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arXiv:quant-ph/0412199

The internal logic of Bell's states

Giulia Battilotti, Paola Zizzi

Published Dec 26, 2004• 6 citations
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Abstract

We investigate the internal logic of a quantum computer with two qubits, in the two particular cases of non-entanglement (separable states) and maximal entanglement (Bell's states). To this aim, we consider an internal (reversible) measurement which preserves the probabilities by mirroring the states. We then obtain logical judgements for both cases of separable and Bell's states.

DOI
10.48550/arxiv.quant-ph/0412199

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We investigate the internal logic of a quantum computer with two qubits, in the two particular cases of non-entanglement (separable states) and maximal entanglement (Bell's states). To this aim, we consider an internal (reversible) measurement which preserves the probabilities by mirroring the states. We then obtain logical judgements for both cases of separable and Bell's states.
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Our investigation aims to study the specific role played by entanglement in the quantum computation process, by elaborating an entangled spin model developed within the 'hidden measurement approach' to quantum mechanics. We show that an arbitrary tensor product state for the entity consisting of two entangled qubits can be described in a complete way by a specific internal constraint between the ray and density states of the two qubits. For the individual qubits we use a sphere model representation, which is a generalization of the Bloch or Pauli representation, where also the collapse and noncollapse measurements are represented. We identify a parameter r in [0,1], arising from the Schmidt diagonal decomposition, that is a measure of the amount of entanglement, such that for r = 0 the system is in the singlet state with 'maximal' entanglement, and for r = 1 the system is in a pure product state.
Perfect quantum state mirroring in a chain of N spins is defined as the condition in which the state |i〉 of the chain is swapped into the state |N — i〉 within a time evolution interval τ. Such a phenomenon is an interesting way of transfering entanglement. An expressions for the perfect mirroring of a single qubit contained in a spin chain were proposed in the past. We exploit such an expressions for calculating the evolution times in chains of both two and three spins. In the case of a chain of two qubits, we derive conditions under which the associated four Bell states diagonalize the Hamiltonian. It is found that for the two Bell states |Φ + 〉 and |Φ − 〉, perfect mirroring does not occur (i.e. entanglement is not preserved under swapping). On the other hand, perfect single qubit mirror effect (entanglement preservation) indeed occurs for the other two Bell states |Ψ + 〉 and |Ψ − 〉 which are mapped into |Φ + 〉 and |Φ − 〉 respectively. For the case of a chain of three qubits, the effects of a perfect single qubit mirroring on a set of four maximally entangled three qubit states ψ 1 , ψ 2 , χ 1 , and χ 2 are studied. Due to the fact that quantum mirroring preserves maximal entanglement, the states ψ 1 and ψ 2 are not altered. However, quantum mirroring changes the states χ 1 and χ 2 only if we apply perfect quantum state mirroring in the site a = 1 of the three qubits spin chain. The above constrains the preservation of maximal entanglement under qubit mirroring of such a state. Due to the fact that swapping has already been experimentally tested, a posible experimental implementations of single qubit mirroring is possible.