The Quantum Condition Space

Abstract

The fundamental properties of quantum physics are exploited to evaluate event probabilities with projection measurements. Next, to study what events can be specified by quantum methods, the concept of the condition space is introduced, which is found to be the dual space of the classical outcome space of bit strings. Like the classical outcome space generates the quantum state space, the condition space generates the quantum condition space being the central idea of this work. The quantum condition space permits the existence of entangled conditions having no classical equivalent. In addition, the quantum condition space is related to the quantum state space by a Fourier transform guaranteed by the Pontryagin duality, and therefore an entropic uncertainty principle can be defined. The quantum condition space offers a novel perspective of understanding quantum states with the duality picture. Furthermore, the quantum conditions have physical meanings and realizations of their own and thus may be studied for purposes beyond the original motivation of characterizing events for probability evaluation. Finally, the relation between the condition space and quantum circuits provides insights into how quantum states are collectively modified by quantum gates, which may lead to deeper understanding of the complexity of quantum circuits.

Characterizing quantum circuits with qubit functional configurations

“…As discussed in Ref. 41 the collection of all these bit strings can be considered as a 3-dimensional linear space V over the binary field and any bit string can be expressed as a linear combination of the three basis vectors (100), (010), and (001): , where the coefficients ’s take values 0 or 1, and the addition “ + ” is bit-wise addition modulo 2. Now by the theory of linear vector spaces, the dual space of V is formed by the linear functionals over V : .…”

“…In Ref. 41 we have associated each functional with a “0” condition that specifies a half-set of the bit strings and studied the quantum condition space generated by these “0” conditions. In this work we call these functionals “qubit functionals” (because they are functionals on the qubit values) and focus on their roles in how the entries of the quantum state vectors are modified collectively by quantum gates.…”

“…( 2 ) we apply and then to produce , where in the subscripts means is the control and is the target. By studying either the truth table of the CNOT gates or the results from previous works 41 , 47 , 48 , the calculates and stores its value on , and afterwards the calculates and stores its value on . In other words, after and have been applied, the value on now represents the value of , and the value on now represents the value of , as compared to the state before the two CNOT gates are applied.…”

“…As detailed in Ref. 41 , the duality between the linear functional space and the bit string space V leads to a one-to-one correspondence between each functional and a unique partition of the bit strings—defined by two half-sets specified by setting “0” and “1” for the functional value respectively. When acts on , the partition of specifies which entries are “0” and which are “1”, while the partitions of all other functionals define how the “0” entries are paired with the “1” entries.…”

“…For a single layer, because the functional space is the dual of the bit string space with elements, there are functionals (for details see Ref. 41 ). Excluding the “0” functional, there are to choose from, and n locations to allocate them.…”