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Decoherence time

Sergei Viznyuk

I critically assess some published values for decoherence times. I show there

are two characteristic times inversely proportional to each other: decoherence

time, and probability decay time, with second often mistaken for the first. I

present formulas for decoherence and decay times

While decoherence has been extensively studied, surprisingly little is available in terms of

numeric results, either calculated from formulas [1, 2] or measured in experiments [3, 4]. The

reason for researchers shying away from publishing numbers becomes apparent once these

numbers are estimated from widely quoted expressions [5, 1]. The estimated values of decoherence

times span an incredible range from for a large molecule immersed into cosmic

microwave background [1], to for a “canonical” classical object [5] with mass ,

characteristic length , temperature , and assumed characteristic “relaxation” time of .

An interval of cannot conceivably correspond to any physical process in a macroscopic

object, as no parameter can undergo a change in any meaningful way during such time. To note, it

takes times longer, i.e. for light to cross proton radius. Due to time-energy

uncertainty, the duration of of a change in object’s state would result in energy

uncertainty of , enough to evaporate object. Such improbable times call into question

the premise decoherence estimates are based on.

Part of the problem, at least for some authors, is that … no clear, unambiguous and universally

accepted deﬁnition of coherence (which is supposed to get lost in the process) is available [6].

Which leads to a confusion as far as which process’ characteristic time to take as decoherence

time. The common view links coherence with presence of interference terms in density matrix,

and their reduction – with decoherence [1]. With this understanding, can the characteristic time of

reduction of interference be taken as decoherence time? As I explain below, that is not the case.

The reduction of object’s density matrix is achieved by tracing out entangled ancilla system,

often assumed to be the environment [1]. As has been shown elsewhere [7], the “tracing out”

operation is nothing else but a measurement performed on ancilla

1

. The decoherence time is,

therefore, the interval between state preparation

2

, and subsequent measurement, even if only on

ancilla part of the state. The classical information extracted by measurement on ancilla reduces

ambiguity in measurement of entangled object’s state [7], i.e. reduces interference terms in density

matrix. The reduction of density matrix signifies decay of object’s probability distribution.

One would intuitively expect the faster ancilla is measured, the faster will interference decay,

but, in fact, the relation between decoherence time defined above, and the decay of probability

distribution is inverse, i.e. the smaller is , the slower is the decay

3

. One can understand this by

considering measurements on ancilla as random walk from the surface of generalized Bloch ball

[8] towards its interior. A point on the surface of Bloch ball corresponds to the initial pure state,

and the point in the center corresponds to the complete mixture. The larger are the steps (i.e. the

larger is ), the fewer steps are needed to reduce density matrix, the faster is the probability decay.

It can also be schematically proven as follows. The distance from the surface of Bloch ball signifies

1

and not by ancilla, as often mistakenly claimed [1]

2

the preparation is also measurement, just with a different device, in a different measurement basis

3

this is also known as quantum Zeno effect [12]

the probability decay. The random walk is described by binomial distribution with probability

to make a step in either direction. The variance of binomial distribution after random

steps is given by:

(1)

The standard deviation , multiplied by the length of each step, gives the distance gained from

the surface of Bloch ball, i.e. the probability decay. If elapsed time is , then

. Using (1):

(2)

From (2),

, i.e. the probability decay time is inversely proportional to

decoherence time . The expression for probability decay has been obtained in Section 4 of [9]:

(3)

, where is the elapsed time; is the characteristic decay time; is explained below; is

calculated from object’s energy spectrum as:

;

(4)

A generic formula for decoherence time has been obtained in [9] using Fermi’s golden rule:

(5)

, where is density of states, i.e. the number of states per unit energy interval around energy

of the object. From (3,5) the characteristic probability decay time is:

(6)

For a two-level system, is simply equal to the difference in energy levels: . Assuming

no degeneracy, there is 1 state per energy interval, i.e.

. Then, from (5,6):

(7)

Thus, for non-degenerate two-level system, the probability decay time and decoherence time are

both equal to Margolus-Levitin bound [10]. A sensible value for decoherence time of has been

obtained in [4] from experimental data on spectral linewidth, effectively using formula (7), even

though (7) is only applicable to non-degenerate two-level systems.

The value of can be quite large for macroscopic objects. That explains why probability

decay time (6) can be incredibly small. One has to keep in mind, the probability decay (3) does

not describe a change in any particular object’s state. It only means the reduction in correlation

between measurements outcomes obtained by different devices when performing measurements

on ensemble of identically prepared objects. Each measurement and ensuing [partial] decoherence

has characteristic time (5). One cannot measure probability distribution (3) on timescales shorter

than (5). On such timescales (3) is a mere abstraction.

I shall now expound on the value of parameter in (3). A measurement outcome is one of

possible events . The measurement event sample represents the collected information about

object’s state [9]. Here is the number of occurrences of event in the sample. There are

distinct ways to collect event sample ; where equals statistical weight of the sample:

(8)

, where is Boltzmann’s entropy of the sample. Different ways to collect the sample (i.e.

different event sequences) relate to different correlations between events. One of the ways to

collect event sample would have the same sequence of events as that of the measurement sample

collected on initial pure state. In full decoherence, all ways to collect event sample have equal

probability

. Thus,

is the minimum probability the measured object is in initial pure state,

with confidence (fidelity) provided in [11].

References

[1]

M. Schlosshauer, "Quantum Decoherence," arXiv:1911.06282 [quant-ph], 2019.

[2]

J. Anglin, J. Paz and W. Zurek, "Deconstructing Decoherence," arXiv:quant-ph/9611045,

1996.

[3]

A. Berkley, H. Xu, R. Ramos, M. Gubrud, F. Strauch, P. Johnson, J. Anderson, A. Dragt,

C. Lobb and F. Wellstood, "Entangled Macroscopic Quantum States in Two

Superconducting Qubits," Science, vol. 300, no. 1548-1550, p. 5625, 2003.

[4]

M. Steffen, M. Ansmann, R. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E.

Weig, A. Cleland and J. Martinis, "Measurement of the Entanglement of Two

Superconducting Qubits via State Tomography," Science, vol. 313, no. 5792, pp. 1423-

1425, 2006.

[5]

W. Zurek, "Reduction of the Wavepacket: How Long Does it Take?," arXiv:quant-

ph/0302044, 2003.

[6]

E. Okon and D. Sudarsky, "Less Decoherence and More Coherence in Quantum Gravity,

Inﬂationary Cosmology and Elsewhere," arXiv:1512.05298 [quant-ph], 2015.

[7]

S. Viznyuk, "No decoherence by entanglement," 2020. [Online]. Available:

https://www.academia.edu/43260697/No_decoherence_by_entanglement.

[8]

D. Massimiliano and S. Bianchi, "The extended Bloch representation of quantum

mechanics and the hidden-measurement solution to the measurement problem," Annals of

Physics, vol. 351, pp. 975-1025, 2014.

[9]

S. Viznyuk, "From QM to KM," 2020. [Online]. Available:

https://www.academia.edu/41619476/From_QM_to_KM.

[10]

N. Margolus and L. Levitin, "The maximum speed of dynamic evolution," arXiv:quant-

ph/9710043, 1998.

[11]

S. Viznyuk, "The measurement and the state evaluation," 2020. [Online]. Available:

https://www.academia.edu/42855863/The_measurement_and_the_state_evaluation.

[12]

B. Misra and E. Sudarshan, "The Zeno's paradox in quantum theory," Journal of

Mathematical Physics, vol. 18, no. 4, pp. 756-763, 1977.