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Revocation and Reconstruction of Shared Quantum States

Prakash Mudholkar, Chiranjeevi Vanarasa, Indranil Chakrabarty, Srinathan Kannan

TL;DR

This work addresses revocation of a quantum state after sharing but before reconstruction by introducing a four-qubit entangled resource $|G_{abcd} angle$ in a CSR setting where the dealer holds a quantum share. The protocol uses Bell-basis sharing to distribute a three-qubit entangled state among Alice, Bob, and Charlie, and then employs the dealer’s quantum share together with Hadamard-basis measurements to revoke the state if the shareholders are dishonest, or to enable reconstruction if they are not. It provides an explicit demonstration with a four-qubit state $|G_{abcd}^E angle$ and derives parameter regimes—via Theorems 1 and 2—where both revocation and reconstruction are possible, highlighting ranges on $a,b,c,d$ (real/imaginary) that ensure the corrective unitaries are independent of the secret amplitudes $| heta angle=ig(egin{smallmatrix}\alpha\ etaig)$, and thus practical viability. The results advance quantum secret sharing by reducing qubit requirements for revocation compared to prior five-qubit schemes and by outlining concrete conditions under which secure, controllable state reconstruction can be achieved in a multi-party setting.

Abstract

The problem of revocation of quantum states after sharing is interesting and we ask: Is it possible for a dealer to revoke the state once shared, before the reconstruction process? Additional resources like bell states are used to help the dealer to get back the state. In a three-party scenario, we show an independent way to revoke, if, for any reason, the dealer is not sure about the intention of the/any reconstructor. In general, the classical outcomes of the dealer in sharing phase are needed, to be able to reconstruct the state perfectly. When both the shareholders are dishonest, and without the dealer's knowledge, collude to reconstruct, they always have some chance of succeeding. This is addressed by giving more control to the dealer by making him/her) to have a quantum share as well. We give a sharing and revocation protocol with a four-qubit entangled resource shared among three parties (two qubits with the dealer and one each with the shareholders). We further consider a class of four qubit pure entangled states as resource and explicitly find the range of parameters for which the protocol will be successful.

Revocation and Reconstruction of Shared Quantum States

TL;DR

This work addresses revocation of a quantum state after sharing but before reconstruction by introducing a four-qubit entangled resource in a CSR setting where the dealer holds a quantum share. The protocol uses Bell-basis sharing to distribute a three-qubit entangled state among Alice, Bob, and Charlie, and then employs the dealer’s quantum share together with Hadamard-basis measurements to revoke the state if the shareholders are dishonest, or to enable reconstruction if they are not. It provides an explicit demonstration with a four-qubit state and derives parameter regimes—via Theorems 1 and 2—where both revocation and reconstruction are possible, highlighting ranges on (real/imaginary) that ensure the corrective unitaries are independent of the secret amplitudes , and thus practical viability. The results advance quantum secret sharing by reducing qubit requirements for revocation compared to prior five-qubit schemes and by outlining concrete conditions under which secure, controllable state reconstruction can be achieved in a multi-party setting.

Abstract

The problem of revocation of quantum states after sharing is interesting and we ask: Is it possible for a dealer to revoke the state once shared, before the reconstruction process? Additional resources like bell states are used to help the dealer to get back the state. In a three-party scenario, we show an independent way to revoke, if, for any reason, the dealer is not sure about the intention of the/any reconstructor. In general, the classical outcomes of the dealer in sharing phase are needed, to be able to reconstruct the state perfectly. When both the shareholders are dishonest, and without the dealer's knowledge, collude to reconstruct, they always have some chance of succeeding. This is addressed by giving more control to the dealer by making him/her) to have a quantum share as well. We give a sharing and revocation protocol with a four-qubit entangled resource shared among three parties (two qubits with the dealer and one each with the shareholders). We further consider a class of four qubit pure entangled states as resource and explicitly find the range of parameters for which the protocol will be successful.
Paper Structure (9 sections, 2 theorems, 120 equations, 1 figure, 3 tables)

This paper contains 9 sections, 2 theorems, 120 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. The Revocation and Reconstruction Protocol: Quantum Share Of The State
  4. Going Beyond Three Qubit State Sharing
  5. Our Protocol: A New Quantum State Sharing Protocol
  6. Characterization of Four Qubit States useful for Protocol
  7. Revocation of State on Alice's location
  8. Reconstruction of State on Charlie's location
  9. Conclusion:

Key Result

Theorem 1

For ${G}_{abcd}$ state, the necessary conditions for Alice to retrieve the state are as follows: 1. If $a$ and $c$ are chosen as real numbers, then we have $b$ and $d$ as purely imaginary numbers such that: (a) $a^2 \leq \frac{1}{2}$ or $-\frac{1}{\sqrt{2}} \leq a \leq \frac{1}{\sqrt{2}}$ (b) $c^2 \

Figures (1)

  • Figure 1: $A(i.)$ Alice with the secret state (depicted in red), Alice, Bob and Charlie with the 4 qubit resource (depicted in black). $A(ii.)$ Three qubit entanglement among Alice, Bob and Charlie obtained after splitting of quantum information. $A(iii.)$ Two qubit entanglement between Alice and Charlie after Bob's measurement. $A(iv.)$ Revocation by Alice. $B(i.)$ Alice with the secret state (depicted in red), Alice, Bob and Charlie with the 4 qubit resource (depicted in black). $B(ii.)$ Three qubit entanglement among Alice, Bob and Charlie obtained after splitting of quantum information. $B(iii.)$ Two qubit entanglement between Bob and Charlie after Alice's measurement. $B(iv.)$ Charlie with the Reconstructed State.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2