No-Go Theorem for Generic Simulation of Qubit Channels with Finite Classical Resources

Introduction.– Classical physics, rooted in intuitive and objective principles, offers deterministic descriptions of the physical phenomena we encounter in daily life (though see [1]). In stark contrast, the quantum realm defies classical reasoning, exhibiting phenomena that challenge conventional intuition. Quantum mechanics—formulated within the Hilbert space framework—delivers an extraordinarily precise mathematical account of these phenomena, but it refrains from offering clear physical intuition about their nature [2, 3, 4]. Nonetheless, the advent of quantum information theory has highlighted practical advantages of quantum resources over their classical counterparts in tasks such as computation, communication, and cryptography [5, 6, 7, 8, 9, 10, 11, 12, 13]. In this context, simulating quantum processes with classical resources promises a compelling research avenue [14, 15, 16]. Such investigations serve a dual purpose: quantifying the computational and communicational power of quantum resources while deepening our understanding of the unique features that distinguish quantum phenomena from classical intuitions.

A hallmark of quantum mechanics, underscored by Bell’s theorem [17] and corroborated through decades of experiments [18, 19, 20, 21, 22, 23, 24], is the emergence of nonlocal correlations among the outcomes of local measurements performed on entangled states. These correlations defy any local realistic explanation [25, 26, 27, 28, 29]. Furthermore, entangled states shared among distant parties cannot be prepared through local quantum operations and classical communication (LOCC) [30]. Despite their inherent nonlocality, the local measurement statistics of entangled states can often be faithfully reproduced through finite classical communication between distant parties holding parts of the composite system [31, 32, 33, 34, 35, 36, 37, 38, 39]. This paradigm extends naturally to quantum channel simulation, where a receiver (Bob) aims to replicate the statistics of arbitrary measurements on a quantum state unknown to him but fully known to a sender (Alice), who aids Bob while minimizing the classical communication required [40, 41, 42, 43, 44]. In particular, the result by Toner and Bacon demonstrated that the statistics of any projective measurement, also called the von Neumann measurement, on a qubit state can be simulated using just two classical bits of communication [41]. Subsequent work extended this result to more general settings, including positive operator-valued measures (POVMs) [45], further illustrating the feasibility of classical simulation with finite communication [44].

In this work, we argue that quantum channel simulation must go beyond replicating local measurement statistics to address more general scenarios. Specifically, simulations must account for the statistics of joint measurements—including entangled basis measurements—on Alice’s system and an ancillary system held by Bob. This requirement remains valid even when Bob’s ancillary system is unknown or entangled with a larger system. Considering an entangled basis measurement, we demonstrate that such a general simulation is impossible with finite classical communication for a perfect qubit channel. On the other hand, given classical description of any finite dimensional quantum system to Alice and another finite dimensional unknown quantum state to Bob, we show that the statistics of any joint measurement consisting of only product or separable effects can always be simulated at Bob’s end with finite amount of classical communication from Alice. We also investigate classical simulation of noisy qubit channel, and show that statistics of any measurement consisting of separable as well as entangled effect can always be simulated with finite communication for qubit depolarizing channels. Notably, the communication cost increases as the noise diminishes, underscoring a subtle interplay between the level of noise in the channel and the classical resources required for its simulation.

Classical simulation of quantum channels.– A quantum channel is a physical device, such as an optical fiber, that transmits quantum states from a sender to a receiver, even when the state is unknown or a part of a larger entangled system. Mathematically, a quantum channel is described by a completely positive trace-preserving (CPTP) map [45, 46]. The seminal quantum teleportation protocol illustrates that the action of a quantum channel can be perfectly replicated using classical communication, provided Alice and Bob share prior quantum entanglement [9]. However, when the quantum state is known to the sender, as in the case of Remote State Preparation [47, 48, 49], Alice and Bob can attempt to simulate the channel’s action using classical communication supplemented by pre-shared classical correlations.

The classical simulation of a quantum channel, as investigated in [43, 41, 40, 42, 44], is formally defined as follows: Alice, holding the classical description of the quantum state $\psi\equiv\ket{\psi}\in\mathbb{C}^{d}$, generates a variable $m$ according to a probability distribution $p(m|x,\psi)$. This distribution depends on the quantum state $\psi$ and a shared random variable $x$, drawn from a distribution $p(x)$. Alice communicates $m$ to Bob, who then aims to reproduce the outcome statistics of an arbitrary POVM $\mathrm{M}=\{\mathrm{E}^{k}\mid\mathrm{E}^{k}\geq 0,\sum\mathrm{E}^{k}=\mathbf{I}\}$ based on a conditional probability $p(k|m,x,\mathrm{M})$. Crucially, the simulation should succeed even if Alice does not know which POVM Bob will perform. By $\mathrm{P}_{\psi}$, we denote the projector onto the state $\ket{\psi}$, which for a qubit is uniquely specified by the Bloch vector $\hat{\psi}\in\mathbb{R}^{3}$, i.e., $\mathrm{P}_{\hat{\psi}}=\frac{1}{2}(\mathbf{I}_{2}+\hat{\psi}\cdot\sigma)$. The protocol successfully simulates a quantum channel $\Lambda:\mathcal{D}(\mathbb{C}^{d})\to\mathcal{D}(\mathbb{C}^{d})$ if the combined process reproduces the quantum probabilities, i.e. $\sum_{m}\int dx\,p(k|m,x,\mathrm{M})\,p(m|x,\psi)\,p(x)=\operatorname{Tr}[\Lambda(\mathrm{P}_{\psi})\mathrm{E}^{k}]$, for any quantum state $\psi$ and measurement $\mathrm{M}$; $\mathcal{D}(\star)$ denotes the set of density operators on the corresponding Hilbert space.

The communication cost, $\mathrm{C}$, of such a classical simulation can be defined as the Shannon entropy of the distribution $p(m|x):=\int d\psi\,p(m|x,\psi)\,p(\psi)$, averaged over the shared variable $x$, where $p(\psi)$ is assumed to be uniformly distributed. The communication complexity, $\mathrm{C}_{\text{min}}$, for one-shot channel simulation, corresponds to the minimal classical communication required to exactly simulate the quantum channel. Building on the results from [50], which analyze the communication complexity of correlations in one-shot settings (see [51] for asymptotic analysis), and leveraging the maximally $\psi$-epistemic Kochen-Specker model for qubits [52], the author in [43] refined the communication cost of a perfect qubit channel, improving upon the Toner-Bacon result [41]. While the Kochen-Specker model applies only to projective measurements, more recent work [44] demonstrates that the statistics of any POVM can also be simulated using purely classical resources—specifically, the protocol invokes just two bits of communication augmented with shared randomness.

Channel simulation, generic setup.– The simulation of a quantum channel must faithfully reproduce all possible measurement outcome statistics as dictated by the Born rule. This encompasses broader scenario of channel simulation that involves reproducing the statistics of a joint measurement $\mathrm{M}_{AB}\equiv\{\mathrm{E}^{k}_{AB}\}$, applied to system $A$, known to Alice, and system $B$, provided to Bob (see Fig. 1). Importantly, the state of system $B$ may be unknown to both parties and could even form part of a larger entangled system, such as $BC$. This generalized context introduces significant challenges for classical simulation protocols. The communication cost for such a simulation, denoted as $\mathrm{C}^{G}_{\min}$, can be defined analogously to simpler scenarios but incorporates the additional complexity of joint measurements and entanglement. As a relevant aside, here we recall the semi-quantum Bell scenario introduced in [53]. In the traditional bipartite Bell game, spacelike-separated Alice and Bob generate classical outputs $a$ and $b$ in response to classical inputs $s$ and $t$, producing an input-output probability distribution $\{p(a,b|s,t)\}$. In contrast, the semi-quantum Bell scenario replaces these classical inputs with quantum states $\phi_{s}$ and $\chi_{t}$, which may be unknown to the players. This framework has revealed the nonlocality of all entangled quantum states [53] and has spurred various applications [54, 55, 56, 57]. The quantumness of preparation also find application in verifiability of blind quantum computation [58, 59, 60]. Coming back to the generalized channel simulation problem, we establish that the minimum communication cost $\mathrm{C}^{G}_{\min}$ diverges to infinity, even for the case of qubit channels.

A natural question is whether the use of entangled basis measurements is essential to establish the no-go result in Theorem 1. Specifically, if the joint measurement consists solely of product or separable effects, can its statistics be simulated with finite communication? To address this question we start by observing that the statistics of computational basis measurements, $\mathrm{M}_{comp}\equiv\{\mathrm{P}_{\hat{z}}\otimes\mathrm{P}_{\hat{z}},\mathrm{P}_{\hat{z}}\otimes\mathrm{P}_{\hat{z}^{\perp}},\mathrm{P}_{\hat{z}^{\perp}}\otimes\mathrm{P}_{\hat{z}},\mathrm{P}_{\hat{z}^{\perp}}\otimes\mathrm{P}_{\hat{z}^{\perp}}\}$, can be simulated using only 1 bit of classical communication from Alice to Bob: Alice measures her state in $\sigma_{z}$ basis and communicates measurement outcome to Bob, who also measures his unknown state in $\sigma_{z}$ basis. A similar approach works for the twisted measurement $\mathrm{M}^{(B)}_{twist}\equiv\{\mathrm{P}_{\hat{z}}\otimes\mathrm{P}_{\hat{z}},\mathrm{P}_{\hat{z}}\otimes\mathrm{P}_{\hat{z}^{\perp}},\mathrm{P}_{\hat{z}^{\perp}}\otimes\mathrm{P}_{\hat{x}},\mathrm{P}_{\hat{z}^{\perp}}\otimes\mathrm{P}_{\hat{x}^{\perp}}\}$: Alice measures her state in $\sigma_{z}$ basis and communicates the result to Bob, who performs either $\sigma_{z}$ or $\sigma_{x}$ measurement on his qubit, depending on Alice’s communication. The simulation becomes slightly more intricate when the twist is on Alice’s side, namely for the measurement $\mathrm{M}^{(A)}_{twist}\equiv\{\mathrm{P}_{\hat{z}}\otimes\mathrm{P}_{\hat{z}},\mathrm{P}_{\hat{z}^{\perp}}\otimes\mathrm{P}_{\hat{z}},\mathrm{P}_{\hat{x}}\otimes\mathrm{P}_{\hat{z}^{\perp}},\mathrm{P}_{\hat{x}^{\perp}}\otimes\mathrm{P}_{\hat{z}^{\perp}}\}$. In this case, Alice performs measurements in both the $\sigma_{z}$ and $\sigma_{x}$ bases on her state and sends the outcomes to Bob using two separate 1-bit classical channels. This is possible as the state is known to her, and thus she can make copies of it. Bob measures his qubit in the $\sigma_{z}$ basis and selects the appropriate bit from Alice’s communication based on his measurement result. These observations leads us to the following general result.

Although simulation of the twisted measurement $\mathrm{M}^{(B)}_{\text{twist}}$ can be achieved with 1 bit of communication from Alice to Bob, discussion on simulability of product von Neumann measurement we end with the following observation.

The proof is obtained utilizing the fact that in random-access-code (RAC) task, a qubit provides an advantage over the classical bit [62, 63] (see Appendix I).

Theorem 2, however, does not fully resolve the question of whether all product POVMs can be simulated with a finite amount of classical communication from Alice to Bob, as there exist measurements involving only rank-1 product effects, but not LOCC implementable. For instance, inspired by an construction in [64], we consider the following POVM:

where $\kappa:=3/4$. We call this the twisted-butterfly POVM $\mathrm{M}_{tb}$, a name justified by its structure (see Fig.2).

Although the measurement $\mathrm{M}_{tb}$ is not LOCC implementable, quite interestingly, it turns out that the statistics of this measurement on a qubit state known to Alice and an unknown qubit state provided to Bob can be simulated at Bob’s end with finite classical communication from Alice. Instead of proving this particular claim, in the following we establish a more generic result (proof provide in Appendix II).

Theorem 3 is important as it establishes that the no-go result in Theorem 1 necessitates considering a measurement involving entangled effects on the joint system of Alice and Bob. In Supplemental material we argue that statistics of the measurement $\mathrm{M}_{tb}$ can be simulated at Bob’s end by 1-bit of communication from Alice.

Simulating noisy qubit channels.– Thus far, we have focused on the simulation of perfect qubit channel. A natural extension is to ask whether the no-go result of Theorem 1 applies to imperfect qubit channels. To address this, we consider the qubit depolarizing channel $\mathrm{D}_{\eta}:\mathcal{D}(\mathbb{C}^{2})\to\mathcal{D}(\mathbb{C}^{2})$, defined as $\mathrm{D}_{\eta}(\rho):=\eta\rho+(1-\eta)\frac{\mathbf{I}_{2}}{2}$, where $\eta\in[0,1]$. We now analyze the classical simulability of this particular class of channels.

In general, deriving an exact expression for $\eta(\theta_{m})$ for arbitrary $m$ is challenging, as it depends on the specific choices of Bloch vectors $\{\hat{\omega}_{i}\}_{i=1}^{2^{m}}$. However, for small $m$’s we can have some natural choices of Bloch vectors – $(m$=$1)$: 2 diametrically opposite vectors, yielding $\theta_{1}=\pi/2$ and $\eta(\theta_{1})=1/2$, $(m$=$2)$: 4 vectors forming a regular tetrahedron, yielding $\theta_{2}=0.5\times\cos^{-1}(-1/3)$ and $\eta(\theta_{2})=(3+\sqrt{3})/6\approx 0.789$, and $(m$=$3)$: 8 vectors forming the vertices of a cube, yielding $\theta_{3}=0.5\times\cos^{-1}(1/3)$ and $\eta(\theta_{3})=(3+\sqrt{6})/6\approx 0.908$.

Discussions.– We have generalized the channel simulation task which has a long history in literature [43, 41, 40, 42, 44]. While the standard simulation scenario allows efficient classical protocols, in the generalized task we have shown that simulation of a perfect qubit channel requires an unbounded amount of classical communication from Alice to Bob, even when augmented with arbitrary pre-shared classical correlation. In particular, even though Alice has complete classical knowledge of her qubit state, the unknown state provided to Bob prohibits an efficient classical simulation.

This finding raises some deep foundational questions. For instance, in the standard simulation scenario, it has been shown that simulating any POVM at Bob’s end with finite classical communication from Alice is possible if and only if there exists a $\psi$-epistemic model underlying quantum theory, where quantum wavefunctions represent an agent’s knowledge about the system [43]. Extension of this result to the generalized simulation scenario along with our no-go results (Theorems 1) would suggest a $\psi$-ontic nature of the qubit wavefunction. That is, wavefunctions correspond to intrinsic properties of the system rather than merely an observer’s knowledge. Such a conclusion would align with the claims of the Pusey-Barrett-Rudolph (PBR) theorem [66]. Additionally, it would offer a pathway to weaken the Preparation Independence assumption used in the PBR theorem, an assumption that has faced criticism [67]. On the other hand, inspired by studies like [68], it would be intriguing to examine the status of Theorem 1 when Bob’s unknown state is restricted to a predefined set.