Nonlinear and non-CP gates for Bloch vector amplification

Michael R Geller

Center for Simulational Physics,University of Georgia,Athens,GA 30602,United States of America

Abstract Any state r=(x,y,z) of a qubit,written in the Pauli basis and initialized in the pure state r=(0,0,1),can be prepared by composing three quantum operations:two unitary rotation gates to reach a pure state r=(x 2+y2+×(x,y ,z)on the Bloch sphere,followed by a depolarization gate to decrease |r|.Here we discuss the complementary state-preparation protocol for qubits initialized at the center of the Bloch ball,r=0,based on increasing or amplifying|r|to its desired value,then rotating.Bloch vector amplification increases purity and decreases entropy.Amplification can be achieved with a linear Markovian completely positive trace-preserving(CPTP)channel by placing the channel’s fixed point away from r=0,making it nonunital,but the resulting gate suffers from a critical slowing down as that fixed point is approached.Here we consider alternative designs based on linear and nonlinear Markovian PTP channels,which offer benefits relative to linear CPTP channels,namely fast Bloch vector amplification without deceleration.These gates simulate a reversal of the thermodynamic arrow of time for the qubit and would provide striking experimental demonstrations of non-CP dynamics.

Keywords: nonlinear channels,non-completely positive channels,Bloch vector amplification

Several papers have explored the use of real or effective quantum nonlinearity for information processing [1–17].Nonlinear master equations have also been frequently discussed in open systems theory [18–32].In this paper we go beyond the paradigm of linear CPTP maps to design singlequbit gates that increase the length of the Bloch vector without changing its direction.It is well known that this operation can be implemented using linear Markovian completely positive trace-preserving (CPTP) channels,via the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation[33,34].This provides a baseline which we call the linear CPTP amplification gate.The nonunital channel behind it is entropy decreasing,which is possible in an open system that compensates by producing enough environmental entropy as to not violate the second law.In addition to the linear CPTP gate,we also consider alternatives based on noncompletely positive (non-CP) and nonlinear channels.The channels considered here are Markovian normalized PTP channels taking the formX↦φ(X)tr[φ(X)],where φ(X)is a continuous 1-parameter positive linear or nonlinear map satisfying tr[φ(X)]≠ 0for all positive semidefinite (PSD)operators X.Normalized PTP channels fall into 4 classes,yielding 3 distinct forms of nonlinearity [35]:

(i) Linear PTP: linear φ and tr [φ(X)]=1 for allX;

(ii) NINO: linear φ and tr [φ(X)]≠1 for someX;

(iii) State-dependent PTP: nonlinear φ and tr[φ(X)]=1 for allX;

(iv) General normalized PTP: nonlinear φ and tr [φ(X)]≠1 for someX.

The linear CPTP gate belongs to class (i).A non-CP gate from class (i) will also be considered.Class (ii) leads to a restricted form of nonlinearity,where a diagonal nonlinear term is added to the master equation to conserve trace.This type of evolution equation extends a pure-state nonlinear Schrödinger equation first introduced by Gisin [18] in 1981,to mixed states [23,24,26,30,35].Rembieliński and Caban[36] recently argued that this type of nonlinearity is causal(does not support superluminal signaling) and should not be excluded from a fundamental theory.We call these channels nonlinear in normalization only (NINO) to emphasize their restricted form of nonlinearity.In section 2,several amplification gates based on NINO channels are investigated.Class(iii)channels include unitary mean field theories such as the Gross–Pitaevskii equation for weakly interacting bosons,and they support Bloch-ball torsion,believed to be a powerful computational resource [1–3,8,14,35].Our main result is a non-CP gate from class (i) and we do not discuss channels from class (iii) or (iv) in this paper.

Quantum channels that are positive but not completely positive,or non-CP,are well known in open systems theory[25,37–40],and are used to detect entanglement [41].However the question of whether non-CP channels could provide a computational advantage over linear CPTP channels appears to be largely unexplored[25],although they have been shown to increase channel capacity [42–44] in communication settings.Here we find an advantage for Bloch vector amplification,also called repolarization [25],and propose that an experimental demonstration of ‘fast’ amplification would constitute a striking demonstration of a physical non-CP map.

1.PSD cone

For the analysis of linear qubit channels it is sufficient to take,as the state space of a qubit,the Bloch sphere or ball,and to study the dynamics within that space.Here we will work in the larger space of PSD operators X0 with strictly positive traceτ≔ tr(X),a convex but noncompact set called the PSD cone.1The PSD condition X 0 implies tr(X) ≥0.We further require that tr(X)≠ 0,excluding the state X=0 at the apex of the cone.Allowing density matrices to have a trace differing from the canonical value τ=1 is a straightforward extension of pure state quantum mechanics with squareintegrable but unnormalized wave functions,and is equivalent to the canonical formulation as long as expectation values〈A〉≔ tr(XA) tr(X)are properlydefined,and tr(X)≠ 0.

Fig. 1.Extended state space of a qubit.On the left,the subspace with fixed trace is shown as a green circle,but it is really a Bloch ball with radius τ.

The condition defining pure states is also modified.Let ρ=X/τ=ρ2be a canonically normalized pure state.Therefore X is pure if and only if

Using(1)we see that these pure states lie on the surface of the PSD cone |r|=τ,as expected.

2.Linear and NINO channels

In this paper we examine Bloch vector amplification gates based on the following NINO model [23,24,26,30,35]:

Table 1.Jump operators used in this paper.The σ1,σ2,σ3 are Pauli matrices.Them ∊R are constants determining mean jump frequencies,which we assume to be individually controllable.

The anti-Hermitian part L-generates unitary evolution.Because amplification is purely nonunitary,we set L-to zero.TheBα∊B(H,C)are a set of linearly independent jump operators.The ζα=±1 are signs of the Choi matrix eigenvalues (all nonnegative for CPTP channels).Any jump operator with ζα=-1 indicates a non-CP channel [37–40].g∊ R controls the strength of the nonlinear term.The observable Ω ∊ Her (H,C)is chosen to conserve trace and plays an important role in NINO channels because it governs the dynamics in the τ direction of the PSD cone.

The trace equation in(4)shows that there are two distinct ways to achieve trace-conservation dτ/dt=0:The first is the linear option,g=0,which requires Ω=0 and leads to the GKSL equation and the linear CPTP gate (if the ζαare positive).The linear option conserves trace for any initial τ.The second option is to make use of the nonlinearity and fix g=1,leading to the NINO gates.In this case

This conserves trace if τ starts with the canonical value τ=1.The τ=1 plane in the PSD cone is a fixed plane of the channel.The fixed plane is locally stable wherever tr(XΩ) ≤0.The operator Ω can lead to an intricate fixed point structure in the PSD cone,including instabilities in dynamically inaccessible regions of the cone that nevertheless leave their imprint on the accessible regions in an intuitive way.

The NINO model (4) has a continuous symmetry that is absent (pushed to infinity) in the linear model: Under

so the equation of motion is invariant if

The condition(11)is the same as that for trace conservation in(4).Note that the jump operators do not change under this transformation.In certain cases this symmetry can be used to map a NINO channel to a dual linear channel (section 2.4).

We will consider a sequence of increasingly complex NINO channels and amplification gates constructed from a set of jump operators{B0,B1,B2,B3}listed in table 1,combined with specific values of L+.By amplification we mean a process that smoothly increases |r| from 0 to τ.Without loss of generality we can amplify along the x axis of the Bloch ball.The first jump operator B0in table 1 is chosen to produce x-axis amplification in the linear CPTP limit(when combined with an appropriate L+).The additional jump operators allow for increased control over the fixed points of the map.The amplification gates assume that the qubit is initially prepared in the stateX=,so they require a nonunital channel or an unstable fixed point.Note that

In the linear theory,g=Ω=0,this expression recovers the well known result that a nonunital Markovian channel requires one or more nonnormal jump operators.However in a NINO channel with Ω ≠0 we can implement nonunital maps with normal jump operators or even with no jump operators (section 2.2).

In the Pauli basis (4) becomes

from one or more jump operators.Expanding anyB∊C2×2in complex coordinates

2.1.Linear CPTP

If g=0,the trace equation in(4)requires that Ω=0,leading to the linear channel

If the ζα=1 then (16) reduces to the GKSL master equation and the map is CPTP.Here we will consider a model with a single jump operator B0from table 1 with ζ0=1:

The requirement Ω=0 imposes conditions ℓ0=-m2,ℓ1=m2,ℓ2=0,and ℓ3=0 on L+.In the Pauli basis

leading to the equations of motion

The solution giving the desired Bloch vector amplification gate is

The channel has a single stable fixed point at rfp=(τ,0,0),but the velocity decreases as this fixed point is approached.Due to this critical slowing down,it takes infinitely long to reach the pure state at x=τ,although it becomes exponentially close for t ≫1/4m2.To quantify the critical slowing down,let δ=τ-x ≥0.Thus dx/dt vanishes linearly with δ at the fixed point δ=0.

2.2.No-jump NINO

The first NINO gates we consider are variations on the linear CPTP gate.The simplest is an amplification gate is based on(4) with g=1 and no jump operators.The single model parameter is L+,which we take to be L+=ℓ0I2+ℓ1σ1,ℓ0,ℓ1∊ R,as in(17).The equation of motion for the no-jump NINO model is

a nonlinear channel with a purely non-Hermitian Hamiltonian H=iL+and no jump operators.In the Pauli basis,

The equations of motion are

In the τ=1 plane,

2.3.One-jump NINO

Next we consider a NINO gate using the jump operator B=B0from the linear CPTP gate,but with L+=0 and trace conserved nonlinearly.The equation of motion for the onejump NINO channel is

In the τ=1 plane,

which has a single fixed point at rfp=(1,0,0).To determine its stability,let δ=τ-x ≥0 and rewrite (31) as

Inside the PSD cone,δ>0 and the motion is stable,but the critical slowing down is worse now because dx/dt vanishes as δ2at the fixed point.Outside the PSD cone,however,the motion is unstable,and x increases without bound.The unstable region does not appear to be accessible dynamically when starting from the initial stateX=

2.4.Pseudo-liner NINO

Next we consider a special subclass of NINO channels with g=1 and Ω=κI2proportional to the identity.In this case the observable Ω vanishes except for a component in the I2direction.The equation of motion for the pseudo-liner NINO channel is

Here the operatorsL+∊Her (H,C)andBα∊B(H,C)areno longer independent;they are constrained to satisfy

The condition g=1 is required by trace conservation(applied to a τ=1 initial state).In this subclass the nonlinearity is effectively invisible,because

which acts linearly when restricted to the τ=1 plane.This implies a duality between pseudo-linear NINO channels and linear PTP channels.Assuming(33)and(34),the dual model is obtained by

We should think of(36)as being composed of two physically distinct steps.In the first step we shiftL+↦L++while keeping g=1,which [according to (9)] rescales Ω ↦0.It is important that g=1 during this first step,as required by(11).However now that Ω=0,the trace can be conserved linearly,while violating condition (11).So in the second step we switch off the nonlinearity.

Consider the following example of a pseudo-linear NINO channel for a qubit: Combine the jump operator B0from the linear CPTP gate with L+=ℓ1σ1,where ℓ1=m2.Then Ω=κI2with κ=-2m2.In the Pauli basis

In the τ=1 plane this leads to

with same solution (22) as the linear CPTP gate.The nonlinearity in the pseudo-liner model is invisible in the Bloch ball picture but is apparent in the PSD cone,where it produces a fixed plane at τ=1.The fixed plane is lineary stable if m ≠0.This form of nonlinearity does not appear to offer any computational advantage over the linear CPTP gate,as expected from the duality (36).

2.5.Three-jump NINO

Here we examine a three-jump NINO channel with a particularly striking fixed point structure due to the presence of fixed lines in the PSD cone.These fixed lines,when properly controlled,enable fast Bloch vector amplification without deceleration.The equation of motion for the three-jump NINO channel is

Next we set the jump operator strengths to be

With these settings

Next we choose L+=-(M/2)σ3to tune the σ3component of Ω to zero,leading to a pseudo-linear channel:

According to (12),however,when g=1 the resulting channel satisfies

and is therefore unital.This would seem to preclude its use for Bloch vector amplification,but this conclusion does not apply if the fixed point is unstable.In the pseudo-linear case (51) we have

Finally,upon restriction to the τ=1 plane and g=1 we obtain the following qubit equation of motion in the three-jump NINO model:

Let us examine the equations of motion for the model(54),which is similar to a model investigated in[35]but now without torsion:

If M ≠Γ the channel has a single fxied point=(0,0,0)at the center of the Bloch ball.To examine its stability,switch to rotated coordinates

Note that the ξ-direction is always stable in this model.This is a great improvement over the linear CPTP gate because it does not decelerate.

An amplification gate based on the three-jump NINO model(54)requires a small modification due to the instability at the starting pointX=.The simplest modification would be to initialize the qubit slightly away from r=(0,0,0)before switching on the nonlinearity.This can be achieved by applying the linear CPTP amplification gate for a short duration to pre-amplify the state to r=(x,0,0) with small positive x.In the rotated frame the pre-amplified state is

Applying the three-jump NINO channel then successfully amplifies the qubit state.

2.6.Linear non-CP

The final model we consider is motivated by the three-jump gate (58),which supports fast Bloch vector amplification without deceleration.As explained above,the NINO model(4) is invariant under a shift (8) of the dissipative part L+of the non-jump component of the linear infinitesimal generator.2Recall that iL+ is an anti-Hermitian but otherwise arbitrary qubit Hamiltonian.The Hermitian part of the Hamiltonian vanishes here because amplification is nonunitary.Furthermore,in the special case of a pseudo-linear NINO channel (section 2.4),where the observable Ω is proportional to the identity,the invariance leads to the duality(36) between pseudo-linear NINO channels with g=1 and strictly linear channels with g=0.Here we use this duality to construct a linear non-CP channel and gate equivalent to those of section 2.5,specifically to the pseudo-linear model (51).The equivalent linear non-CP model,an instance of (16),is

Here the sum is over α=1,2,3,with signs ζ1=1,ζ2=1,ζ3=-1,indicating a non-CP channel [37–40].The jump operators B1,B2,B3are given in table 1 and are the same as in section 2.5.In addition to these jump operators we include the L+specified in (60).Then

where,after using (45),

as in (54).This supports the fast amplification gate (58)without requiring nonlinearity.

3.Conclusions

In this paper we have investigated several designs for Bloch vector amplification gates based on linear and nonlinear PTP channels,which offer benefits relative to linear CPTP channels for this application.We do not consider microscopic models for these channels,but instead think of them as effective Markovian models for engineered strongly correlated quantum materials coupled to their environment.Thus,the models only satisfy the minimal properties of positivity and trace preservation.Our results indicate that,while non-CP dynamics is essential for fast Bloch vector amplification,NINO-type nonlinearity offers no additional computational benefit.This is because the instability underlying the gate(58)does not result from nonlinearity but instead from a competition between gain M and dissipation Γ.

Although we have only considered channels from class(i) and (ii),this was sufficient to achieve a significant improvement over the linear CPTP gate.In the future it would be interesting to consider amplification gates from classes(iii)and (iv) as well.Such gates might provide additional design benefits,such as robustness to noise.Finally,we note that the linear non-CP gate proposed here should be practical to realize,as it only requires linear operations on an open system with initial system-environment entanglement.It is well known that time-reversal transformations can be simulated on a quantum computer by implementing complex conjugation or reversing the sign of a simulated Hamiltonian [45,46].Similarly,fast Bloch vector amplification gates can be used to simulate a reversal of the thermodynamic arrow of time[47–50] for a qubit,and would constitute a striking demonstration physical non-CP dynamics.

Acknowledgments

This work was partly supported by the NSF under Grant No.DGE-2152159.

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