An investigation of time-symmetric quantum mechanics, Hardy’s paradox and its experimental realisation

Research Area

Time symmetric quantum mechanics was first proposed - Aharonov et al. (1964) - in the same year that John Bell published his famous inequality theorem for a local realistic interpretation of quantum mechanics using hidden variables. Since then it has led to the development of weak measurements which are naturally defined within its formalism. Contrary to strong measurements in standard quantum mechanics that cause wavevector collapse, weak measurements may be performed any number of intermediate times to give detailed knowledge of a system of particles. Hardy (1992) published his paradox as a way of applying the non-locality and non-lorentz invariance properties explored by Bell to an entangled pair of particles. This was achieved through the use of counterfactual argument where, under the assumption that the particles follow real paths, contradictions arise. Weak measurements have allowed further investigation of these counterfactual statements by reinterpreting the paradox experimentally with entangled photon pairs - Lundeen and Steinberg (2009). The experiment is a form of entanglement distillation for continuous variable systems, where correlations of photon pairs are measured together with their polarisation angle. The experimental validity of weak measurements is generally accepted, but their interpretation is still open to debate. Nevertheless they have proved a valuable tool in the nascent field of quantum computing where they have led to quantum logic gates in linear optics - Knill et al. (2001). Weak values may vary over a far larger range and this allows very high amplification most recently seen in the more accurate measurement of quantum dot spectra. The importance of pre- and post-selected states has only recently been recognised with growing attention leading to the discovery of new applications, many of which are again in the field of quantum computing for instance in cluster states. Hardy’s paradox could be investigated in future with experimental versions using fermions where the negative weak values will be associated with particle properties exhibiting opposite effects to those associated with their positive weak counterparts. The realisation of newer techniques such as ion-traps proposed by Molmer (2001) will also allow it to be investigated under different applications. Indeed the techniques required are the same as those that will lead to further developments in quantum computing with the measuring of pre- and post-selected qubits - Brun (2008). It is only by investigating our understanding of quantum mechanics at its most fundamental level with the aid of thought experiments and challenging our current notions that we can extend our knowledge to gain a deeper insight into the true nature of quantum mechanics.

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Aharonov Y, Bergmann P.G. and Lebowitz J.L. (1964) Time symmetry in the quantum process of measurement Physical Review 134 B1410.

Brun T. A. Diosi L, Strunz W. T. (2008) Test of weak measurement on a two- or three- qubit computer Physical Review A 77, 032101.

Hardy, L. (1992) Quantum Mechanics, Local Realistic Theories, and Lorentz-Invariant Realistic Theories Physical Review Letters 68, 2981.

Knill E., Laflamme R. and Milburn G.J. (2001) A scheme for efficient quantum computation with linear optics Nature 409, 46.

Lundeen, J.S. and Steinberg A.M. (2009) Experimental joint weak measurement on a photon pair as a probe of Hardy’s paradox Physical Review Letters 102, 020404.

Molmer K. (2001) Counterfactual statements and weak measurements: an experimental proposal Physics Letters A 292, 151.