Bidirectional Remote State Preparation (BRSP)

• BRSP is a protocol where two parties simultaneously prepare known quantum states using pre-shared entanglement and classical communication.
• It supports probabilistic, deterministic, and controlled variants that employ structured multipartite resources for secure state reconstruction.
• Extensions to high-dimensional systems and quantum-walk generated entanglement demonstrate BRSP’s adaptability and resilience under noise.

Bidirectional Remote State Preparation (BRSP) is a two-way remote state preparation protocol in which two parties simultaneously prepare quantum states at each other’s locations, using pre-shared entanglement and classical communication. In RSP, the sender knows the full classical description of the target state but does not physically possess it; the receiver reconstructs it using feed-forward corrections. In the qubit setting emphasized in the generalized controlled framework, the target single-qubit state is parametrized as

$$|\psi(\theta,\phi)\rangle=\cos(\theta/2)|0\rangle+e^{i\phi}\sin(\theta/2)|1\rangle.$$

The 2014 generalized treatment established probabilistic, deterministic, and joint forms of controlled bidirectional remote state preparation (CBRSP), identified a broad resource-state structure that guarantees controllability and bidirectionality, and analyzed amplitude-damping and phase-damping effects on the probabilistic case (Sharma et al., 2014). Subsequent work extended the subject in two distinct directions: deterministic bidirectional controlled remote state preparation in arbitrary $N$-dimensional systems using two generalized GHZ states and no high-dimensional CNOT operations during the protocol (Du et al., 6 Jan 2025), and BRSP generated dynamically by coined quantum walks, with and without a controller, without any pre-shared initial entanglement (Naseeda et al., 8 Feb 2026).

1. Core definitions and task structure

Remote state preparation (RSP) is the task in which a sender prepares a known quantum state at a receiver using preshared entanglement and classical communication, without transmitting the quantum state itself. In the original Pati protocol, the process is probabilistic, with success probability $1/2$ for a maximally entangled Bell pair and 1 classical bit. Deterministic RSP variants succeed with unit probability but require more classical communication. BRSP is the corresponding simultaneous two-way task: Alice and Bob each act as sender and receiver in the same round, and each remotely prepares a known single-qubit state at the other’s side (Sharma et al., 2014).

Controlled BRSP introduces a third party, Charlie, whose role is supervisory rather than informational about the prepared states. Charlie prepares and distributes a structured multipartite entangled state; later he measures his system and reveals a classical bit that identifies which pair of entanglement resources is active in each direction. Without this bit, the receivers cannot select the correct correction unitaries, so remote state preparation cannot be completed (Sharma et al., 2014).

The operational distinction between probabilistic, deterministic, and joint variants is central. In the probabilistic case, success occurs with probability $1/2$ per direction using a Bell pair and 1 cbit from sender to receiver. In deterministic RSP, success probability is 1 using an augmented procedure such as an ancilla plus a CNOT to create a GHZ state, and 2 cbits from sender to receiver. In joint remote state preparation (JRSP), the knowledge of the target state is split between two senders; deterministic JRSP with GHZ states uses 1 cbit from each sender, 2 cbits total. The generalized controlled bidirectional framework extends each of these to the controlled, bidirectional setting: probabilistic CBRSP, deterministic CBRSP, and deterministic controlled joint BRSP (CJBRSP) (Sharma et al., 2014).

A later high-dimensional formulation preserves the same basic logic while changing the target class. In that setting, Alice and Bob simultaneously prepare equatorial single-quNit states with equal amplitudes and arbitrary phases at each other’s sites, under Charlie’s control, and the success probability is 1 once Charlie participates (Du et al., 6 Jan 2025). A different line of work reformulates BRSP as a coined-quantum-walk process in which the required entanglement is generated during the walk itself rather than assumed beforehand (Naseeda et al., 8 Feb 2026).

2. Entangled resources and the control condition

The key resource for qubit CBRSP is a 5-qubit entangled state with a particular structure: $$|\Psi\rangle_{S_1R_1S_2R_2C} = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle_{S_1R_1}|\psi_2\rangle_{S_2R_2}|a\rangle_C \pm |\psi_3\rangle_{S_1R_1}|\psi_4\rangle_{S_2R_2}|b\rangle_C \right),$$ where $|a\rangle,|b\rangle$ are orthonormal states of the controller, each $|\psi_i\rangle$ is a Bell state chosen from

$$|\psi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\phi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}},$$

and the constraints $|\psi_1\rangle\neq|\psi_3\rangle$ and $|\psi_2\rangle\neq|\psi_4\rangle$ hold (Sharma et al., 2014).

These constraints enforce control. Measuring Charlie in the $N$0 basis collapses the shared state into a product of two definite Bell states, one for each direction. The receivers’ correction rules depend on which Bell state is active. Because the two branches use different Bell pairs in each direction, there is no single correction that works for both branches, so Charlie’s disclosure is indispensable (Sharma et al., 2014). The generalized treatment also notes that, for fixed $N$1, there are 144 valid assignments of $N$2.

An explicit example is the Cao–An channel recast in this form,

$N$3

with $N$4 (Sharma et al., 2014).

For deterministic CJBRSP, local ancillas and CNOT operations convert the Bell-pair-based resource into a 7-qubit channel,

$N$5

with $N$6 and $N$7, preserving the same control logic (Sharma et al., 2014).

The high-dimensional BCRSP program uses a different resource: two tripartite generalized GHZ states of local dimension $N$8,

$N$9

and

$1/2$0

Alice holds $1/2$1, Bob holds $1/2$2, and Charlie holds $1/2$3 (Du et al., 6 Jan 2025).

A markedly different resource model appears in the quantum-walk formulation. There, no pre-shared initial entanglement is assumed. Instead, coined quantum walk dynamics on two independent graphs create the entanglement needed for BRSP during the protocol itself (Naseeda et al., 8 Feb 2026). This suggests that, within the literature covered here, “resource state” can mean either an explicitly distributed multipartite entangled channel or a dynamically generated entanglement structure.

3. Protocol families in the qubit framework

The generalized 2014 treatment organizes qubit BRSP into three protocol families (Sharma et al., 2014).

Variant	Success property	Classical communication per bidirectional round
Probabilistic CBRSP	$1/2$4 per direction, $1/2$5 jointly	3 cbits
Deterministic CBRSP	Success probability 1	5 cbits
Deterministic CJBRSP	Success probability 1	5 cbits

In probabilistic CBRSP, Alice and Bob each measure their sender-side qubit in a basis derived from the state they want to prepare. With $1/2$6, $1/2$7, and phase $1/2$8, define

$1/2$9

Alice succeeds if her outcome is $1/2$0, and Bob succeeds if his corresponding outcome is $1/2$1. For a maximally entangled Bell resource, the success probability is $1/2$2 per direction. Conditional on Charlie’s disclosure of the active Bell states, the receiver corrections are: $1/2$3 If the sender’s outcome is $1/2$4, the attempt fails because the receiver cannot convert $1/2$5 without knowing $1/2$6 (Sharma et al., 2014).

Deterministic CBRSP builds on the one-directional deterministic RSP procedure associated with the Ba An protocol. Starting from a Bell pair, the sender locally adds an ancilla and applies CNOT to create a GHZ state. The sender then performs two measurements: first in the amplitude basis

$1/2$7

then, after a conditional application of $1/2$8 on the ancilla, in the phase basis

$1/2$9

The receiver applies a single-qubit Pauli correction according to the pair of outcomes and the initial shared Bell state. For initial $$|\Psi\rangle_{S_1R_1S_2R_2C} = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle_{S_1R_1}|\psi_2\rangle_{S_2R_2}|a\rangle_C \pm |\psi_3\rangle_{S_1R_1}|\psi_4\rangle_{S_2R_2}|b\rangle_C \right),$$0, the corrections are $$|\Psi\rangle_{S_1R_1S_2R_2C} = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle_{S_1R_1}|\psi_2\rangle_{S_2R_2}|a\rangle_C \pm |\psi_3\rangle_{S_1R_1}|\psi_4\rangle_{S_2R_2}|b\rangle_C \right),$$1, $$|\Psi\rangle_{S_1R_1S_2R_2C} = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle_{S_1R_1}|\psi_2\rangle_{S_2R_2}|a\rangle_C \pm |\psi_3\rangle_{S_1R_1}|\psi_4\rangle_{S_2R_2}|b\rangle_C \right),$$2, $$|\Psi\rangle_{S_1R_1S_2R_2C} = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle_{S_1R_1}|\psi_2\rangle_{S_2R_2}|a\rangle_C \pm |\psi_3\rangle_{S_1R_1}|\psi_4\rangle_{S_2R_2}|b\rangle_C \right),$$3, and $$|\Psi\rangle_{S_1R_1S_2R_2C} = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle_{S_1R_1}|\psi_2\rangle_{S_2R_2}|a\rangle_C \pm |\psi_3\rangle_{S_1R_1}|\psi_4\rangle_{S_2R_2}|b\rangle_C \right),$$4 for $$|\Psi\rangle_{S_1R_1S_2R_2C} = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle_{S_1R_1}|\psi_2\rangle_{S_2R_2}|a\rangle_C \pm |\psi_3\rangle_{S_1R_1}|\psi_4\rangle_{S_2R_2}|b\rangle_C \right),$$5, $$|\Psi\rangle_{S_1R_1S_2R_2C} = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle_{S_1R_1}|\psi_2\rangle_{S_2R_2}|a\rangle_C \pm |\psi_3\rangle_{S_1R_1}|\psi_4\rangle_{S_2R_2}|b\rangle_C \right),$$6, $$|\Psi\rangle_{S_1R_1S_2R_2C} = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle_{S_1R_1}|\psi_2\rangle_{S_2R_2}|a\rangle_C \pm |\psi_3\rangle_{S_1R_1}|\psi_4\rangle_{S_2R_2}|b\rangle_C \right),$$7, and $$|\Psi\rangle_{S_1R_1S_2R_2C} = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle_{S_1R_1}|\psi_2\rangle_{S_2R_2}|a\rangle_C \pm |\psi_3\rangle_{S_1R_1}|\psi_4\rangle_{S_2R_2}|b\rangle_C \right),$$8, respectively (Sharma et al., 2014).

Deterministic CJBRSP distributes the classical knowledge of the state between two senders per direction. One sender, who knows amplitudes, measures in $$|\Psi\rangle_{S_1R_1S_2R_2C} = \frac{1}{\sqrt{2}} \left( |\psi_1\rangle_{S_1R_1}|\psi_2\rangle_{S_2R_2}|a\rangle_C \pm |\psi_3\rangle_{S_1R_1}|\psi_4\rangle_{S_2R_2}|b\rangle_C \right),$$9. The other, who knows the phase, conditionally applies $|a\rangle,|b\rangle$0 on the ancilla and measures in $|a\rangle,|b\rangle$1. The receiver correction depends on the pair of outcomes and the GHZ label, and Charlie’s disclosure specifies which GHZ resource is active in each direction (Sharma et al., 2014).

4. Noise model, fidelity, and comparative behavior

The 2014 analysis studies probabilistic CBRSP under two noise models: amplitude damping and phase damping. The noise acts independently only on the four travel qubits $|a\rangle,|b\rangle$2; Charlie’s qubit $|a\rangle,|b\rangle$3 is not transmitted and is assumed noiseless. The two qubits going to Alice experience identical noise, and the two qubits going to Bob do likewise (Sharma et al., 2014).

The Kraus operators are

$|a\rangle,|b\rangle$4

for amplitude damping, and

$|a\rangle,|b\rangle$5

for phase damping (Sharma et al., 2014).

For a 5-qubit resource $|a\rangle,|b\rangle$6, the noisy state is

$|a\rangle,|b\rangle$7

with $|a\rangle,|b\rangle$8. Successful probabilistic CBRSP is modeled by postselection on sender outcomes $|a\rangle,|b\rangle$9 and a fixed Charlie outcome, followed by normalization, tracing out the measured systems, and applying the appropriate Pauli corrections (Sharma et al., 2014).

For the specific channel

$|\psi_i\rangle$0

the target receiver state in the noiseless successful case is

$|\psi_i\rangle$1

and the fidelity is defined by

$|\psi_i\rangle$2

The analytical fidelities are

$|\psi_i\rangle$3

and

$|\psi_i\rangle$4

The paper’s key observations are that the phases $|\psi_i\rangle$5 drop out of the fidelity, that fidelities depend on amplitudes $|\psi_i\rangle$6 and decoherence parameters, and that for equal parameters $|\psi_i\rangle$7 and $|\psi_i\rangle$8, $|\psi_i\rangle$9 for all $$|\psi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\phi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}},$$0; moreover, $$|\psi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\phi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}},$$1 decreases monotonically with $$|\psi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\phi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}},$$2, while $$|\psi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\phi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}},$$3 can be non-monotonic (Sharma et al., 2014).

The same Kraus-operator method and postselection formalism are stated to apply to deterministic CBRSP and CJBRSP by using the appropriate 7-qubit resource and applying Kraus maps to the travel qubits only; the locally created ancillas are not sent and thus are not affected (Sharma et al., 2014). In the high-dimensional setting, a separate robustness study for $$|\psi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\phi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}},$$4 evaluates qudit-flip noise, dephasing noise, and qudit-phase-flip noise, with explicit fidelity expressions and the observation that the equal-amplitude equatorial structure can produce symmetry-based resilience patterns (Du et al., 6 Jan 2025).

5. High-dimensional and dynamically generated BRSP

A major generalization replaces qubits by arbitrary $$|\psi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\phi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}},$$5-dimensional systems. In this deterministic bidirectional controlled remote state preparation scheme, Alice and Bob simultaneously prepare equatorial single-quNit states

$$|\psi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\phi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}},$$6

with $$|\psi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\phi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}},$$7 and $$|\psi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\phi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}},$$8, under Charlie’s control (Du et al., 6 Jan 2025).

Let $$|\psi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\phi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}},$$9. Alice measures $|\psi_1\rangle\neq|\psi_3\rangle$0 in the orthonormal basis

$|\psi_1\rangle\neq|\psi_3\rangle$1

Bob measures $|\psi_1\rangle\neq|\psi_3\rangle$2 in

$|\psi_1\rangle\neq|\psi_3\rangle$3

and Charlie measures $|\psi_1\rangle\neq|\psi_3\rangle$4 and $|\psi_1\rangle\neq|\psi_3\rangle$5 in the generalized Fourier basis

$|\psi_1\rangle\neq|\psi_3\rangle$6

After the announced outcomes, the residual states on the receivers are $|\psi_1\rangle\neq|\psi_3\rangle$7 and $|\psi_1\rangle\neq|\psi_3\rangle$8, and the required corrections are diagonal phase operators: $|\psi_1\rangle\neq|\psi_3\rangle$9 The success probability is 1 once Charlie performs the prescribed measurements and broadcasts his outcomes. The classical communication cost is $|\psi_2\rangle\neq|\psi_4\rangle$0 bits total, and a stated advantage is that high-dimensional CNOT operations are not required during the protocol (Du et al., 6 Jan 2025).

The quantum-walk-assisted BRSP formulation changes the underlying mechanism more radically. Alice and Bob each hold a position register and two coin registers; in the controlled variant, Charlie holds two additional coins. The protocol is implemented on two independent one-dimensional lattices or two independent cycles with two and four vertices. A coined quantum walk step has the standard form

$|\psi_2\rangle\neq|\psi_4\rangle$1

with $|\psi_2\rangle\neq|\psi_4\rangle$2, and the conditional shift correlates the coin value with the walker’s move. This coin–position correlation generates entanglement dynamically. For example, on the line,

$|\psi_2\rangle\neq|\psi_4\rangle$3

whose reduced coin density matrix has von Neumann entropy 1 bit (Naseeda et al., 8 Feb 2026).

The uncontrolled BRSP uses four walk steps and 6 qubits total; the controlled BRSP uses six walk steps and 8 qubits total. Alice and Bob measure positions and first coins in bases tied to the target coefficients $|\psi_2\rangle\neq|\psi_4\rangle$4 and $|\psi_2\rangle\neq|\psi_4\rangle$5, then apply local Pauli corrections on their second coins. The resulting prepared states are exact conditional on success, with $|\psi_2\rangle\neq|\psi_4\rangle$6. The typical successful branches reported have $|\psi_2\rangle\neq|\psi_4\rangle$7 for uncontrolled line, $|\psi_2\rangle\neq|\psi_4\rangle$8, and $|\psi_2\rangle\neq|\psi_4\rangle$9; $N$00 for controlled $N$01 and controlled $N$02; and $N$03 for the controlled line example (Naseeda et al., 8 Feb 2026).

These developments show two non-equivalent generalization strategies. One retains the entangled-channel viewpoint and extends it to quNits with deterministic correction by diagonal phase operators (Du et al., 6 Jan 2025). The other eliminates the preshared-entanglement assumption entirely by generating the relevant entanglement during a walk, at the cost of probabilistic postselection and a restriction to real-amplitude single-qubit targets (Naseeda et al., 8 Feb 2026).

6. Related protocols, boundaries of the term, and open directions

The literature surrounding BRSP includes protocols that are adjacent but not identical to the strict Alice$N$04Bob formulation. A notable example is simultaneous asymmetric remote state preparation, where a single sender prepares a 1-qubit state for one receiver and a 2-qubit Bell-like state for another in one round using a five-qubit entangled state. That protocol is “two-directional outward” from Alice but is not reciprocal, so it is not bidirectional in the strict Alice$N$05Bob sense (Choudhury et al., 2023). This distinction is important because BRSP, as defined in the controlled and uncontrolled works discussed above, requires simultaneous preparation in opposite directions.

A second boundary case is the bidirectional controlled hybrid of teleportation and RSP. In that protocol, Alice teleports an unknown $N$06-qubit state to Bob while Bob remotely prepares a known $N$07-qubit product state at Alice’s site, under Charlie’s authorization, using a $N$08-qubit channel. In its native form, it is a bidirectional hybrid rather than true BRSP, because one direction is teleportation and the other is RSP (Valeh et al., 29 Jan 2025). The distinction matters because BRSP presumes that both target states are known to their respective senders.

Within the strict BRSP line, the comparative discussion in the generalized 2014 treatment is explicit. Relative to BRSP without control, CBRSP adds a single classical bit from Charlie and replaces two independent Bell pairs with a structured five-partite resource. Relative to one-way probabilistic RSP, bidirectionality doubles the sender communications and adds Charlie’s bit. Relative to bidirectional teleportation, probabilistic RSP can save classical bits because the sender knows the state; deterministic RSP removes this advantage and matches teleportation’s classical cost per direction, although the protocol structure and correction rules differ (Sharma et al., 2014).

Practical feasibility is also treated in a bounded way. Bell and GHZ sources, single-qubit rotations, CNOTs, single-qubit projective measurements, and classical broadcast are described as being within current experimental reach in photonic and trapped-ion platforms for the qubit framework (Sharma et al., 2014). In the high-dimensional scheme, if the quNit is encoded in the spatial mode of single photons, the protocol can be accomplished solely using only linear optical elements, including multiport interferometers and phase shifters (Du et al., 6 Jan 2025). In the quantum-walk approach, physical implementations are associated with trapped ions, photonic platforms, and NMR, under assumptions of negligible decoherence across the walk duration and high-fidelity single-qubit Hadamards and projective measurements (Naseeda et al., 8 Feb 2026).

Open directions are also stated explicitly. For the generalized qubit framework, full noise analysis beyond the probabilistic case is outlined but not carried out analytically; the same Kraus/postselection method is said to apply directly to deterministic CBRSP and CJBRSP (Sharma et al., 2014). Extensions to multi-qubit target states and higher-dimensional systems, security modeling beyond control, and experimental demonstrations of controlled bidirectional variants under realistic noise are identified as open problems (Sharma et al., 2014). The high-dimensional work is currently tailored to equatorial states with uniform amplitudes; extending to arbitrary amplitude states may require additional resources or operations (Du et al., 6 Jan 2025). The quantum-walk formulation identifies the extension to arbitrary complex amplitudes, deterministic walk-engineered BRSP, and rigorous robustness analysis under realistic noise models as open questions (Naseeda et al., 8 Feb 2026).