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arXiv:1310.6002

Weak Values with Remote Postselection and Shared Entanglement

Arun Kumar Pati, Uttam Singh

Published Oct 22, 2013• 1 citation
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Abstract

We propose a new protocol for the weak measurement of any observable with remote pre and postselections. We show that if two parties share a pure entangled state, then by using local operations and classical communication they can preselect and postselect at distant locations leading to the weak value of an observable as a shift in the pointer of the apparatus at one location in the process of the weak measurement. This can be achieved with either sharing of a pure maximally or non-maximally entangled state. We generalize the protocol for realizing the weak value of any observable with remote pre and postselection of mixed states. Finally, we show how the weak value is modified in the remote pre and postselection setting if Alice and Bob share a mixed entangled state.

DOI
10.48550/arxiv.1310.6002

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We propose a new protocol for the weak measurement of any observable with remote pre and postselections. We show that if two parties share a pure entangled state, then by using local operations and classical communication they can preselect and postselect at distant locations leading to the weak value of an observable as a shift in the pointer of the apparatus at one location in the process of the weak measurement. This can be achieved with either sharing of a pure maximally or non-maximally entangled state. We generalize the protocol for realizing the weak value of any observable with remote pre and postselection of mixed states. Finally, we show how the weak value is modified in the remote pre and postselection setting if Alice and Bob share a mixed entangled state.
It is argued that a weak value of an observable is a robust property of a single pre- and postselected quantum system rather than a statistical property. During an infinitesimal time a system with a given weak value affected other systems as if it had been in an eigenstate with eigenvalue equal to the weak value. This differs significantly from the action of a system preselected only and possessing a numerically equal expectation value. The weak value has a physical meaning beyond a conditional average of a pointer in the weak measurement procedure. The difference between the weak value and the expectation value has been demonstrated on the example of photon polarization. In addition, the weak values for systems pre- and postselected in mixed states are considered.
Weak measurement is a novel quantum measurement scheme, which is usually characterized by the weak value formalism. To guarantee the validity of the weak value formalism, the fidelity between the preselection and the postselection should not be too small generally. In this work, we study the weak measurement on a qubit system with exactly or asymptotically orthogonal pre- and postselections. We shall establish a general rigorous framework for the weak measurement beyond the weak value formalism, and obtain the average output of a weak measurement when the pre- and postselections are exactly orthogonal. We shall also study the asymptotic behavior of a weak measurement in the limiting process that the pre- and postselections tend to be orthogonal.
Entanglement and uncertainty relation are two focuses of quantum theory. We relate entanglement sharing to the entropic uncertainty relation in a $(d\ifmmode\times\else\texttimes\fi{}d)$-dimensional system via weak measurements with different pointers. We consider both the scenarios of one-sided sequential measurements in which the entangled pair is distributed to multiple Alices and one Bob and two-sided sequential measurements in which the entangled pair is distributed to multiple Alices and Bobs. It is found that the maximum number of observers sharing the entanglement strongly depends on the measurement scenarios, the pointer states of the apparatus, and the local dimension ddd of each subsystem, while the required minimum measurement precision to achieve entanglement sharing decreases to its asymptotic value with the increase of ddd. The maximum number of observers remain unaltered even when the state is not maximally entangled but has strong-enough entanglement.
We revisit the problem of detection of entanglement of an unknown two-qubit state using minimal resources. Using weak values and just two copies of an arbitrary two-qubit state, we present a protocol where a postselection measurement in the computational basis provides enough information to identify if the state is entangled or not. Our protocol enables complete state identification with a single-setting postselection measurement on two copies of the state. It follows that by restricting to pure states, the global interaction required for determining the weak values can be realized by local operations. We further show that our protocol is robust against errors arising from inappropriate global interactions applied during weak value determination.
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=Rz(γ)Rx(β)Rz(α)U = R_z(γ)R_x(β)R_z(α)U=Rz​(γ)Rx​(β)Rz​(α) using pre-shared Bell states. We then construct a protocol for the two-qubit operation Rzz(θ)R_{zz}(θ)Rzz​(θ) using a single pre-shared Bell state and extend it to Rxx(θ)R_{xx}(θ)Rxx​(θ) and Ryy(θ)R_{yy}(θ)Ryy​(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.
We propose a communication protocol exploiting correlations between two events with a definite time ordering: (a) the outcome of a weak measurement on a spin, and (b) the outcome of a subsequent ordinary measurement on the spin. In our protocol, Alice first generates a ``code'' by performing weak measurements on a sample of N spins. The sample is sent to Bob, who later performs a postselection by measuring the spin along either of two certain directions. The results of the postselection define the ``key,'' which he then broadcasts publicly. Using both her previously generated code and this key, Alice is able to infer the direction chosen by Bob in the postselection. Alternatively, if Alice broadcasts publicly her code, Bob is able to infer from the code and the key the direction chosen by Alice for her weak measurement. Two possible experimental realizations of the protocols are briefly mentioned.