# Two-state vector formalism

The two-state vector formalism (TSVF) is a description of quantum mechanics in terms of a causal relation in which the present is caused by quantum states of the past and of the future taken in combination.

## Theory

The two-state vector formalism is one example of a time-symmetric interpretation of quantum mechanics (see Minority interpretations of quantum mechanics). Time-symmetric interpretations of quantum mechanics were first suggested by Walter Schottky in 1921,[1] and later by several other scientists. The two-state vector formalism was first developed by Satosi Watanabe[2] in 1955, who named it the Double Inferential state-Vector Formalism (DIVF). Watanabe proposed that information given by forwards evolving quantum states is not complete; rather, both forwards and backwards evolving quantum states are required to describe a quantum state: a first state vector that evolves from the initial conditions towards the future, and a second state vector that evolves backwards in time from future boundary conditions. Past and future measurements, taken together, provide complete information about a quantum system. Watanabe's work was later rediscovered by Yakir Aharonov, Peter Bergmann and Joel Lebowitz in 1964, who later renamed it the Two-State Vector Formalism (TSVF).[3] Conventional prediction, as well as retrodiction, can be obtained formally by separating out the initial conditions (or, conversely, the final conditions) by performing sequences of coherence-destroying operations, thereby cancelling out the influence of the two state vectors.[4]

The two-state vector is represented by:

where the state ${\displaystyle \langle \Phi |}$ evolves backwards from the future and the state ${\displaystyle |\Psi \rangle }$ evolves forwards from the past.

In the example of the double-slit experiment, the first state vector evolves from the electron leaving its source, the second state vector evolves backwards from the final location of the electron on the detection screen, and the combination of forwards and backwards evolving state vectors determines what occurs when the electron passes the slits.

The two-state vector formalism provides a time-symmetric description of quantum mechanics, and is constructed such as to be time-reversal invariant.[5] It can be employed in particular for analyzing pre- and post-selected quantum systems. Building on the notion of two-state, Reznik and Aharonov constructed a time-symmetric formulation of quantum mechanics that encompasses probabilistic observables as well as nonprobabilistic weak observables.[6]

## Relation to other work

In view of the TSVF approach, and in order to allow information to be obtained about quantum systems that are both pre- and post-selected, Yakir Aharonov, David Albert and Lev Vaidman developed the theory of weak measurements.

In TSVF, causality is time-symmetric; that is, the usual chain of causality is not simply reversed. Rather, TSVF combines causality both from the past (forward causation) and the future (backwards causation, or retrocausality).

Similarly as the de Broglie–Bohm theory, TSVF yields the same predictions as standard quantum mechanics.[7] Lev Vaidman emphasizes that TSVF fits very well with Hugh Everett's many-worlds interpretation.[8]

The two-state vector formalism has similarities with the transactional interpretation of quantum mechanics proposed by John G. Cramer in 1986, although Ruth Kastner has argued that the two interpretations (Transactional and Two-State Vector) have important differences as well.[9] [10] It shares the property of time symmetry with the Wheeler–Feynman absorber theory by Richard Feynman and John Archibald Wheeler and with the time–symmetric quantum mechanics of K. B. Wharton.

## References

1. Schottky, Walter. Das Kausalproblem der Quantentheorie als eine Grundfrage der modernen Naturforschung uberhaupt. Naturwissenschaften vol. 9, issue 25, pp. 492-496 (1921); vol. 9, issue 26, pp. 506-511 (1921);
2. Watanabe, Satosi. "Symmetry of physical laws. Part III. Prediction and retrodiction." Reviews of Modern Physics 27.2 (1955): 179.
3. Yakir Aharonov, Lev Vaidman: Protective measurements of two-state vectors, in: Robert Sonné Cohen, Michael Horne, John J. Stachel (eds.): Potentiality, Entanglement and Passion-At-A-Distance, Quantum Mechanical Studies for A. M. Shimony, Volume Two, 1997, ISBN 978-0792344537, pp. 1–8, p. 2
4. Yakir Aharonov, Peter G. Bergmann, Joel L. Lebowitz: Time symmetry in the quantum process of measurement, Physical Review B., vol. 134, no. 6, pp. 1410–1416, 1964
5. Michael Dickson: Non-relativistic quantum mechanics, Jeremy Butterfield, John Earman (eds.): Philosophy of Physics, Handbook of the Philosophy of Science, North-Holland, Elsevier, pp. 275–416, Footnote on p. 327
6. B. Reznik, and Y. Aharonov: Time-symmetric formulation of quantum mechanics, Phys. Rev. A, vol. 52, no. 4, pp. 2538–2550, 1995
7. Yakir Aharonov, Lev Vaidmann: About position measurements which do not show the Bohmian particle position, in: James T. Cushing, Arthur Fine, Sheldon Goldstein (eds.): Bohmian mechanics and quantum theory: an appraisal, Kluwer Academic Publishers, 1996, pp. 141–154, p. 141, 147
8. Yakir Aharonov, Lev Vaidman: The Two-State Vector Formalism of Quantum Mechanics: an Updated Review. In: Juan Gonzalo Muga, Rafael Sala Mayato, Íñigo Egusquiza (eds.): Time in Quantum Mechanics, Volume 1, Lecture Notes in Physics 734, pp. 399–447, 2nd ed., Springer, 2008, ISBN 978-3540734727, DOI 10.1007/978-3-540-73473-4_13, arXiv:quant-ph/0105101v2 (submitted 21 May 2001, version of 10 June 2007) p. 443
9. Ruth E. Kastner, talk presented at Cambridge 2014 Conference, Free Will and Retrocausality in the Quantum World, [1]
10. Avshalom C. Elitzur, Eliahu Cohen: The Retrocausal Nature of Quantum Measurement Revealed by Partial and Weak Measurements, AIP Conf. Proc. 1408: Quantum Retrocausation: Theory and Experiment (13–14 June 2011, San Diego, California), pp. 120-131, DOI http://dx.doi.org/10.1063/1.3663720 (abstract)

• Yakir Aharonov, Lev Vaidman: The Two-State Vector Formalism of Quantum Mechanics: an Updated Review. In: Juan Gonzalo Muga, Rafael Sala Mayato, Íñigo Egusquiza (eds.): Time in Quantum Mechanics, Volume 1, Lecture Notes in Physics, vol. 734, pp. 399–447, 2nd ed., Springer, 2008, ISBN 978-3540734727, DOI 10.1007/978-3-540-73473-4_13, arXiv:quant-ph/0105101v2 (submitted 21 May 2001, version of 10 June 2007)
• Lev Vaidman: The Two-State Vector Formalism, arXiv:0706.1347v1 (submitted 10 June 2007)
• Lev Vaidman: Backward Evolving Quantum States, Journal of Physics A: Mathematical and Theoretical, vol. 40, no. 12, pp. 3275 ff., DOI 10.1088/1751-8113/40/12/S23 arXiv:quant-ph/0606208v1 (submitted 25 June 2006)
• Yakir Aharonov, Eyal Y. Gruss: Two-time interpretation of quantum mechanics, arXiv:quant-ph/0507269v1 (submitted 28 July 2005)
• Eyal Gruss: A Suggestion for a Teleological Interpretation of Quantum Mechanics, arXiv:quant-ph/0006070v2 (submitted 14 June 2000, version of 4 August 2000)
• Yakir Aharonov, Lev Vaidman: Complete description of a quantum system at a given time, Journal of Physics A: Mathematical and General Volume 24 Number 10, pp. 2315 ff., 1991, DOI 10.1088/0305-4470/24/10/018 (abstract)
• Yakir Aharonov, David Albert, Lev Vaidman: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Phys. Rev. Lett. 60, pp. 1351–1354, 1988 (abstract)
• Yakir Aharonov, Peter G. Bergmann, Joel L. Lebowitz: Time symmetry in the quantum process of measurement, Physical Review B., vol. 134, no. 6, pp. 1410–1416, 1964
• Yakir Aharonov, et al. Measurement and collapse within the two-state vector formalism. Quantum Studies: Mathematics and Foundations 1.1-2 (2014): 133-146.