Observing a Changing Hilbert-Space Inner Product

In quantum mechanics, physical states are represented by rays in Hilbert space $\mathscr H$, which is a vector space imbued by an inner product $\langle\,|\,\rangle$, whose physical meaning arises as the overlap $\langle\phi|\psi\rangle$ for $|\psi\rangle$ a pure state (description of preparation) and $\langle\phi|$ a projective measurement. However, current quantum theory does not formally address the consequences of a changing inner product during the interval between preparation and measurement. We establish a theoretical framework for such a changing inner product, which we show is consistent with standard quantum mechanics. Furthermore, we show that this change is described by a quantum channel, which is tomographically observable, and we elucidate how our result is strongly related to the exploding topic of PT-symmetric quantum mechanics. We explain how to realize experimentally a changing inner product for a qubit in terms of a qutrit protocol with a unitary channel.

On the dense subspace of H spanned by finite linear combinations of {|e j ⟩ , j ∈ N} the inverse of η is given by which is clearly not bounded on the whole H .For η to be invertible, we need to enforce an additional constraint, namely that the range of η coincides with H , or in other words, η is surjective.Below we prove the corrected version of Lemma 2, which is stated above.
Invertibility of the metric operator also guarantees that H η defined in the original paper is a Hilbert space with respect to the modified inner product ⟨•|•⟩ η [1, Appendix A].Subsequently, surjectivity of η, in addition to boundedness and positive-definiteness, is required for defining the change in representation R η [Eq.(2) of the original paper] and the inner-product changing operation E η Eq. (4) of the original paper].
Our results concerning the simulation of PT-symmetric Hamiltonians hold without any additional assumptions.This is because we exclusively work with unbroken PT-symmetric Hamiltonians in finite dimensions.The rank-nullity theorem implies that all injective operators with finite-dimensional domains are additionally surjective.Therefore, Lemma 2 as originally stated in the paper is valid for any finite-dimensional Hilbert space H .
We also note that for any unbroken PT-symmetric Hamiltonian in a separable, infinite dimensional Hilbert space H , an η ∈ B(H ) satisfying the quasi-Hermiticity condition that is both surjective and positive-definite can be constructed [4,Thm. 4.3] (see also [5,Eq. (23)]).This is a consequence of the fact that unbroken PT-symmetric Hamiltonians in infinite dimensions are defined to have eigenvectors forming a Riesz basis, in addition to these vectors being invariant under the action of the PT operator [6].
In conclusion, amendment to Lemma 2 mandates that the operations R η [Eq.(2) of the original paper] and E η [Eq.(4) of the original paper] are to be defined only under the additional constraint that η is surjective.The rest of our original paper is correct and remains unchanged after these modifications.

Observing a changing Hilbert space inner product
Salini Karuvade, Abhijeet Alase, and Barry C. Sanders Institute for Quantum Science and Technology, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada In quantum mechanics, physical states are represented by rays in Hilbert space H , which is a vector space imbued by an inner product ⟨ | ⟩, whose physical meaning arises as the overlap ⟨ϕ|ψ⟩ for |ψ⟩ a pure state (description of preparation) and ⟨ϕ| a projective measurement.However, current quantum theory does not formally address the consequences of a changing inner product during the interval between preparation and measurement.We establish a theoretical framework for such a changing inner product, which we show is consistent with standard quantum mechanics.Furthermore, we show that this change is described by a quantum operation, which is tomographically observable, and we elucidate how our result is strongly related to the exploding topic of PT-symmetric quantum mechanics.We explain how to realize experimentally a changing inner product for a qubit in terms of a qutrit protocol with a unitary channel.
Hilbert-space inner product is fundamental to quantum mechanics (QM), and its physicality relates to norm through the Born interpretation and to fidelity and distinguishability through its complex angle [7].The uniqueness of the inner product associated to a quantum system has come under scrutiny following the advent of PT-symmetric QM.PT-symmetric systems are described by non-Hermitian Hamiltonians invariant under the combined action of parity (P) and time (T) inversion symmetries [8][9][10][11], and they are predicted to exhibit novel physical phenomena which have been simulated on a variety of experimental platforms [12][13][14][15][16][17][18][19][20].These phenomena have been explained by observing that non-Hermitian Hamiltonians with unbroken PT symmetry are Hermitian with respect to a different Hilbertspace inner product [9,[21][22][23].Changing Hilbert-space inner-product is valuable for certain quantum information processing (QIP) tasks [24] such as non-orthogonal state discrimination [25], cloning [26] and quantum algorithms [27,28], but perfunctory applications have led to counter-factual conclusions [24,29,30] including violation of the no-signalling principle [31].Our aim is to prescribe the correct procedure for changing Hilbert-space inner product and to devise an experiment to validate our prescription.
Consistency of a changing Hilbert-space inner product with standard QM and the unobservability of such a change in closed systems have been investigated.A C *algebraic approach shows that a set of non-Hermitian operators comprises the observables of a quantum mechanical system if and only if the operators are Hermitian with respect to a new Hilbert-space inner product [1].Such a modified inner product is the key to proving the equivalence of PT-symmetric QM with the Dirac-von Neumann formulation of QM in the case of closed systems i.e., systems in which every time evolution is a unitary operation [6,9,21,23,[32][33][34][35].Furthermore, this equivalence implies that any change in inner product is unobservable in experiments on closed systems [36].Therefore, the above proposals that use the inner-product change for QIP tasks as well as the counter-factual claims are not applicable to closed systems.
PT-symmetric Hamiltonians and a changing Hilbertspace inner product are known to be consistent with standard QM for closed systems, but they are not yet known to be consistent for open systems.To solve these outstanding problems, we construct an operational framework, consistent with the C * -algebraic formulation of QM, which accommodates a change in inner product between preparation and measurement.Furthermore, neither PT symmetry nor a changing Hilbert-space inner product are observable in closed systems, but could be observable in open systems [36].We show our change in inner product is implemented by a quantum operation (henceforth assumed to be completely positive and trace non-increasing), which can be observed using tomography.Next we connect our framework to the burgeoning topic of PT-symmetric QM by explaining how an innerproduct-changing quantum operation can be used to implement PT-symmetric dynamics in an open system.Finally, at the empirical level, we describe a potential experimental simulation for changing the inner product of a qubit by subjecting a qutrit to unitary evolution and neglecting the third Hilbert-space dimension during preparation and measurement but not during evolution.We also extend this simulation procedure to d-dimensional systems.
To construct the operational framework for changing the inner product associated to a quantum system between preparation and measurement, we adopt the C *algebraic framework of QM [53], which provides freedom in representing a given system on different Hilbert spaces following the Gel'fand-Naimark-Segal (GNS) construction [54,55].We employ this representation freedom first to construct representations of the C * algebra on a pair of Hilbert spaces whose inner products are related by a given metric operator η.We then define the change in inner product by η as the identity isomorphism between the two Hilbert spaces.To operationalize the change in inner product, we use commutative diagrams that connect this isomorphism to a quantum operation between the bounded operators on the two Hilbert spaces and finally observe that the quantum operation induces an observable physical transformation on the system.In the operational approach, the operators of a quantum system form a unital C * algebra A = {A}, which is equipped with a * operation that captures the notion of adjoint.The algebra A is representable on a possibly infinite dimensional Hilbert space H = (V , ⟨ | ⟩), comprising a complete vector space V and an inner product ⟨ | ⟩, which is a non-degenerate sesquilinear form.In Fig. 1(a), observables are self-adjoint elements of A and correspond to allowed measurements.A representation of A is a product-preserving linear map where B(H ) denotes the space of bounded linear operators acting on H and † denotes the Hermitian conjugate.Such a representation can be obtained using the GNS construction [54,55].Product preservation ensures that if I is the identity operator in A, then π(I) is the identity operator in B(H ).An operator M ∈ B(H ) satisfying M † = M is called a self-adjoint or a Hermitian operator.
We now define states and explain how to represent states as operators on Hilbert space.States correspond to allowed preparations of the system (Fig. 1(b)).A state ω is a positive linear functional on A that is normalized, i.e. ω(I) = 1.This definition is extended to include any subnormalized positive linear functional, i.e. ω(I) ≤ 1, which corresponds to probabilistic preparation in the state ω/ω(I) with probability ω(I) [56].Supernormalized positive linear functionals are not valid states according to this probabilistic interpretation [57] → ω if and only if the expectation value tr(ρπ(A)) = ω(A) ∀A.As # π is uniquely determined by π, we say that ω is represented by ρ ∈ D(H ) under π.We denote by S = {ω}, the set of all states that are represented by density operators under π.For the special case of pure ω, ρ = |ψ⟩ ⟨ψ| for some |ψ⟩ ∈ H with ⟨ψ|ψ⟩ ≤ 1.The transformation |ψ⟩ lift → |ψ⟩ ⟨ψ| relates Hilbert-space vectors to the density operators in D(H ) [58].
Now that we have explained states and their representations, we now discuss changing representation to being over a different Hilbert space.Given a self-adjoint positive-definite metric operator η ∈ B(H ), a new Hilbert space H η = (V , ⟨ | ⟩ η ) can be constructed [1] such that the inner products of the two Hilbert spaces are related by where R = η −1 /2 is a linear isometry from H to H η .We note that this isometry has been used to prove that Hamiltonians with unbroken PT symmetry are consistent with standard QM, in the case of closed systems [6,24,32,33,45].We refer to the quantum channel R η as a 'change in representation' (Fig. 1b).
In representation π η , the state ω representations π and π η are physically, i.e. observationally, indistinguishable.The right-hand side of Eq. ( 5) can also be interpreted as preparation (state) described in π followed by a change in representation from π to π η effected by R η and finally measurement (observable) described in π η .Change in representation between preparation and measurement sets the stage for our definition of change in inner product.We define a change in inner product by η to be the identity isomorphism I η : H → H η such that every |ψ⟩ → |ψ⟩.For any pair |ψ⟩ , |ϕ⟩ ∈ H , the inner product between the pair of transformed vectors I η |ψ⟩ , I η |ϕ⟩ is ⟨ψ|η|ϕ⟩, and the change is trivial if η = π(I); i.e. for all pairs |ψ⟩ , |ϕ⟩, ⟨ψ|η|ϕ⟩ = ⟨ψ|ϕ⟩.Our definition is motivated by proposals to effect PT-symmetric evolution and measurement by changing the Hilbert-space inner product [25,27] but without a prescription for making such changes operationally or mathematically.Next we explain separately, for the cases η ≤ π(I) and η ≰ π(I), how the isomorphism I η can be physically realized as a quantum operation.
The change in inner product by η ≤ π(I) is physically realizable via the operation The operation E η induces a linear map F η : S → S such that, for any pure ω ∈ S, both ω and F η (ω) are represented by the same |ψ⟩ under the representations π and π η respectively.However, ω and F η (ω) are not necessarily the same state (Fig. 1c).Even in the special case where the two states differ by a scaling factor, they are inequivalent in our setting.The expectation value of I with respect to F η (ω) gives the success probability of the inner-product changing quantum operation on the state ω.E η can be implemented experimentally by lossy purity-preserving operations, i.e., operations that are not necessarily deterministic and transform the set of pure states into itself.In the Heisenberg picture, the operators transform according to the map This transformation E op η could modify commutator relations as we show in Appendix C.
In the case η ≰ π(I), E η is completely positive but trace-increasing for some ρ ∈ B(H ) and hence not a quantum operation.In such cases, a scaled version of change in inner product can be implemented in the following way: choose κ ∈ (0, 1) such that κη ≤ π(I) and observe that E κη = κE η with E κη a quantum operation.Therefore, E κη implements change in inner product by η ≰ π(I) up to a scaling factor κ.Such a scaled version of change in inner product is useful to reverse the effect of operation E η when η ≤ π(I).In this case, the isomorphism I η −1 : H η → H reverses the change in inner product and the corresponding E η −1 is not a valid operation because η −1 ≥ π η (I).Nevertheless, we can choose κ = 1 /∥η −1 ∥, where ∥ • ∥ denotes the operator norm [59], and observe that Therefore, the operation E κη −1 : B(H η ) → B(H ) reverses, with probability κ, the change in inner product by η.
The metric operator η can be estimated via quantum process tomography [60] for η ≤ π(I), or κη if otherwise.The change in inner product by η ≤ π(I) is implemented via the operation (Fig. 2) ) for the Kraus rank-1 operation G η .Then the Kraus operator η 1 /2 and therefore η can be estimated by quantum process tomography for trace non-increasing channels [61].In the other case η ≰ π(I), the change in inner Commutative diagram showing the action of Fη decomposed in terms of Gη and Rη.
product is implemented by the operation E κη from which κη is estimated similarly; however, the above procedure does not yield κ and η separately.We now discuss how to implement dynamics generated by a diagonalizable Hamiltonian H PT with unbroken PT symmetry in finite dimensions over a time t ≥ 0, by building on our framework for changing inner product.The dynamical transformation generated by for some κ ∈ (0, 1), where both ρ, U PT (ρ) ∈ D(H ) represent states under π.We show that this dynamics can be implemented by first changing the inner product, then applying an appropriate unitary channel and finally reversing the change in inner product.To explain this sequence, we consider an arbitrary pure state represented by |ψ⟩ ∈ H , which is to be transformed to that represented by √ κU PT |ψ⟩ ∈ H (lower row of Fig. 3).Next, we compute a metric operator η ≤ π(I) ∈ B(H ) that satisfies the quasi-Hermiticity condition the existence of such an η is guaranteed as H PT has unbroken PT symmetry [62].The Hamiltonian H PT is selfadjoint with respect to the inner product of the new Hilbert space H η .Therefore, U PT represents unitary dynamics on H η , which constitutes the second step of the sequence.Prior to implementing U PT , we transform |ψ⟩ ∈ H to |ψ⟩ ∈ H η via a change in inner product using I η .Finally, the transformation from U PT |ψ⟩ ∈ H η to √ κU PT |ψ⟩ ∈ H is equivalent to reversing the change in inner product using I η −1 with probability κ.This sequence extends to general mixed states by the application of lift, lift η maps and linearity (upper row of Fig. 3); here PT-symmetric dynamics in Eq. ( 9) can be expressed as a sequence of channels acting exclusively on B(H ), thereby paving the way for experimental simulation of PT-symmetric systems.Following the upper row of Fig. 3, we start by expressing U PT (Eq.( 9)) as U PT = E κη −1 • U PT • E η .Similar to Eq. ( 8), we express the reverse change in inner product as Diagram showing implementation of a PT-symmetric dynamics using change in inner product and unitary dynamics.
the change in representation form π η to π.We then rewrite U PT as which is the desired decomposition.The channels G κη −1 , G η have single Kraus operators √ κη − 1 /2 and η 1 /2 respectively.The maps R η , R κη −1 only effect change in representation and operationally are equivalent to no change.Finally, the transformation R κη −1 • U PT • R η implements a channel corresponding to the unitary Kraus-operator η 1 /2 U PT η −1 /2 acting on H , generated by the Hamiltonian which can be verified to be self-adjoint, i.e. h † PT = h PT , using the quasi-Hermiticity condition in Eq. (10).
We now design a qutrit procedure for an agent to simulate successfully the change in inner product by η ≤ π(I) of a qubit system with algebra A, which is represented on a two-dimensional Hilbert space H 2 by π.Our procedure, which shall simulate the operation G η (Fig. 2), uses a unitary operation on the three-dimensional Hilbert space H 3 = H 2 ⊕ H 1 followed by a projective measurement on to H 2 and postselection, as we now explain.For any η ≤ π(I), we first construct the metric operator and the unitary operator where P is the orthogonal projector on H 2 .The matrix representation of U η is (see Appendix D) where [ ] denotes matrix representation, u is the eigenvector of [η] 1 /2 with eigenvalue r and ∥u∥ = √ 1 − r 2 .Furthermore, ū⊤ is the Hermitian conjugate of the vector u.Both θ and the global phase of u are free parameters.The qutrit unitary operator U η is part of the overall simulation procedure (Eq.( 14)).Now we explain how an agent can sequentially apply each operator in Eq. ( 14) to simulate G η (Fig. 4).The agent is provided with a description of 2 × 2 matrix [η], in the logical basis {|0⟩ , |1⟩} and a quantum state σ (Eq.( 14)).
The agent may further estimate the success probability tr (G η (ρ) ⊕ 0), if required, by repeating the simulation procedure on a large number of copies of σ provided to them and then calculating the ratio of non-zero measurement outcomes to the total number of copies used [63].
In Appendix E, we provide an explicit procedure to simulate the dynamics (Eq.( 9)) of the qubit PTsymmetric Hamiltonian [8], by sequentially applying the operators in Eq. (11) and by using the qutrit simulation procedure to implement G η , G κη −1 .In Appendix F, we design a simulation procedure, similar to our qutrit procedure given above, for changing the inner product of a d-dimensional system using a 2d-dimensional system for any positive integer d.Furthermore, we use our procedure to simulate the dynamics of a d-dimensional PT-symmetric Hamiltonian by using only 2d dimensions, instead of d 3 dimensions as required in the Stinespring dilation approach [64].
We also design a scheme to verify tomographically whether a prover can perform an arbitrary change of inner product using our qutrit simulation procedure.Input to the verification scheme is a threshold function D th : B(H 2 ) → (0, 1) given as a black-box.The output is 'accept' if ∥G η ⊕0− Ĝη ∥ 1→1 ≤ D th (η) or 'reject' otherwise, where Ĝη : B(H 3 ) → B(H 3 ) represents a tomographic reconstruction of the qutrit process implemented by the prover, G η ⊕ 0 extends the action of G η to B(H 3 ) and ∥ • ∥ 1→1 is the induced Schatten (1 → 1)-norm [65].The verifier supplies to the prover a randomly chosen valid η, a positive integer N sufficiently large for the process tomography [66] and copies of the quantum states σ i encoding ρ i on demand, where {ρ i } is chosen based on the tomography procedure in use.The prover returns N copies of the qutrit states on which the change of inner product is successful as well as the success ratios for each ρ i , both of which are used by the verifier to reconstruct Ĝη .To ensure that the verifier does not accept the process performed by a dishonest prover implementing only qubitunitary channels and randomly discarding the system, it suffices to set the threshold to D th (η) = 1 /3(λ 1 − λ 2 ), where λ 1 > λ 2 > 0 are the eigenvalues of η (see Appendix G).
In conclusion, we have three major results.First, we have operationalized Hilbert-space inner-product change in a way that is both observable and fully compatible with axiomatic quantum mechanics.Physically we can understand this inner-product change as a lossy quantum operation effecting a change in norm.This lossy operation is reminiscent of how superluminality is reconciled by electromagnetic absorption [67], with loss in our case forbidding past counterfactual claims.Consistency of our work is proven using C * algebra and representations.Alternatively, our claims can be verified experimentally by conducting two physically distinct experiments.One experiment is for the lower-dimensional lossy quantum operation and the other experiment is for the higher-dimensional unitary channel with both realizations yielding the same success ratio and measurement statistics for a given task.Our theory fully explains unbroken PT-symmetric quantum mechanics in all its forms as being about changing Hilbert-space inner product and observing its consequences.Our scheme for simulating qubit PT-symmetric Hamiltonians only requires one extra Hilbert-space dimension and no interaction with the environment, which eliminates the requirements for multiple subsystems and entangling operations used in existing schemes [19,45,48,51,52].We also show how to simulate d-dimensional (d ≥ 2) PT-symmetric Hamiltonians using 2d dimensions, as opposed to using d 3 dimensions in the Stinespring dilation approach.Our results open possibilities for simulating PT-symmetric dynamics on new experimental platforms, such as transmons, where high fidelity qutrit-unitary operations have already been demonstrated [68,69].

ACKNOWLEDGMENTS
This project is supported by the Government of Alberta and by the Natural Sciences and Engineering Research Council of Canada (NSERC).S. K. is grateful for a University of Calgary Eyes High International Doctoral Scholarship and an Alberta Innovates Graduate Student Scholarship.A. A. acknowledges support through a Kil-lam 2020 Postdoctoral Fellowship.

Appendix A: Constructing a representation of the C * algebra on the Hilbert space with a different inner product
In this section, we show the construction of the Hilbert space H η with inner product related to that of H by the metric operator η, the construction of a * -representation of the C * algebra A on this new Hilbert space, and finally the representation of states in S using density operators on H η .

Constructing a new Hilbert space from the metric operator
For a possibly infinte dimensional Hilbert space H , we denote by L(H ) and B(H ) the algebra of linear and bounded linear operators on H respectively.We also denote by D (H ) := {ρ ∈ B(H ) : ρ ≥ 0, ρ † = ρ, trρ ≤ 1} the set of density operators acting on H η .

Definition 1 ([59]). The adjoint of an operator A ∈ B(H ) is the unique operator
The operator

the sesquilinear form
is non-degenerate, therefore an inner product on V .

Constructing a * -representation on the new Hilbert space
We now construct a * -representation of the algebra A on the new Hilbert space H η constructed in Theorem 1.In the following, H and H η are two Hilbert spaces with their inner product related by the metric operator η as in Theorem 1, A is a C * algebra of operators and π : A → B(H ) is a * -representation of A. We first establish some results required for constructing such a new representation.The following lemma, which establishes the inverse of the metric operator η, is adapted from the Appendix A of Ref. [1].Lemma 2. Any self-adjoint and positive-definite operator η ∈ B(H ) is invertible.Furthermore, the inverse η −1 ∈ B(H ) is self-adjoint and positive-definite.
We next show that the bounded operator spaces on H and H η coincide.
We now show that both D(H ) and D(H η ) represent the same set of states, S, under the respective *representations.
Proof.To prove the forward implication, note ω(A) = tr(ρπ(A)) ∀A by the definition of # π.Then tr(ρπ(A)) = tr(ρ η η −1 /2 π(A)η 1 /2 ) = tr(ρ η π η (A)) using the cyclic property of trace and the definition of π η respectively.Therefore ω(A) = tr(ρ η π η (A)) ∀A, and therefore, ρ η # πη → ω.The reverse implication can be proved by following the same steps in reverse order.Finally, we represent pure states in S by vectors in the Hilbert space H η .Recall that a state ω ∈ S has a vector-representation |ψ⟩ ∈ H under π if We now extend this definition to representation π η .In this section, we construct the quantum operation that implements the change in inner product by η ≤ π(I).Change in inner product is defined by the identity isomorphism I η : H → H η (see Fig. 1c in main text).We now show how I η is extended to B(H η ) through the map E η defined below.
3. E η is a quantum operation.
Proof.To prove Statement 1, note that for any M ∈ B (H ), the operator M η ∈ B (H η ) because where the first equality follows from Eq. (A4).The commutative diagram in Statement 2 follows immediately from the action of lift η map in Lemma 8. We now show that E η is a valid quantum operation, i.e. a completely-positive, trace non-increasing map.To prove the positivity of E η , let M ≥ 0, so that it can be expressed as M = AA † [59].Then E η (M ) = AA † η, which can expressed as E η (M ) = BB ‡ with B = Aη To prove that E η is trace non-increasing, let ρ ∈ D(H ) and note that trace is independent of the inner product (Eq.(A12)).We have tr(ρη) = tr(η 1 /2 ρη 1 /2 ) and tr(η where |M | = (M † M ) and we used |M | = M for any M ≥ 0. The first inequality in Eq. ( B3) is a property of the trace norm [59], and the last inequality in Eq. (B3) follows from the fact that η ≤ π(I) and therefore, η 1 /2 2 ≤ 1.

Appendix C: Transformation of the Operators under Changing the Inner Product
An inner product changing channel could modify the commutation relations between the operators.In this section, we demonstrate such a change with an explicit example of a qubit system undergoing an inner product change.Consider a qubit system undergoing change in inner product by The Pauli operators X, Y, Z ∈ B(H ) acting on the original Hilbert space along with the identity operator I 2 ∈ B(H ) generate the u(2) algebra.These operators transform according to Eq. ( 5) in the main text following the inner product change by η.This transformation is given by the map E op η .The transformed operators satisfy the commutation relations where a = 1/(1 + r sin ϕ).These commutation relations are different from those of u(2) algebra for r ̸ = 0, or equivalently a ̸ = 1.
A choice of the qutrit unitary U η2 (see Eq. (D1)) simulating the action of G η2 is , p = r sin ϕ s + r sin ϕ .
(E7) The output of this step is the qutrit state

Simulate the unitary evolution generated by h
) and implements the qutrit unitary operator e −i(h PT ⊕0)t , which is equivalent to simulating the channel (R κη −1

2
• U PT • R η2 ) in Eq. (E3).The output of this deterministic step is the qutrit state 4. Simulate change in inner product by κη −1 2 : Agent applies the qutrit procedure (Fig. 4) to simulate G κη −1 2 , by setting η = κη −1 2 .Note that we have κ = The output of this procedure is the qutrit Therefore, the output of the simulation procedure is the state ⊕ 0 with success probability given by the combined probability of success in Steps 2,4, which is equal to 1 2 Agent then implements U η followed by projective measurement and postselection on to the subspace H (s) d .All steps of the simulation procedure are similar to the qutrit simulation procedure for changing inner product explained in the main text.
We now discuss how this simulation procedure for changing the inner product can be used for simulating PT-symmetric dynamics in d dimensions using a 2ddimensional system.Similar to the d = 2 case discussed in Sec.E, the input to the simulation procedure is ρ ∈ B(H 1.The agent calculates η satisfying the quasi-Hermiticity condition H † PT = ηH PT η −1 with ∥η∥ = 1. 2. The agent implements the procedure described above to simulate G η by setting η = η and for a single copy of σ (Eq.(F1)).
The output of this procedure is the state PT .

Appendix G: Additional details on the verification scheme
We now prove a threshold distance D th for the tomographic verification scheme, for the qutrit procedure simulating the change in inner product by an arbitrary η, provided in the main text.The scheme allows a verifier to distinguish an honest prover implementing the operation G η from a dishonest prover failing to implement the same.We assume that the dishonest prover implements only unitary operations, on the qubit subspace, drawn from the set {U j ⊕ 1 : U j ∈ B (H 2 )}, where each U j is selected with probability p j , and the system is discarded with probability p := 1 − j p j < 1.The quantum operation implemented by the dishonest prover is given by Ĝη (•) = j p j (U j ⊕ 1) † • (U j ⊕ 1) .(G1) We now derive a lower bound for the induced Schatten (1 → 1)-norm distance [65] between the inner-product changing operation G η ⊕0 and the implemented operation Ĝη .Note that The above given value for D th allows the verifier to distinguish an honest prover from a dishonest one, provided the honest prover implements the operation G η with error less than D th , where error is quantified by the induced Schatten (1 → 1)-norm.

FIG. 1 .
FIG. 1.(a) Diagram illustrating the relation between the * operation of A and the † operation of B(H ) under the representation π.(b) Commutative diagram depicting the change of representation from π to πη under Rη.Operationally, Rη represents a trivial transformation, i.e., no change, in S. (c) Commutative diagram illustrating the relation between the maps Fη, Eη and Iη.

1 2 . 2 ⊗
Therefore E η (M ) ∈ B(H η ) is positive if M is positive, which proves the positivity of E η .Complete positivity of E η can be proven by showing that the mapE η ⊗I k : B (H )⊗B(C k ) → B (H η )⊗B(C k )is positive, for every positive integer k, where I k denotes the identity map on B(C k ).The action of the new map is given by [E η ⊗ I k ] (N ) = N (η ⊗ I k ), with I k ∈ B(C k ) the identity operator.Operator [E η ⊗ I k ] (N ) ∈ B (H η ) ⊗ B(C k ) because ∥η ⊗ I k ∥ = ∥η∥ η • ∥I k ∥ = ∥η∥ η .For proving positivity, let N ∈ B(H ) ⊗ B(C k ) be a positive operator, so that N = CC † .Then [E η ⊗ I k ] (N ) = CC † (η ⊗ I k ), which can be expressed as [E η ⊗ I k ] (N ) = DD ‡ with D = C(η 1 I k ),thereby proving positivity of [E η ⊗ I k ] (N ) and consequently positivity of E η ⊗ I k .