Quantum Steering on IBMQ

Let us consider a general bipartite setup where two parties, Alice and Bob, share an unknown quantum state $\rho$. Alice and Bob would like to demonstrate that the shared state is quantum correlated. To this end they perform local measurements on their respective sides and communicate classically with each other. If they trust their measurement devices they can measure locally a set of informationally complete observables to reconstruct the shared state and check whether it is entangled. If they do not trust their devices, they can still verify quantum correlations in the shared state if they are able to violate a Bell inequality with the measured statistics. The former is called a device-dependent and the latter a device-independent scenario.

Quantum steering is an asymmetric semi-device-independent scenario [3, 4]. Only Bob performs trusted quantum measurements while Alice’s measurement device is treated as a black box. Alice can apply different measurement strategies labeled by $x$ on her part of the bipartite state and obtains outcome $a$ with probability $p(a|x)$ from her untrusted apparatus. By doing so she can, in general, prepare different state ensembles $\{p(a|x),\rho_{a|x}\}_{x}$ on Bob’s side. In each run Bob randomly chooses one of Alice’s strategies $x$, asks her to perform the corresponding measurement and share the outcome $a$ with him. He is then – after many runs of the experiment – able to reconstruct the states $\rho_{a|x}$ through state tomography.

It is useful to introduce the corresponding subnormalized states $\sigma_{a|x}=p(a|x)\rho_{a|x}$. The set $\{\sigma_{a|x}\}$ is often called an assemblage [9]. From now on we will drop the brackets and simply refer to $\sigma_{a|x}$ as an assemblage unless otherwise deemed necessary. We say that an assemblage allows a local hidden state model (“$\sigma_{a|x}$ is LHS”) if it can be decomposed into the form [9]

This implies that Bob’s observations can be explained by a classical mixture of hidden states $\rho_{\lambda}$ on his side and the black box on Alice’s side could just output this classical statistics. However, if such a decomposition cannot be found, then $\sigma_{a|x}$ demonstrates quantum steering and the shared state is said to be steerable from Alice to Bob. Steerability implies entanglement, while on the other hand the set of steerable states is a strict superset of the Bell nonlocal states.

An important property of general assemblages is the no-signaling condition

which expresses that averaging the ensemble states over Alice’s outcomes yields the reduced state on Bob’s side $\rho_{B}$ for all choices of measurement $x$. This is important because otherwise Alice could influence Bob’s local reduced state by her measurement.

Given an assemblage $\sigma_{a|x}$, trying to find a decomposition of the form (1) is very difficult in general. In past works different steering inequalities have been proposed [6, 22, 10], whose violation can show that a LHS decomposition does not exist.

In this work we will make use of another witness of steering that is moreover able to quantify steering, the so called steering weight SW [7, 9]. It represents the minimal amount of a generic assemblage $\gamma_{a|x}$ that needs to be added to a LHS assemblage $\sigma^{\text{LHS}}_{a|x}$ in order to reproduce the original assemblage $\sigma_{a|x}$:

This optimization problem can be efficiently solved using semidefinite programming (SDP) [36, 9]. We will use the code provided by the authors of Ref. [9] and apply this scheme for quantifying steering within our experimentally produced assemblages.

In the present work we analyze quantum steering in an open quantum system, i.e., a quantum system $\mathcal{S}$ coupled to an environment $\mathcal{E}$. We will use steering to verify that the system becomes quantum correlated with the environment by the open system dynamics. The asymmetry of a quantum steering task very well reflects the asymmetric roles of the system and the environment. While the open system is usually considered to be well known and manageable, this generally cannot be said about the environment, whose degrees of freedom might be partially unknown. Thus, in order to verify quantum correlations between $\mathcal{S}$ and $\mathcal{E}$, it is reasonable to resort to quantum steering. The open system $\mathcal{S}$ is treated as the trusted part on Bob’s side and the environment $\mathcal{E}$ remains untrusted so that Alice’s measurements are treated as a black box [22, 23, 11].

We will model the open system dynamics by a collision model. Let $\rho$ denote the density operator describing the open quantum system $\mathcal{S}$. The surrounding environment is modeled by discrete subenvironments (ancillas) $\mathcal{A}_{i}$ initially in the state $\rho_{\mathcal{A}_{i}}$ which sequentially interact with the open system. Here, every such collision shall be governed by a unitary operator $\mathcal{W}_{i}$. After the collision the subenvironments do not interact again neither with the system nor with another ancilla. The joint state of the system and the environment after $N$ collisions is then given by

where the unitary $\mathcal{W}_{i}$ acts on the system and the $i$th ancilla only and $\rho_{0}$ is the initial state of the system.

To demonstrate that the system has entangled with the environment during the dynamics, Alice and Bob perform a steering task on the joint state $\rho_{\mathcal{S}\mathcal{E}}$. By implementing different measurement scenarios $x$ on the ancillas, Alice tries to steer Bob’s system into different ensembles that form the assemblage $\sigma_{a|x}$. After a tomography of the assemblage, Bob can calculate the steering weight $\operatorname{SW}$. For $\operatorname{SW}(\sigma_{a|x})>0$ his system is steerable by Alice and the open system must be entangled with the environment.

Let us assume that the collision model correctly describes the quantum channel on Bob’s system and that Alice is indeed able to perform quantum measurements on the ancillas. The subnormalized states in the assemblage are then theoretically given by

where the $\{A^{x}_{a}\}$ form a positive operator valued measure (POVM) corresponding to measurement strategy $x$ on the environment. For practical reasons that will become clear later we restrict Alice’s measurements to observables that are local on each subenvironment. Thus each POVM element is given by

where the $\{A^{x}_{a_{i}}\}$ now form a local POVM on the $i$th ancilla and the measurement outcome $a$ is a tuple of all local outcomes $a=(a_{1},\ldots,a_{N})$. A measurement strategy $x$ on Alice’s side can then be specified by a tuple of POVMs that defines the measurement on each ancilla, respectively.

Crucially, while Alice can use her knowledge about the joint state $\rho_{\mathcal{S}\mathcal{E}}$ to tailor her measurement strategies $x$, the demonstration of steering does not depend on the validity of this information. Bob’s reconstruction of the assemblage only depends on the classical labels of the measurement strategy $x$ and the outcome $a$ without any reference to the underlying state or the quantum measurements on Alice’s side.

In our collision model the ancillas shall start in the $\operatorname{\sigma_{z}}$ eigenstate $\rho_{\mathcal{A}_{i}}=\rho_{\mathcal{A}}=|0\rangle\langle 0|$. A single collision is given by a rotation of the ancilla about the $y$-axis followed by an ancilla-controlled CNOT gate $CX^{\mathcal{S}}_{\mathcal{A}}$ targeting the system qubit: $$\longleftrightarrow\quad\mathcal{W}=CX^{\mathcal{S}}_{\mathcal{A}}\,\quantity(\mathds{1}\otimes R_{y}(g))$$ (7)

The rotation angle $g\in(0,\frac{\pi}{2})$ acts as a coupling parameter and determines the strength of a single collision. The reduced state of Bob’s system qubit $\mathcal{S}$ changes during a single collision as

For convenience we represent the state of the system qubit by its 4-component Bloch vector $\mathbf{k}$. Using the Pauli matrices $\sigma_{1,2,3}$ and defining $\sigma_{0}=\mathds{1}$ we can write the system state $\rho_{\mathcal{S}}$ as

with $\mathbf{k}=(1,\mathbf{r})^{\intercal}$ and $\mathbf{r}=(\Tr[\sigma_{x}\rho_{\mathcal{S}}],\Tr[\sigma_{y}\rho_{\mathcal{S}}],\Tr[\sigma_{z}\rho_{\mathcal{S}}])^{\intercal}$. The map $\mathcal{E}$ in the Bloch vector representation then reads:

To establish a connection to a continuous dephasing process we define the time step $t=-\ln(\cos(g))\in(0,\infty)$ and the total time after $N$ collisions as $T=Nt$. Applying $N$ collision we then obtain

which is independent of the underlying time discretization. Accordingly, our model stroboscopically simulates an exponential decay of the system’s Bloch vector’s $y$- and $z$-component over time, i.e., a dephasing in the $\operatorname{\sigma_{x}}$-basis.

Henceforth, we focus on a specific dephasing channel with fixed time $T=2.0$. As we have seen we can exactly model this channel by an arbitrary number of collisions. Physically speaking we can decide how large the environment is that contributes to the dephasing process by varying the number of ancillas involved. Due to limitations of the available IBMQ devices we will consider $1$ to $4$ collisions.

The system qubit shall start in the ground state $\rho_{0}=|0\rangle\langle 0|$ with Bloch vector $\mathbf{r}_{0}=(0,0,1)^{\intercal}$, which is why the final reduced state predicted by quantum theory is $\mathbf{r}=(0,0,e^{-2})^{\intercal}$. The discrete evolution for each case is depicted in Figure 1 and Table 1 shows the corresponding time steps $t$ and coupling parameters $g$.

We will now establish the measurement scenarios on the ancilla qubits that we subsequently use to demonstrate quantum correlations between the system and the environment.

Producing two different state ensembles would suffice for investigating the presence of quantum steering. However, since we expect to encounter noise distorting the quantum correlations, we apply three different measurement scenarios on Alice’s side in order to enhance the steering weight.

We aim to create ensembles that are as distinct as possible in the Bloch ball. Since the IBMQ devices can only measure each qubit separately we also restrict ourselves to observables that are local on each ancilla. In principle it would be possible to measure also nonlocal observables on multiple ancillas by implementing a global gate before the local measurements. However, implementing a suitable nonlocal unitary would require a large number of error-prone two-qubit gates and introduce so much noise that a steering task became infeasible.

In analogy to the notation of Section II.2, we may represent a measurement strategy by a sequence of POVMs $x=\quantity{A_{1},A_{2},\dots,A_{N}}$. The local measurements in the IBMQ devices have binary outcomes. In the ideal case these measurements are projective and, thus, can be represented by Bloch vectors $\mathbf{a}_{i}$ of unit length. We assume these kind of measurements for constructing expedient scenarios. Since the local POVMs are dichotomic, it suffices to specify the elements associated with outcome “0”. The respective element for outcome “1” is then given by $-\mathbf{a}_{i}$. Therefore, we can represent a measurement scenario on Alice’s side by $x=(\mathbf{a}_{1},\mathbf{a}_{2},\dots,\mathbf{a}_{N})$. We find two scenarios which lead to distinct ensembles easily:

• •

  $x_{1}=\quantity(\mathbf{a}_{i}=(0,0,1)^{\intercal}|i=1,\dots,N)$

• •

  $x_{2}=\quantity(\mathbf{a}_{i}=\quantity(\sin(g),0,\cos(g))^{\intercal}|i=1,\dots,N)$

The red dots in Figures 2(a) and 2(b) show the so produced ensembles in Bob’s system, where the dot sizes encode the state probabilities $p(a|x)$. The ensembles lie on almost orthogonal lines to each other, which makes them great candidates for demonstrating steering. Averaging over all states of an ensemble yields the reduced state due to Eq. (2), which is represented by a red triangle.

The corresponding noiseless assemblage already maximizes the steering weight to unity theoretically. Thus a third ensemble seems unnecessary. However, noisy assemblages show steering weights $\operatorname{SW}<1$ and an increase may be achieved by producing more ensembles.

For finding a suitable third strategy, we first parametrize the Bloch vectors $\mathbf{a}_{i}$ with usual spherical coordinates $(\theta_{i},\varphi_{i})$. Then we (theoretically) compute the full assemblage, manually add a small amount of white noise $\lambda=0.05$ (that is, apply the transformation $\mathbf{r}\mapsto(1-\lambda)\mathbf{r}$ to all states) ¹¹1Without the noise the optimization would fail since the steering weight would already be saturated by the first two pure ensembles. and maximize the steering weight over all angles. We find that for a given number of collisions $N$, all ancillas should be measured with the same local POVM. The first angle is always $\varphi_{i}=\nicefrac{{\pi}}{{2}}$ while the angle $\theta_{i}=\theta_{N}$ depends on the total number of collisions $N$. The values for $\theta_{N}$ can be found in Table 1. The final scenario is then given by:

• •

  $x_{3}=(\mathbf{a}_{i}=\quantity(0,\sin(\theta_{3}),\cos(\theta_{3}))^{\intercal}|i=1,\dots,N)$

Figure 2(c) shows the corresponding ensemble. The most probable states are close to the $y$-poles, which makes this ensemble distinct from the others.

We have theoretically estimated which measurement scenarios Alice should implement to demonstrate steering. In each run she announces to Bob the measurement strategy $x$ she has implemented and the corresponding outcome $a$ she has obtained. In order to reconstruct the ensembles that Alice produces, Bob needs to implement an informationally complete (IC) set of measurements on his side to do quantum state tomography (QST). In general a single IC-POVM would suffice to acquire the necessary statistics. However, since such a POVM is always non-projective, an implementation on the IBMQ computer – which can only measure projectively in the $\sigma_{z}$-basis – would require additional ancillas. Thus, for experimental reasons it is more appropriate to measure a set of three projective POVMs $\Pi_{i}$

which measure the different $\sigma_{i}$-components. Accordingly, for each state in the ensemble, Bob obtains a probability vector $\mathbf{p}_{a|x}=(p_{1}(0),p_{2}(0),p_{3}(0))^{\intercal}_{a|x}$, where $p_{i}(0)$ is the probability of the outcome “0” if POVM $\Pi_{i}$ was measured. Here we use the convention that outcome “0” refers to the POVM element $B_{i}=\frac{1}{2}\quantity(\mathds{1}+\sigma_{i})$.

The POVMs in the set $\mathcal{B}_{\text{id}}$ describe perfect projective measurements. This is an idealization which is certainly not true in a real experiment. In other words, when Bob implements a $\Pi_{i}$-measurement, the POVM actually performed by the apparatus will slightly deviate from the ideal projective one. Quantum steering and QST, however, generally require a perfect characterization of the involved measurements. We will address this subtlety in detail in Sec. V.1 when we discuss the measured data.

The raw data for each scenario $x$ obtained from a single experiment consists of count statistics for the joined outcome space of Alice and Bob. We use Alice’s $N$-bit outcome sequence $a$ to label each ensemble state, group Bob’s POVM measurement outcomes in bins accordingly and estimate the probability vector $\mathbf{p}_{a|x}$. To reconstruct the ensemble states Bob performs QST for each of these vectors. Once the assemblage is reconstructed the steering weight can be computed with a semidefinite program as mentioned in Section II.1.

The measurements on Bob’s side are theoretically described by the idealized set of projective POVMs $\mathcal{B}_{\text{id}}$ in Eq. (12). The actual measurements in the real experiment will not be perfectly projective. Therefore, we assume for the moment that the three measurements on Bob’s side are given by general POVMs $\Pi_{i}=\quantity{B_{i},\mathds{1}-B_{i}}$. Each $B_{i}$ can be represented by a three-component vector $\mathbf{b}_{i}$ and a bias term $b^{0}_{i}$:

We stress that the vector length $b_{i}=\absolutevalue{\mathbf{b}_{i}}$ and the bias term $b^{0}_{i}$ are constrained by the relation:

This guarantees that the set $\Pi_{i}$ consists of positive operators and actually describes a POVM.

According to Born’s rule the probability of measuring “0” when performing $\Pi_{i}$ on a qubit state with Bloch vector $\mathbf{r}$ is given by:

To reconstruct the Bloch vector $\mathbf{r}$ from the measured data $\mathbf{p}=(p_{1}(0),p_{2}(0),p_{3}(0))$ we have to solve this system of three linear equations:

Here $\mathbf{b^{0}}=\quantity(b^{0}_{1},b^{0}_{2},b^{0}_{3})$. The chosen set of POVMs is informationally complete if $\mathbf{B}$ is invertible. This QST approach is usually referred to as linear inversion method [38, 39].

Figure 2 shows exemplary ensembles reconstructed from the $N=4$ experiment on ibmq_santiago, depicted with blue dots. For this visualization we assumed in the tomography that the measurements were indeed given by the ideal POVM set $\mathcal{B}=\mathcal{B}_{\text{id}}$, so that it is comparable to the ideal theory. The blue triangles represent the ensemble averages, which should all be equal, and ideally, equivalent to the reduced state predicted by quantum theory (red triangles). This is only approximately true, which indicates that the no-signaling condition (2) is slightly violated. This behavior is expected on a real machine due to typical crosstalk errors in quantum computers [40].

To verify quantum steering, we cannot simply assume the idealized measurements $\mathcal{B}_{\textrm{id}}$ in the matrix $\mathbf{B}$ of the QST. Instead we would need to perfectly characterize the noise impacting the $\operatorname{\sigma_{z}}$-measurement and the preceding qubit rotation on each device’s physical qubits to obtain the actual set of measurements $\mathcal{B}$. This is of course impossible in a real experiment. Furthermore, the finite data measured in a QST can only approximate the underlying true probability distributions. As a consequence, the states obtained by the QST are in general not the actual states in the produced ensembles. This can most easily be seen by the well known fact that QST with the linear inversion method almost certainly predicts unphysical, i.e., non-positive density matrices. To cure these states, several techniques that enforce positive states have been proposed in the literature, such as maximum likelyhood methods [41, 42, 43], Bayesian approaches [44, 45, 46], least square methods [47, 48] and linear regression estimation [49, 50, 51]. For example, in the maximum likelihood method the non-positive states, which lie outside the Bloch sphere, are projected onto the surface, thereby altering the reduced state of the ensemble. This directly leads to a violation of the no-signaling condition (2), which is problematic in a steering scenario. Furthermore, the purity of the states predicted by the tomography is then often overestimated. While this is usually not a big issue for local observables, it severely influences the estimation of device-dependent non-local properties such as entanglement. For instance, in Ref. [52] it has been shown that maximum likelihood and least square methods systematically overestimate the entanglement in the measured state. Recently a method that allows to verify steering without full characterization of the measurement apparatus has been proposed [20]. However, this approach relies on a source of trusted quantum states and is therefore also not applicable to our experimental setup.

To circumvent these problems we make use of a more practical approach. Strictly speaking we are not interested in the quantum states on Bob’s side but these are only a means of demonstrating steering. Therefore, we stick to linear inversion but do not rely on a specific reconstruction of Bob’s states. Instead we search for a lower bound $\mathrm{LB}$ for every assemblage, which we realize by minimizing the steering weight over all valid sets of POVMs $\mathcal{B}$ that determine the state tomography:

Here we say that a set of POVMs $B$ is valid if all assemblage states obtained from the tomography are valid quantum states. In particular, assuming for example compatible POVMs on Bob’s side (a choice which would not be able to show steering), the tomography would in general produce non-positive states. Thus, the assumption of such POVMs is incompatible with the measured data.

Bob’s POVMs are described by three three-component vectors $\mathbf{b}_{i}$ and three bias terms $b^{0}_{i}$, making a total of twelve parameters. However, we do not need to distinguish between all of these sets. The definition of the steering weight (3) suggests a unitary freedom of the input assemblage, which translates to a unitary freedom of the tomography POVM set due to Born’s rule. This again translates to a rotational freedom of the vectors ${\mathbf{b}_{i}}\in\mathbb{R}^{3}$. Therefore, w.l.o.g., we can set

Here $\theta_{1},\theta_{2}\in(0,\pi)$ are the usual polar angles, while the azimuthal angle $\varphi_{2}\in(-\pi,\pi)$ is measured from the $y$-axis. The ideal POVM set $\mathcal{B}_{\text{id}}$ translates to $\theta_{1,\text{id}}=\theta_{2,\text{id}}=\nicefrac{{\pi}}{{2}}$ and $\varphi_{2,\text{id}}=0$. Thus we are left with nine parameters and our minimization problem becomes ²²2The validity constraint is implied and will not be noted henceforth.:

This minimization problem is not convex. For the small assemblages ($N=1,2$) we compute the lower bound $\mathrm{LB}$ using several global minimization algorithms as well as a gradient descent approach for multiple initial states. As all these methods converge to the same minimum we consider the problem well-behaved. Furthermore, we find that the minima always correspond to projective measurements, i.e. $b_{i}=b^{0}_{i}=1$. This is reasonable, because vector lengths $b_{i}<1$ in effect increase state purity through the tomography, which causes an increase of the steering weight. The measurements on the IBMQ devices are certainly not perfectly projective. Moreover, due to quantum decay, one generally observes a bias towards the “0” outcome. However, the full characteristics of the performed measurements are unknown and, therefore, assuming projectors ensures not to overestimate the steerability. Then the final form of our minimization problem comprises only three parameters (the relative angles between the measurement directions) and simply reads:

For the larger data sets of $N=3,4$ collisions we then stick to the gradient descent method in order to reduce the number of calls of the expensive steering weight SDP, which allows us to process the data on a conventional workstation in reasonable time.

We visualize the minimization results in Figure 4 and present the numbers in Table 2. Interestingly, ibmq_santiago produces the largest steering weight in all cases, despite providing the most unfavorable qubit layout. This indicates that this device is affected by the least noise, which coincides with IBM reporting it to have the largest quantum volume out of our selection [25]. Still, the impact of the necessary $\mathrm{SWAP}$ operation is apparent when going from two to three collisions. here the lower bound of the steering weight drops significantly, as does when going from three to four collisions.

This effect is even more prominent when adding the first $\mathrm{SWAP}$ for the 3-collisions experiment on ibmq_16_melbourne. This device performs poorly in the case of four collisions as there is barely any detectable steering left.

ibmq_belem presents itself as the average device of our selection, with a decent amount of steering left even after four collisions. Unfortunately, as with ibmq_santiago, we cannot increase the number of collisions on this machine as there are not enough physical qubits available. While ibmq_16_melbourne would in principle allow up to 15 collisions, the noise impact in this machine prevents us from detecting steering for more than four collisions.

We point out that the minimizing POVM sets deviate from the ideal case by only a few degrees in each angle. However, keeping in mind the freedom of a global unitary rotation on the set of POVMs, these angles merely describe the relative position of the POVM elements with respect to each other.

The analysis of the steerability of Bob’s system by Alice’s measurements is of course motivated by the asymmetric scenario of system and environment. Nevertheless, one might wonder if the measured data could also violate a Bell inequality. For our measured data the answer is negative which is due to the fact that the angles between the measurement directions on Bob’s side are approximately $90^{\circ}$. For the violation of the CHSH inquality, for example, one would choose an angle of $45^{\circ}$ [54]. Nevertheless, using the measured data we can estimate what Bob would have obtained if he had measured with different angles. These estimations – which are of course device-dependent and therefore do not replace a Bell test – suggest that the scenario could also violate a Bell inequality, at least on the best device ibmq_santiago.