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The measurement and the state evaluation

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Abstract

I critically analyze the fidelity measure used for state estimation. I discuss the impossibility of complete determination. As an alternative to traditional fidelity, I suggest a figure of merit called confidence in the knowledge of an arbitrary state
The measurement and the state evaluation
Sergei Viznyuk
Abstract
I critically analyze the fidelity measure used for state estimation. I discuss
the impossibility of complete determination. As an alternative to traditional
fidelity, I suggest a figure of merit called confidence in the knowledge of an
arbitrary state
The question of how well one can determine the state of an object by performing measurement
is of rather fundamental nature, laying at the base of most scientific disciplines. Here I shall provide
a general approach to evaluation of degree of confidence in the knowledge of an arbitrary object,
based on the fact that any measurement and associated knowledge is represented by a sample of
events (symbols), each symbol being the outcome of a measurement event.
Consider the measurements are done in preparation + measurement cycles (PMC). The input
to each PMC is , and the output is one of output events , ; where is the dimension
of measurement basis. If PMC is repeated times, the full input is represented by the tensor
product , and the output by


, where
 . Two
questions can be asked:
1. how close is to , or, alternatively, how reliably one can determine from
2. what is the probability of given
A figure of merit, called Uhlmann-Jozsa fidelity [1, 2] has been defined to answer question 1:

(1)
, where , are density matrices of input , and of output . If or is pure, (1) becomes:

(2)
, which is a case of an expression for the expectation value of an operator :

(3)
, with in (2) being the probability POVM. Expression (2) is Born rule, postulated [3] to be the
answer to question 2. When the same measure is used as the answer to both questions, it leads to
some issues I discuss below.
The proposition the fidelity can be used for determination [4] of from lays at the foundation
of several technologies, such as quantum state tomography (QST), quantum process tomography
(QPT) [5]. Due to non-linearity of (1), its practical use for QST is nearly impossible. Linear
inversion of (2), or of alternative fidelity measures [6], is used in all situations, even when both
inputs and outputs are mixtures [4]. Even as (1) is touted as a measure of closeness between
and , specifically for mixtures, (1) does not make sense from standpoint of closeness of states.
For example, if 
then . However, it should be  
. The
reason is, output 
is a mixture with no correlation to input. Thus, the
output is either  or  with  
probability, independent of the input . Hence, is not the
measure of closeness of states, but a measure of closeness of density matrices. For mixtures,
density matrix is not synonymous with state but rather with distribution of states. From this
prospective, makes sense, because . There is an example given in [6] of

and , when (2) gives  
. In authors’ opinion, that is
incorrect. However, that is the expected outcome of the measurement. To assume the Uhlmann-
Jozsa fidelity (1) provides the figure of merit for closeness of states means accepting possibility of
(1) telling is the same as , while measurements show is different from half the times. I shall
conclude, from standpoint of closeness of states, fidelity (2) is the correct measure.
Even as (2) is the correct measure of closeness of states, its use in QST for determination of
is not faultless, for the following reasons:
1. It is impossible to determine a state in a single-device measurement due to no-cloning
theorem [7, 8]. Therefore, (2) implies an ensemble-average. Hence, calculated in QST
is a mixture, even if the input is pure. For evidence, the calculated in QST density
matrices invariably have multiple non-zero eigenvalues, while pure state density matrix
would only have 1 non-zero eigenvalue equal to 1
2. The measure (2) itself cannot be precisely determined in a finite number of measurements,
resulting in uncertainty relation formulated below
The optimal state evaluation involves finding a measurement basis which maximizes (2).
From basic geometric consideration it is clear, that in optimal basis [9]:


(4)
, where 
 is the minimum possible Bures [10] angle between and ; and are
dimensions [9] of the input and output vector spaces; and are equal to the number of
ways to distribute and identical balls into distinguishable cells:
;
(5)
, where is the number of input events, and is the number of output events (measurements),
. The difference is the number of future measurements, given already performed
PMCs. The expression (4) gives the maximum probability that in future 
measurements the result will be the same as in already performed measurements. From (4, 5):

(6)


(7)

(8)
Defining   as the minimum possible uncertainty in
state determination, it follows:

(9)
The expression (4), being sensible as probability measure, has an issue from standpoint of fidelity
of state determination: the fidelity depends on number of measurements already performed, but
it cannot depend on the number of future measurements; e.g. (7) does not make sense. This is a
conceptual issue of identifying fidelity of state determination with measurement probability (2).
In practice [11], the fidelity of optimal state estimation (4) is only used with .
To resolve the conceptual issue, I propose an alternative to fidelity measure, which I call the
confidence in the knowledge of state. The concept of knowledge is based on entropy as the measure
of missing information. The entropy is the amount of unknown. The maximum entropy state, i.e.
equilibrium, has zero information content, i.e. zero known. Thus, the amount of known, i.e. the
knowledge, equals the difference between entropy of equilibrium, and the entropy of the estimated
state. From here I obtain the expression for knowledge [12]:


(10)
, where
  

(11)
 is Boltzmann’s entropy;
 
  is entropy of equilibrium.
The knowledge obtained per measurement event is:
 
(12)
Knowledge (12) has its maximum for the given when :


(13)
 grows with number of measurements , toward limit:
 
(14)
As expected,  equals maximum per-event entropy, i.e. maximum Shannon’s entropy [13].
Once equipped with the notion of knowledge, I define the notion of confidence [12] as:

 

(15)
The fidelity measure (4) for optimal state estimation corresponds to maximum confidence :

 


(16)
To summarize, the fidelity (2) is the probability of measurement outcome given input . The
fidelity (4) is the probability of measurement outcome given input in optimal state estimation.
The knowledge (12) is the obtained information (in nats) about estimated state, per measurement
event. The confidence (15) is the ratio of information obtained per measurement event to the
maximum possible information per event, which could have been obtained under optimal state
estimation with infinite number of measurements.
I shall compare the confidence (16) to fidelity of optimal state estimation (4). Re-normalizing
(4) to the same  domain as confidence, I obtain:



(17)
The calculation of re-normalized fidelity (17) and confidence (16) vs number of input events
is presented on Figure 1, for varying ; and dimension of the
measurement basis. The figure demonstrates the confidence (16) is close to fidelity (6) of optimal
state estimation, i.e. when . It also shows (4) loses its meaning of fidelity of optimal state
estimation when . I conclude the confidence (15, 16) provides the correct figure of merit
for state estimation.
Figure 1
Graphs of confidence (16) and fidelity (17) vs number of measurements .
Blue line: confidence (16).
Red lines: re-normalized fidelity (17) for several values of .
The calculation was done for dimension of measurement basis .
The MATLAB code used for calculation:
http://www.phystech.com/download/fidelity.m



References
[1]
A. Uhlmann, "The "transition probability" in the state space of a *-algebra," Reports on
Mathematical Physics, vol. 9, no. 2, pp. 273-279, 1976.
[2]
R. Jozsa, "Fidelity for Mixed Quantum States," Journal of Modern Optics, vol. 41, no. 12,
pp. 2315-2323, 1994.
[3]
L. Masanes, T. Galley and M. Müller, "The measurement postulates of quantum mechanics
are operationally redundant," Nature Communications, vol. 10, no. 1361, 2019.
[4]
R. Blume-Kohout, "Optimal, reliable estimation of quantum states," arXiv:quant-
ph/0611080, 2006.
[5]
M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge
University Press, 2010.
[6]
Y. Liang, Y. Yeh, P. Mendonca, R. Teh, M. Reid and P. Drummond, "Quantum fidelity
measures for mixed states," arXiv:1810.08034 [quant-ph], 2019.
[7]
W. Wootters and W. Zurek, "A single quantum cannot be cloned," Nature, vol. 299, p.
802803, 1982.
[8]
J. Park, "The concept of transition in quantum mechanics," Foundations of Physics, vol. 1,
pp. 23-33, 1970.
[9]
R. Werner, "Optimal Cloning of Pure States," arXiv:quant-ph/9804001, 04 1998.
[10]
D. Bures, "An extension of Kakutani's theorem on infinite product measures to the tensor
product of semifinite w*-algebras," Transactions of the American Mathematical Society,
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[11]
J. Ren, P. Xu, H. Yong, L. Zhang, S. Liao, J. Yin, W. Liu, W. Cai, M. Yang, L. Li, K.
Yang, X. Han, Y. Yao, J. Li, H. Wu, S. Wan, L. Liu and D. Liu, "Ground-to-satellite
quantum teleportation," arXiv:1707.00934 [quant-ph], 2017.
[12]
S. Viznyuk, "From QM to KM," 2020. [Online]. Available:
https://www.academia.edu/41619476/From_QM_to_KM.
[13]
S. Viznyuk, "Shannon's entropy revisited," arXiv:1504.01407 [cs.IT], 2015.
[14]
A. Winter, "On the fidelity of two pure states," J.Phys. A: Math. Gen., vol. 34, no. 35, pp.
7095-7101, 2001.
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