Abstract
In this chapter, we will take a trip around several hot-spots where Bohmian mechanics and its capacity to describe the microscopic reality, even in the absence of measurements, can be harnessed as computational tools, in order to help in the prediction of phenomenologically accessible information (also useful for the followers of the Copenhagen theory). As a first example, we will see how a Stochastic Schrödinger Equation, when used to compute the reduced density matrix of a non-Markovian open quantum system, necessarily seems to employ the Bohmian concept of a conditional wavefunction. We will see that by dressing these conditional wavefunctions with an interpretation, the Bohmian theory can prove to be a useful tool to build general quantum simulation frameworks, such as a high-frequency electron transport model. As a second example, we will explain how a Copenhagen “observable operator” can be related to numerical properties of the Bohmian trajectories, which within Bohmian mechanics, are well-defined even for an “unmeasured” system. Most importantly in practice, even if these numbers are given no ontological meaning, not only we will be able to simulate (thus, predict and talk about) them, but we will see that they can be operationally determined in a weak value experiment. Therefore, they will be practical numbers to characterize a quantum system irrespective of the followed quantum theory.
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Notes
- 1.
- 2.
If only \(\mathbb {I}m\{\mathfrak {W}\}\) vanished, the CWF would already seem to be ruled by a unitary Schrödinger Equation of a closed system, with a real potential energy field defined as \(V(\vec {x},t):=U(\vec {x},\vec {y}^{\,\xi }(t))+\mathbb {R}e\{\mathfrak {W}(\vec {x},\vec {y}^{\,\xi }(t),t)\}\). Computationally though, in order to evaluate \(\mathbb {R}e\{\mathfrak {W}\}\) and the trajectory \(\vec {y}^\xi (t)\), a quantum description of the environment would still be required, making the CWF of S not independent of the environment’s evolution and thus, not an EWF.
- 3.
If the measurement was for the position operator of A, \( | {\phi _m} \rangle _S\) would be the CWFs of the system for the state of \(\hat{U}_{AS} | {\theta _0} \rangle _A\otimes | {\psi } \rangle _S\) as it was before the strong measurement of A, otherwise, they will only be CWFs of the collapsed \( | {\theta _m} \rangle _A\otimes \hat{M}_m | {\psi } \rangle _S\).
- 4.
At each time a different generalized measurement is performed on S, meaning the stochastic trajectory w(t) reflects the Bohmian positions of different measurement pointers at each \(\Delta t\). Its non-differentiability is thus unproblematic.
- 5.
To allow non-Markovian SSEs “unravelled” through non-position variables, consider the positions of environment “pointers” coupled with non-position observables of the ancillas around the system. Else, consider the associated unmeasured system information \(\mathfrak {B}^\psi \) presented in Sect. 3, or the modal theory corresponding to the unravelled observable.
- 6.
An alternative definition could be the real part of the (complex) expectation \(\langle \hat{B}(t_2)\hat{F}(t_1)\rangle \) in the Heisenberg formalism, which turns out to be the correlation of a weak measurement [10] of F at time \(t_1\) and a strong measurement of B at time \(t_2\). Yet, as shown in Ref. [35], even an ideally weak measurement is in fact contextual (in the sense of footnote 8).
- 7.
A system that is not being measured, e.g. a closed system evolving without quantum interaction with its environment.
- 8.
Contextual means it depends and implies the particular environment used to convey the information to the observer.
- 9.
The information \(\mathfrak {B}^\psi \) will evolve continuously as long as the wavefunction evolves unitarily (which in Bohmian mechanics always does, as we saw). Then, if the system evolves from an eigenstate \( | {b_1} \rangle \) to another \( | {b_2} \rangle \) with eigenvalues \(b_1\ne b_2\), \(\mathfrak {B}^\psi \) will take all the intermediate values not necessarily among the eigenvalues of \(\hat{B}\). This suggests an interpretation in which the “quantization” of quantum mechanics is an apparent property, due to the fact that for a “proper” measurement, we require that a pointer saying b is compatible with a wavefunction \( | {b} \rangle \) that yields for a strong measurement the result b with probability 1. That is, a wavefunction which has all its Bohmian trajectories with value b for \(\mathfrak {B}^\psi \). Then, we would call it “quantum” because this delicate orchestration can only happen for a certain “quantized number” of wavefunctions (the eigenstates).
- 10.
A number that can be obtained in a laboratory with a well-defined protocol.
- 11.
There is a (quite important) exception. Identical particles are always ontologically distinguishable by their trajectories in Bohmian mechanics. In the laboratory however, there are no means to tag each individual particle under many-body wavefunctions with exchange symmetry. In consequence, if we follow our weak value protocol to “measure” the information \(\mathfrak {B}_{(k)}^\psi :=\mathbb {R}e\{ \left\langle {\vec {x}_1, \ldots ,\vec {x}_M} \right| \hat{Id}_{(1)}\cdots \hat{Id}_{(k-1)}\hat{B}_{(k)}\hat{Id}_{(k+1)}\cdots \hat{Id}_{(M)} | {\psi } \rangle / \left\langle {\vec {x}_1,\ldots ,\vec {x}_M} \right| {\psi }\}\) related to the observable B of the k-th electron, in a system of M electrons of positions \(\vec {x}_k\) with many-body wavefunction \( | {\psi } \rangle \), what we will get instead is the average: \(\sum _{k=1}^M \frac{1}{M}\mathfrak {B}^\psi _{(k)}(\vec {x}_1,\ldots ,\vec {x}_M)\). Thus, the average \(\mathfrak {B}_{(k)}^\psi \) for a multi-particle Bohmian trajectory is operational (say, the sum of the current contributions of the active region electrons, as discussed in the next paragraph), but the individual indistinguishable particle \(\mathfrak {B}^\Psi _{(k)}\) (like the individual electron current contributions) are not, even if they might be ontic properties within Bohmian mechanics.
References
X. Oriols, D.K. Ferry, Proc. IEEE 109, 955 (2021)
D. Ferry, The Copenhagen Conspiracy (Jenny Stanford Publishing, 2019)
D. Bohm, Phys. Rev. 85, 166 (1952)
P.R. Holland, The quantum theory of motion: an account of the de Broglie-Bohm causal inter- pretation of quantum mechanics (Cambridge University Press, 1993)
S. Teufel, D. Dürr, Bohmian Mechanics: The Physics and Mathematics of Quantum Theory (Springer, Berlin, 2009)
X. Oriols, J. Mompart, Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology (CRC Press, 2019)
D. Marian, N. Zanghì, X. Oriols, Phys. Rev. Lett. 116, 110404 (2016)
G. Albareda, F. Traversa, A. Benali, X. Oriols, Fluct. Noise Lett. 11, 1242008 (2012)
D. Pandey, E. Colomés, G. Albareda, X. Oriols, Entropy 21, 1148 (2019)
Y. Aharonov, D.Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)
J. Gambetta, H. Wiseman, Phys. Rev. A 68, 062104 (2003)
H.M. Wiseman, J.M. Gambetta, Phys. Rev. Lett. 101, 140401 (2008)
D. Dürr, S. Goldstein, N. Zanghi, J. Stat. Phys. 67, 843 (1992)
D. Dürr, S. Goldstein, N. Zanghì, J. Stat. Phys. 116, 959 (2004)
X. Oriols, Phys. Rev. Lett. 98, 066803 (2007)
J. Von Neumann, Mathematische grundlagen der quantenmechanik, vol. 38 (Springer, Berlin, 2013)
H.M. Wiseman, G.J. Milburn, Quantum Measurement and Control (Cambridge University Press, 2009)
D. Dürr, S. Goldstein, R. Tumulka, N. Zanghì, Found. Phys. 35, 449 (2005)
J. M. Gambetta, H.M. Wiseman, in Fluctuations and Noise in Photonics and Quantum Pptics, vol. 5111 (SPIE, 2003), pp. 313–324
H.M. Wiseman, Quant. Semiclass. Opt. Euro. Opt. Soc. Part B 8, 205 (1996)
L. Li, M.J. Hall, H.M. Wiseman, Phys. Rep. 759, 1 (2018)
K. Jacobs, D.A. Steck, Contemp. Phys. 47, 279 (2006)
G. Albareda, H. López, X. Cartoixa, J. Suné, X. Oriols, Phys. Rev. B 82, 085301 (2010)
G. Albareda, J. Suné, X. Oriols, Phys. Rev. B 79, 075315 (2009)
H. López, G. Albareda, X. Cartoixá, J. Suñé, X. Oriols, J. Comput. Electron. 7, 213 (2008)
E. Colomés, Z. Zhan, D. Marian, X. Oriols, Phys. Rev. B 96, 075135 (2017)
M. Villani, G. Albareda, C. Destefani, X. Cartoixá, X. Oriols, Entropy 23, 408 (2021)
Z. Zhan, X. Kuang, E. Colomés, D. Pandey, S. Yuan, X. Oriols, Phys. Rev. B 99, 155412 (2019)
Z. Zhan, E. Colomes, X. Oriols, IEEE Trans. Electron Devices 64, 2617 (2017)
G. Orlando, C. McDonald, N. Protik, G. Vampa, T. Brabec, J. Phys. B 47, 204002 (2014)
E. Hauge, J. Støvneng, Rev. Mod. Phys. 61, 917 (1989)
M. Perarnau-Llobet, E. Bäumer, K.V. Hovhannisyan, M. Huber, A. Acin, Phys. Rev. Lett. 118, 070601 (2017)
J.M. Vilar, J.M. Rubi, Phys. Rev. Lett. 100, 020601 (2008)
W. Niedenzu, M. Huber, E. Boukobza, Quantum 3, 195 (2019)
D. Pandey, R. Sampaio, T. Ala-Nissila, G. Albareda, X. Oriols, Phys. Rev. A 103, 052219 (2021)
N.D. Mermin, Rev. Mod. Phys. 65, 803 (1993)
J. Bell, Phys. World 3, 33 (1990)
D. Pandey, R. Sampaio, T. Ala-Nissila, G. Albareda, X. Oriols (2019)
H. Wiseman, New J. Phys. 9, 165 (2007)
J.S. Lundeen, B. Sutherland, A. Patel, C. Stewart, C. Bamber, Nature 474, 188 (2011)
D.H. Kobe, J. Phys. A: Math. Theor. 40, 5155 (2007)
R. Sampaio, S. Suomela, T. Ala-Nissila, J. Anders, T. Philbin, Phys. Rev. A 97, 012131 (2018)
L. Becker, Br. J. Philos. Sci. 55 (2004)
A. Oldofredi, M. Esfeld, Quantum 8, 68 (2019)
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Oianguren-Asua, X., Destefani, C.F., Villani, M., Ferry, D.K., Oriols, X. (2024). Bohmian Mechanics as a Practical Tool. In: Bassi, A., Goldstein, S., Tumulka, R., Zanghì, N. (eds) Physics and the Nature of Reality. Fundamental Theories of Physics, vol 215. Springer, Cham. https://doi.org/10.1007/978-3-031-45434-9_9
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