegative matrices, Vb(l 1) Db(l 1) Vb(l+1) Vb(l) Db(l) we also have the inequality + √1 ⊥ † ⊥ † P (l+1) V (l+1) Vb(l) P (l) V (l) , 2 b b b b † † 2 + 2 Vb(l) Db(l) Vb(l) Vb(l 1) Db(l+1) Vb(l+1) , where the third equality uses Eq. (B4). Similarly,

+ 1 † −1 1 † ⊥ † | | = + + √ + = + + √ + bl+1 Ub(l) 0 Kb(l 1) M (l) Wb(l 1) V (l+1) Qb(l) Kb(l 1) Vb(l) D (l) Pb(l) Wb(l 1) V (l+1) Vb(l) P (l) V (l) b 2 b b 2 b b b

−1 1 † ⊥ † ⊥ † = + + √ + + Kb(l 1) Vb(l) D (l) Pb(l) Wb(l 1) V (l+1) Vb(l 1) P (l+1) V (l+1) Vb(l) P (l) V (l) b 2 b b b b b −1 1 ⊥ † ⊥ † = + + √ + Kb(l 1) Vb(l) D (l) Pb(l) Wb(l 1) P (l+1) V (l+1) Vb(l) P (l) V (l) . b 2 b b b b To prove Eq. (B1)forl = 0,1,...,L− 2, we use † (bl+1|Ub(l) |0) bl+1|Ub(l) |0 b + =0,1 l 1 ⎡ ⎤ ⎣ † −1 † † −1 1 ⊥ † ⊥ † ⊥ † ⊥ ⎦ † = V (l) D (l+1) V + V (l) D P (l) D (l+1) V + V (l) D P (l) + (P (l+1) V + V (l) P (l) ) P (l+1) V + V (l) P (l) V b b b(l 1) b b(l) b b b(l 1) b b(l) b 2 b b(l 1) b b b b(l 1) b b b(l) bl+1=0,1 bl+1=0,1 ⎡ ⎛ ⎞ ⎤ ⎣ −1 † ⎝ 2 † ⎠ −1 1 ⊥ † ⊥ † ⊥ ⎦ † = V (l) P (l) D V V (l+1) D + V + V (l) D P (l) + P V V (l+1) P + V + V (l) P V b b b(l) b(l) b b(l 1) b(l 1) b b(l) b 2 b(l) b(l) b b(l 1) b(l 1) b b(l) b(l) bl+1=0,1 bl+1=0,1 = + ⊥ † = † = Vb(l) [Pb(l) Pb(l) ]Vb(l) Vb(l) IVb(l) I, ⊥ = where the ﬁrst equality uses the orthogonality property Pb(l) Pb(l) 0, the third equality uses Eqs. (B3) and (B5). Similarly, we can prove Eq. (B1)forl = L − 1.

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To prove Eq. (B2)wehave

bL|Ub(L−1) |0···b2|Ub(1) |0b1|Ub(0) |0 −1 † † −1 † † = − − ··· + + ··· (Kb(L) Vb(L 1) Db(L−1) Pb(L 1) Vb(L−1) ) (Vb(l 1) Db(l 1) Vb(l+1) Vb(l) Db(l) Pb(l) Vb(l) ) (Vb(2) Db(1) Vb(1) ) † † † = − − ··· ··· Kb(L) (Vb(L 1) Pb(L 1) Vb(L−1) ) (Vb(l) Pb(l) Vb(l) ) (Vb(1) Db(1) Vb(1) ) † = − − Kb(L) (Vb(L 1) Pb(L 1) Vb(L−1) ) † † = − − (Wb(L) Db(L) Vb(L) )(Vb(L 1) Pb(L 1) Vb(L−1) ) † † = − − (Wb(L) Db(L) Pb(L) Vb(L) )(Vb(L 1) Pb(L 1) Vb(L−1) ) † † † = − − (Wb(L) Db(L) Vb(L) Vb(L) Pb(L) Vb(L) )(Vb(L 1) Pb(L 1) Vb(L−1) ) = † † Wb(L) Db(L) Vb(L) Vb(L) Pb(L) Vb(L) = † Wb(L) Db(L) Pb(L) Vb(L)

= Kb(L) , ⊥ = where the ﬁrst equality only has one nonzero product, because all other terms vanish due to the orthogonality property Pb(l) Pb(l) 0 ⊥ = † = −1 = = = = and Pb(0) 0, the second equality exploits Vb(l) Vb(l) I, Db(l) Pb(l) Db(l) Pb(l) , and Vb(0) Db(0) Pb(0) I, and the third and the † † † − − = last but two equalities require the projection relation (Vb(l) Pb(l) Vb(l) )(Vb(l 1) Pb(l 1) Vb(l−1) ) (Vb(l) Pb(l) Vb(l) ). Therefore, we have proven both Eqs. (B1) and (B2), which fully justify our explicit construction of the CPTP map.

APPENDIX C: EXPLICIT CIRCUITS FOR AN EXAMPLE EXOTIC CHANNEL We show an explicit construction of the isometries needed for the construction of the exotic channel ρTc + I Tr(ρ) T (ρ) = 1 + d for the case of d = 3, ⎛√ √ ⎞ 10+ 2 ⎜ 4 ⎟ ⎜ 2+ √1 ⎟ ⎜ 2 ⎟ ⎜ ⎟ ⎜ 2 √ √ ⎟ | | ⎜ 10+ 2 ⎟ 0 Ub(0) 0 = ⎜ √ 4 ⎟ | | ⎜ √ ⎟, 1 Ub(0) 0 ⎜ 6− 2 ⎟ ⎜ 4 ⎟ ⎜ 2− √1 ⎟ ⎜ 2 ⎟ ⎝ 2 √ √ ⎠ 6− 2 4 ⎛√ ⎞ √ ⎛ ⎞ 29+2 2 √ ⎜ ⎟ ⎜ 7 √ ⎟ ⎜ 2(6 + 2)/17 ⎟ ⎜ + ⎟ ⎜ ⎟ ⎜ (3 2)/7 √ ⎟ ⎜ 0 ⎟ ⎜ √ ⎟ ⎜ √ ⎟ ⎜ 29+2 2 ⎟ ⎜ ⎟ | (1) | | (1) | 2(6 + 2)/17 0 Ub =0 0 = ⎜ 7 ⎟ 0 Ub =1 0 = ⎜ ⎟ | | ⎜ √ 2 ⎟, | | ⎜ √ ⎟, 1 Ub(1)=0 0 ⎜ √ ⎟ 1 Ub(1)=1 0 ⎜ − ⎟ ⎜ 10+ 2 ⎟ ⎜ (5 2 2)/17 ⎟ ⎜ 1 ⎟ ⎜ 1 ⎟ ⎜ 2+ √1 ⎟ ⎝ ⎠ ⎝ 2 ⎠ √ 2 √ √ (5 − 2 2)/17 10+ 2 ⎛ √ ⎞ + ⎛ ⎞ ⎜ (5 2 2)/17 ⎟ ⎜ ⎟ 001 ⎜ 1 ⎟ ⎜ ⎟ ⎜ √ ⎟ ⎜000⎟ 0|U (2)= |0 ⎜ (5 + 2 2)/17⎟ 0|U (2)= |0 ⎜100⎟ b 00 = ⎜ ⎟, b 01 = ⎜ ⎟, 1|U (2)= |0 ⎜ −√ 2 ⎟ 1|U (2)= |0 ⎜000⎟ b 00 ⎜ √ ⎟ b 01 ⎝ ⎠ ⎜ 6+ 2 ⎟ 000 ⎝ 0 ⎠ 2 010 √ √ 6+ 2

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⎛ √ ⎞ ⎛ ⎞ + 010 ⎜0 (4 2)/70⎟ ⎜ ⎟ ⎜ ⎟ ⎜001⎟ ⎜000⎟ 0|U (2)= |0 ⎜000⎟ 0|U (2)= |0 ⎜000⎟ b 10 = ⎜ ⎟, b 11 = ⎜ ⎟. 1|U (2)= |0 ⎜000⎟ 1|U (2)= |0 ⎜1 ⎟ b 10 ⎝ ⎠ b 11 ⎜ √ ⎟ 100 ⎝ ⎠ − (3 − 2)/7 000 1

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