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arXiv:0704.1962 · Journal of Physics A Mathematical and Theoretical

A simple test of quantumness for a single system

Robert Alicki, Nicholas Van Ryn

Published Jan 29, 2008• 37 citations
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Abstract

We propose a simple test of quantumness which can decide whether for a given set of accessible experimental data the classical model is insufficient. Take two observables A , B such that for all states ψ their mean values satisfy 0 ⩽ ⟨ψ| A |ψ⟩ ⩽ ⟨ψ| B |ψ⟩ ⩽ 1. If there exists a state ϕ such that the second moments fulfill the inequality ⟨ϕ| A 2 |ϕ⟩ > ⟨ϕ| B 2 |ϕ⟩ then the system cannot be described by a classical probabilistic scheme. An example of an optimal triple ( A , B , ϕ) in the case of a qubit is given.

DOI
10.1088/1751-8113/41/6/062001

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We propose a simple test of quantumness which can decide whether for a given set of accessible experimental data the classical model is insufficient. Take two observables A , B such that for all states ψ their mean values satisfy 0 ⩽ ⟨ψ| A |ψ⟩ ⩽ ⟨ψ| B |ψ⟩ ⩽ 1. If there exists a state ϕ such that the second moments fulfill the inequality ⟨ϕ| A 2 |ϕ⟩ > ⟨ϕ| B 2 |ϕ⟩ then the system cannot be described by a classical probabilistic scheme. An example of an optimal triple ( A , B , ϕ) in the case of a qubit is given.
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.
We study the problem of deterministic transformations of an initial pure entangled quantum state, |ψ〉, into a target pure entangled quantum state, |ϕ〉, by using local operations and classical communication (LOCC). A celebrated result of Nielsen (1999) gives the necessary and sufficient condition that makes this entanglement transformation process possible. Indeed, this process can be achieved if and only if the majorization relation ψ≺ϕ holds, where ψ and ϕ are probability vectors obtained by taking the squares of the Schmidt coefficients of the initial and target states, respectively. In general, this condition is not fulfilled. However, one can look for an approximate entanglement transformation. Vidal et al. (2000) have proposed a deterministic transformation using LOCC in order to obtain a target state |χopt〉 most approximate to |ϕ〉 in terms of maximal fidelity between them. Here, we show a strategy to deal with approximate entanglement transformations based on the properties of the majorization lattice. More precisely, we propose as approximate target state one whose Schmidt coefficients are given by the supremum between ψ and ϕ. Our proposal is inspired on the observation that fidelity does not respect the majorization relation in general. Remarkably enough, we find that for some particular interesting cases, like two-qubit pure states or the entanglement concentration protocol, both proposals are coincident.
In this letter, we propose and experimentally test a quantumness criterion for single systems. The criterion is based on the violation of an already reported classical inequality. This inequality is expressed in terms of joint probabilities that we identify as realizations of preparation-and-measurement schemes. Our analysis points out that superposition is the key notion to grasp the quantum nature of physical systems. We verify our criterion in an all-optical experimental setup with heralded single photons generated via spontaneous parametric down conversion and use exclusively projective preparations and projective measurements.
Entanglement in the S-Field 1. Ontic basis of entanglement In the S-framework, each particle is a two-mode S-oscillator: Φ = (Φ_sb , Φ_bs). Entanglement occurs when two or more S-oscillators share a common S-phase structure. This is not information transmission between particles in B-space — it is a single, joint S-configuration projected into multiple B-locations. Thus entanglement is a multi-projection of one S-state, not a correlation between two separ ate B-states. 2. Shared S-Phase Condition (the core definition) Let particles A and B have states: Φ_A = (A_sb exp(i φ_A_sb), A_bs exp(i φ_A_bs)) Φ_B = (B_sb exp(i φ_B_sb), B_bs exp(i φ_B_bs)). They are entangled if and only if: Δφ_A = Δφ_B, and the bs-components satisfy a joint coherence: Φ_A_bs = λ Φ_B_bs, where λ is a complex ratio with |λ| fixed during the entangled interval. This means: The S-components form a single connected oscillatory object. Mathematically, define the Joint S-State: Ψ_S = Φ_A_bs ⊗ Φ_B_bs. Entanglement corresponds to: rank(Ψ_S) = 1. This is the most compact mathematical expression of entanglement in this theory. 3. Why entanglement is instantaneous Inside the S-domain: • there is no spatial metric • there is no propagation • there is only phase structure. Thus a change in Φ_A_sb or Φ_A_bs that forces a new Δφ_A immediately alters Δφ_B because both projections originate from the same Ψ_S. This explains experimental observations: • zero-time correlations • infinite effective signaling speed • independence from spatial separation • no energy or momentum transfer. All of them follow automatically from: S has no distance and no finite propagation speed. 4. Collapse as B-projection selection Measurement is interpreted as: P_B Φ_A → specific eigen-projection. Because A and B share the same Ψ_S, selecting one B-projection forces the entire Ψ_S to project consistently. Thus: Measurement collapse is not a physical signal. It is selection of one branch of a shared S-object. No superluminal influence is needed. 5. Conservation laws Because both particles share the same S-state: • Spin correlations • Momentum conservation • Polarization complements appear automatically from the structure of Ψ_S: Ψ_S = Φ_A_bs ⊗ Φ_B_bs. Constraints on Φ_A_bs impose the complementary constraints on Φ_B_bs. Thus conservation laws are not maintained through 3D interaction, but through S-symmetry of the joint state. 6. Entanglement swapping and teleportation Teleportation in the S-framework is simple: 1. Two particles share Ψ_S₁. 2. One particle from Ψ_S₁ interacts with a particle from Ψ_S₂. 3. Their S-phases align (Δφ matching). 4. This merges Ψ_S₁ and Ψ_S₂ into a new unified Ψ_S. The B-space interpretation is “entanglement swapping,” but the S-interpretation is: merging or re-partitioning of phase-coherent S-objects. Teleportation = transfer of B-state through S-resonance alignment, not physical movement. 7. Unified S-Equation of Entanglement The full formal criterion for entanglement is: P_S (Φ_A ⊗ Φ_B) ≠ (P_S Φ_A) ⊗ (P_S Φ_B). This inequality means: “The S-projection of the joint system cannot be expressed as a product of the S-projections of its parts.” Or verbally: Entanglement is non-factorability of the S-projection. This matches all known experimental results but gives a far clearer ontology. Summary in one sentence Entanglement is the multi-location projection of a single S-phase coherent structure into baryonic space, explaining the instantaneous correlations and nonlocal behavior naturally.