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arXiv:quant-ph/0511172

On Classical Teleportation and Classical Nonlocality

Tal Mor

Published Nov 17, 2005• 2 citations
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Abstract

An interesting protocol for classical teleportation of an unknown classical state was recently suggested by Cohen, and by Gour and Meyer. In that protocol, Bob can sample from a probability distribution P that is given to Alice, even if Alice has absolutely no knowledge about P. Pursuing a similar line of thought, we suggest here a limited form of nonlocality - "classical nonlocality". Our nonlocality is the (somewhat limited) classical analogue of the Hughston-Jozsa-Wootters (HJW) quantum nonlocality. The HJW nonlocality tells us how, for a given density matrix rho, Alice can generate any rho-ensemble on the North Star. This is done using surprisingly few resources - one shared entangled state (prepared in advance), one generalized quantum measurement, and no communication. Similarly, our classical nonlocality presents how, for a given probability distribution P, Alice can generate any P-ensemble on the North Star, using only one correlated state (prepared in advance), one (generalized) classical measurement, and no communication. It is important to clarify that while the classical teleportation and the classical non-locality protocols are probably rather insignificant from a classical information processing point of view, they significantly contribute to our understanding of what exactly is quantum in their well established and highly famous quantum analogues.

DOI
10.48550/arxiv.quant-ph/0511172

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An interesting protocol for classical teleportation of an unknown classical state was recently suggested by Cohen, and by Gour and Meyer. In that protocol, Bob can sample from a probability distribution P that is given to Alice, even if Alice has absolutely no knowledge about P. Pursuing a similar line of thought, we suggest here a limited form of nonlocality - "classical nonlocality". Our nonlocality is the (somewhat limited) classical analogue of the Hughston-Jozsa-Wootters (HJW) quantum nonlocality. The HJW nonlocality tells us how, for a given density matrix rho, Alice can generate any rho-ensemble on the North Star. This is done using surprisingly few resources - one shared entangled state (prepared in advance), one generalized quantum measurement, and no communication. Similarly, our classical nonlocality presents how, for a given probability distribution P, Alice can generate any P-ensemble on the North Star, using only one correlated state (prepared in advance), one (generalized) classical measurement, and no communication. It is important to clarify that while the classical teleportation and the classical non-locality protocols are probably rather insignificant from a classical information processing point of view, they significantly contribute to our understanding of what exactly is quantum in their well established and highly famous quantum analogues.
An interesting protocol for classical teleportation of an unknown classical state was recently suggested by Cohen, and by Gour and Meyer. In that protocol, Bob can sample from a probability distribution [Formula: see text] that is given to Alice, even if Alice has absolutely no knowledge about [Formula: see text]. Pursuing a similar line of thought, we suggest here a limited form of non-locality — "classical non-locality." Our non-locality is the (somewhat limited) classical analogue of the Hughston–Jozsa–Wootters (HJW) quantum non-locality. The HJW non-locality (also known as "quantum remote steering") tells us how, for a given density matrix ρ, Alice can generate any ρ-ensemble on the North Star. This is done using surprisingly few resources — one shared entangled state (prepared in advance), one generalized quantum measurement, and no communication. Similarly, our classical non-locality (which we call "classical remote steering") presents how, for a given probability distribution [Formula: see text], Alice can generate any [Formula: see text]-ensemble on the North Star, using only one correlated state (prepared in advance), one (generalized) classical measurement, and no communication. It is important to clarify that while the classical teleportation and the classical non-locality protocols are probably rather insignificant from a classical information processing point of view, they significantly contribute to our understanding of what exactly is quantum in their well established and highly famous quantum analogues.
We investigate whether it is possible to teleport the coherence of an unknown quantum state from Alice to Bob by communicating a smaller number of classical bits in comparison to what is required for teleporting an unknown quantum state. We find that we cannot achieve perfect teleportation of coherence with one bit of classical communication for an arbitrary qubit. However, we find that if the qubit is partially known, i.e., chosen from the equatorial and polar circles of the Bloch sphere, then teleportation of coherence is possible with the transfer of one cbit of information when we have maximally entangled states as a shared resource. In the case of the resource being a nonmaximally entangled state, we can teleport the coherence with a certain probability of success. In a general teleportation protocol for coherence, we derive a compact formula for the final state at Bob's laboratory in terms of the composition of the completely positive maps corresponding to the shared resource state and a joint positive operator-valued measure (POVM) performed by Alice on her qubit and the unknown state. Using this formula, we show that teleportation of the coherence of a partially known state with real matrix elements is perfectly possible with the help of a maximally entangled state as a resource. Furthermore, we explore the teleportation of coherence with Werner states and show that even when Werner states become separable, the amount of teleported coherence is nonzero, implying the possibility of teleportation of coherence without entanglement.
We investigate whether it is possible to teleport the coherence of an unknown quantum state from Alice to Bob by communicating a lesser number of classical bits in comparison to what is required for teleporting an unknown quantum state. We find that we cannot achieve perfect teleportation of coherence with one bit of classical communication for an arbitrary qubit. However, we find that if the qubit is partially known, i.e., chosen from the equatorial and polar circles of the Bloch sphere, then teleportation of coherence is possible with the transfer of one cbit of information when we have maximally entangled states as a shared resource. In the case of the resource being a non-maximally entangled state, we can teleport the coherence with a certain probability of success. In a general teleportation protocol for coherence, we derive a compact formula for the final state at Bob's lab in terms of the composition of the completely positive maps corresponding to the shared resource state and joint POVM performed by Alice on her qubit and the unknown state. Using this formula, we show that teleportation of the coherence of a partially known state with real matrix elements is possible perfectly with the help of a maximally entangled state as a resource. Furthermore, we explore the teleportation of coherence with the Werner states and show that even when the Werner states become separable, the amount of teleported coherence is non-zero, implying the possibility of teleportation of coherence without entanglement.
Quantum information theory has revolutionized the way in which information is processed using quantum resources such as entangled states, local operations and classical communications. Two important protocols in quantum communications are quantum teleportation and remote state preparation. In quantum teleportation neither the sender nor the receiver know the identity of a state. In remote state preparation the sender knows the state which is to be remotely prepared without ever physically sending the object or the complete classical description of it. Using one unit of entanglement and one classical bit Alice can remotely prepare a photon (from special ensemble) of her choice at Bob's laboratory. In remote state measurement Alice asks Bob to simulate any single particle measurement statistics on an arbitrary photon. In this talk we will present these ideas and discuss the latest developments and future open problems.
The hitherto unexplored nonlocality of quantum correlations in the information transfer protocol of remote state preparation (RSP) is investigated in terms of a unified nonlocality argument formulated for teleportation as well as for two distinct RSP schemes. The argument differs only in the respective types of local measurements used in each scenario. One of the RSP schemes uses single particle projective measurements, while the other uses joint Bell-basis measurements. Interestingly, the quantum mechanical violations of the Bell-CHSH type inequalities formulated for each of the RSP setups, while being equal to each other, also turn out to be equal to that of the corresponding Bell-CHSH type local realist inequality for teleportation, using the common resource of non-maximally entangled state. This reveals that for such a resource state, the maximum amount of extractable nonlocality of correlations involved in the transfer of information is the same for the standard teleportation protocol and the two RSP schemes considered here, thereby implying that the essential difference between teleportation and remote state preparation lies in the amount of classical information required to be exchanged between parties, and not in the nonlocality of underlying correlations.
During the quantum teleportation protocol, Alice sends two bits of classical information to Bob, which somehow transports quantum information from Alice's location to Bob's location. The local account of teleportation from Deutsch and Hayden reveals that these bits are actually qubits instantiating locally inaccessible information, and appear classical because they carry information in a form that is robust to decoherence. If the classical bits have absolutely no underlying quantum properties, teleportation will fail - challenging all current and future variants of quantum theory that explain teleportation using only classical properties of the physical bits being transported. A successful teleportation via apparently classical bits therefore indicates that either: (1) the fundamental principle of locality, which says that isolated systems are independent from each other, is false, or (2) the apparently classical bits actually have some quantum properties. We propose using existing models for quantum-classical dynamics, such as Sudarshan's, and the dynamics-independent Constructor Theory, to determine exactly what features a system with unknown dynamics needs to carry information for teleportation. Our proposal may characterise a new form of non-classicality, giving novel insights into hybrid quantum-classical systems. This could lead to new experimental proposals to test the properties of systems with unknown dynamics.