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arXiv:2610.04995

Blind Remote Implementation of Arbitrary Two-Qubit Unitary Operation via Pre-Shared Entanglement

Plaban Saha, Binayak S. Choudhury

quant-phSubmitted Oct 4, 2026• Announced Oct 6, 2026• 0 citations

Abstract

We propose a protocol for the remote implementation of an arbitrary two-qubit unitary operation using pre-shared entanglement. In our protocol, Bob possesses an unknown two-qubit state, while Alice implements the required operations using calibrated black-box devices. Alice does not need to know the parameters of these operations, and Bob's unknown state never leaves his laboratory. The protocol is based on the Euler decomposition of single-qubit unitary operations and the Cartan decomposition of two-qubit unitary operations. First, we construct protocols for the remote implementation of the single-qubit operation $U = R_z(γ)R_x(β)R_z(α)$ using pre-shared Bell states. We then construct a protocol for the two-qubit operation $R_{zz}(θ)$ using a single pre-shared Bell state and extend it to $R_{xx}(θ)$ and $R_{yy}(θ)$ through local unitary transformations. These elementary protocols are finally combined to implement an arbitrary two-qubit unitary operation. A third party, Charlie, determines the required decomposition and provides Alice with the corresponding black-box devices before the protocol begins. Thus, neither Alice nor Bob requires the classical description of the unitary operation.

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We propose a protocol for the remote implementation of an arbitrary two-qubit unitary operation using pre-shared entanglement. In our protocol, Bob possesses an unknown two-qubit state, while Alice implements the required operations using calibrated black-box devices. Alice does not need to know the parameters of these operations, and Bob's unknown state never leaves his laboratory. The protocol is based on the Euler decomposition of single-qubit unitary operations and the Cartan decomposition of two-qubit unitary operations. First, we construct protocols for the remote implementation of the single-qubit operation

U = R_z(γ)R_x(β)R_z(α)

using pre-shared Bell states. We then construct a protocol for the two-qubit operation

R_{zz}(θ)

using a single pre-shared Bell state and extend it to

R_{xx}(θ)

and

R_{yy}(θ)

through local unitary transformations. These elementary protocols are finally combined to implement an arbitrary two-qubit unitary operation. A third party, Charlie, determines the required decomposition and provides Alice with the corresponding black-box devices before the protocol begins. Thus, neither Alice nor Bob requires the classical description of the unitary operation.

Consider two remote laboratories: Alice holds an unknown input |ξ⟩ and a private single-qubit unitary UA, while Bob holds |η⟩ and UB. We present an exact local-operations-and-classical-communication protocol that produces UB|ξ⟩ at Alice’s site and UA|η⟩ at Bob’s site only after the controller labels are released. A five-qubit channel first relocates the input states; operation delivery then uses a five-particle channel of local dimension d, a 2d-outcome nonsymmetric qubit–qudit measurement, and label-dependent inverse maps. We give the qutrit case explicitly and one correction formula for arbitrary d. A dual-layer construction lets Charlie authorize both directions, while David and Fiona authorize one direction each. In the ideal lossless model, all admissible branches have recovery maps and their probabilities sum to unity. The basic qutrit protocol uses ten entangled carriers and 13 fixed-length classical bits; the dual-layer protocol uses fourteen carriers and 19 bits. Thus, dimensional flexibility and direction-selective control are obtained at increased resource-protection and signaling costs. This is an exact theoretical remote-operation construction, not an experimentally mature or composably secure communication protocol.

We consider the implementation of an arbitrary unitary operation U upon a distant quantum system. This teleportation of U can be viewed as quantum remote control. We investigate protocols that achieve this using local operations, classical communication, and shared entanglement. Lower bounds on the necessary entanglement and classical communication are determined using causality and the linearity of quantum mechanics. We examine in particular detail the resources required if the remote control is to be implemented as a classical black box. Under these circumstances, we prove that the required resources are, necessarily, those needed for implementation by bidirectional state teleportation.

We study the remote implementation of a unitary transformation on the state of a qubit. We show the existence of nontrivial protocols (i.e., using less resources than bidirectional state teleportation) that allow the perfect remote implementation of certain continuous sets of quantum operations. We prove that, up to a local change of basis, only two subsets exist that can be implemented remotely with a nontrivial protocol: Arbitrary rotations around a fixed direction $\stackrel{\ensuremath{\rightarrow}}{n}$ and a $\ensuremath{\pi}$ rotation about an arbitrary direction lying in a plane orthogonal to $\stackrel{\ensuremath{\rightarrow}}{n}.$ The former operations effectively constitute the teleportation of arbitrary angles. The overall classical information and distributed entanglement cost required for the remote implementation depends on whether it is known, a priori, in which of the two teleportable subsets the transformation belongs. If it is known, the optimal protocol consumes one e-bit of entanglement and one c-bit in each direction. If it is not known in which subset the transformation belongs, two e-bits of entanglement need to be consumed and the classical channel becomes asymmetric with two c-bits being conveyed from Alice to Bob but only one from Bob to Alice.

A scheme to teleport an unknown two qubit state from Alice (the sender) to Bob (the receiver) using two Einstein Podolsky Rosen (EPR) pairs is presented, each EPR pair being shared by both Alice and Bob. Firstly, Alice combines each of the two particles in the teleported state with an EPR particle and makes Bell state measurement on each combination. Then she transmits the outcomes of her measurements to Bob classically. According to Alice′s measurement results, Bob can perform appropriate unitary operations on his two EPR particles to retrieve the initial state.

Distributed computing is a fastest growing field -- enabling virtual computing, parallel computing, and distributed storage. By exploiting the counterfactual techniques, we devise a distributed blind quantum computation protocol to perform a universal two-qubit controlled unitary operation for any input state without using preshared entanglement and without exchanging physical particles between remote parties. This distributed protocol allows Bob to counterfactully apply an arbitrary unitary operator to Alice's qubit in probabilistic fashion, without revealing the operator to her, using a control qubit -- called the counterfactual concealed telecomputation (CCT). It is shown that the protocol is valid for general input states and that single-qubit unitary teleportation is a special case of CCT. The quantum circuit for CCT can be implemented using the (chained) quantum Zeno gates and the protocol becomes deterministic with simplified circuit implementation if the initial composite state of Alice and Bob is a Bell-type state.