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Broadly speaking, physical information in the context of Classical and Quantum mechanics consists of pure states and mixed states.

This answer addresses the difference between Quantum pure and mixed states

This paper explains pure and mixed states in Quantum systems rigorously

For your last question: Yes.

First, you would need a binary number that represented all possible colors. So for instance, if you have 5 total possible colors that any individual ball can be then "color" can be represented by a 3-bit binary number. However, a 3-bit binary number can reach up to $7$ so the last three possible combinations (representing $5$,$6$ and $7$) of the 3-bit number would be unused (which is fine). Three are unused instead of two because we can use $0$ to represent a color. This means, $111_2$ ($7$), $110_2$ ($6$) and $101_2$ ($5$) would never show up.

Next, we would need separate binary numbers to represent the state, or color, of each ball.

If red was $0$ then it would be represented by $000_2$ and if yellow was $5$ then it would be represented by $101_2$. Then for two balls with these results we would get:

$000_2$ (red)

$101_2$ (yellow)

...

and so on if we had more balls and assuming order didn't matter (meaning the balls are identical apart from color). Thus, we'd get a string of binary numbers representing the outcome of your experiment measuring the color of a system of balls.

Broadly speaking, physical information in the context of Classical and Quantum mechanics consists of pure states and mixed states.

This addresses the difference between Quantum pure and mixed states

This explains pure and mixed states in Quantum systems rigorously

For your last question: Yes.

First, you would need a binary number that represented all possible colors. So for instance, if you have 5 total possible colors that any individual ball can be then "color" can be represented by a 3-bit binary number. However, a 3-bit binary number can reach up to $7$ so the last three possible combinations (representing $5$,$6$ and $7$) of the 3-bit number would be unused (which is fine). Three are unused instead of two because we can use $0$ to represent a color. This means, $111_2$ ($7$), $110_2$ ($6$) and $101_2$ ($5$) would never show up.

Next, we would need separate binary numbers to represent the state, or color, of each ball.

If red was $0$ then it would be represented by $000_2$ and if yellow was $5$ then it would be represented by $101_2$. Then for two balls with these results we would get:

$000_2$ (red)

$101_2$ (yellow)

...

and so on if we had more balls and assuming order didn't matter (meaning the balls are identical apart from color). Thus, we'd get a string of binary numbers representing the outcome of your experiment measuring the color of a system of balls.

Classical information: represents pure states which are individual states, not a mixture or superposition of states. Classical information can describe both Quantum and Classical systems.

Quantum information: represents a mixture/superposition of states OR pure states contained within $|\Psi\rangle$, the wave vector, representing the system (obviously describes only Quantum systems). The meaning of a pure state in a Quantum system is different than a pure state in a Classical system.

This answer addresses the difference between Quantum pure and mixed states

This paper explains pure and mixed states in Quantum systems rigorously

For your last question: Yes.

First, you would need a binary number that represented all possible colors. So for instance, if you have 5 total possible colors that any individual ball can be then "color" can be represented by a 3-bit binary number. However, a 3-bit binary number can reach up to $7$ so the last three possible combinations (representing $5$,$6$ and $7$) of the 3-bit number would be unused (which is fine). Three are unused instead of two because we can use $0$ to represent a color. This means, $111_2$ ($7$), $110_2$ ($6$) and $101_2$ ($5$) would never show up.

Next, we would need separate binary numbers to represent the state, or color, of each ball.

If red was $0$ then it would be represented by $000_2$ and if yellow was $5$ then it would be represented by $101_2$. Then for two balls with these results we would get:

$000_2$ (red)

$101_2$ (yellow)

...

and so on if we had more balls and assuming order didn't matter (meaning the balls are identical apart from color). Thus, we'd get a string of binary numbers representing the outcome of your experiment measuring the color of a system of balls.

Broadly speaking, physical information in the context of Classical and Quantum mechanics consists of pure states and mixed states.

This answer addresses the difference between Quantum pure and mixed states

This paper explains pure and mixed states in Quantum systems rigorously

For your last question: Yes.

First, you would need a binary number that represented all possible colors. So for instance, if you have 5 total possible colors that any individual ball can be then "color" can be represented by a 3-bit binary number. However, a 3-bit binary number can reach up to $7$ so the last three possible combinations (representing $5$,$6$ and $7$) of the 3-bit number would be unused (which is fine). Three are unused instead of two because we can use $0$ to represent a color. This means, $111_2$ ($7$), $110_2$ ($6$) and $101_2$ ($5$) would never show up.

Next, we would need separate binary numbers to represent the state, or color, of each ball.

If red was $0$ then it would be represented by $000_2$ and if yellow was $5$ then it would be represented by $101_2$. Then for two balls with these results we would get:

$000_2$ (red)

$101_2$ (yellow)

...

and so on if we had more balls and assuming order didn't matter (meaning the balls are identical apart from color). Thus, we'd get a string of binary numbers representing the outcome of your experiment measuring the color of a system of balls.

Classical information: represents pure states which are individual states, not a mixture or superposition of states. Classical information can describe both Quantum and Classical systems. https://www.quantiki.org/wiki/pure-states

Quantum information: represents a mixture/superposition of states OR pure states contained within $|\Psi\rangle$, the wave vector, representing the system (obviously describes only Quantum systems). The meaning of a pure state in a Quantum system is different than a pure state in a Classical system. This paper explains pure and mixed states well in Quantum systems: https://pages.uoregon.edu/svanenk/solutions/Mixed_states.pdf

Hopefully someone else will come along and explain that in more detail than myself.

For your last question: Yes.

First, you would need a binary number that represented all possible colors. So for instance, if you have 5 total possible colors that any individual ball can be then "color" can be represented by a 3-bit binary number. However, a 3-bit binary number can reach up to $7$ so the last three possible combinations (representing $5$,$6$ and $7$) of the 3-bit number would be unused (which is fine). Three are unused instead of two because we can use $0$ to represent a color. This means, $111_2$ ($7$), $110_2$ ($6$) and $101_2$ ($5$) would never show up.

Next, we would need separate binary numbers to represent the state, or color, of each ball.

If red was $0$ then it would be represented by $000_2$ and if yellow was $5$ then it would be represented by $101_2$. Then for two balls with these results we would get:

$000_2$ (red)

$101_2$ (yellow)

...

and so on if we had more balls and assuming order didn't matter (meaning the balls are identical apart from color). Thus, we'd get a string of binary numbers representing the outcome of your experiment measuring the color of a system of balls.

Classical information: represents pure states which are individual states, not a mixture or superposition of states. Classical information can describe both Quantum and Classical systems.

Quantum information: represents a mixture/superposition of states OR pure states contained within $|\Psi\rangle$, the wave vector, representing the system (obviously describes only Quantum systems). The meaning of a pure state in a Quantum system is different than a pure state in a Classical system.

This answer addresses the difference between Quantum pure and mixed states

This paper explains pure and mixed states in Quantum systems rigorously

For your last question: Yes.

First, you would need a binary number that represented all possible colors. So for instance, if you have 5 total possible colors that any individual ball can be then "color" can be represented by a 3-bit binary number. However, a 3-bit binary number can reach up to $7$ so the last three possible combinations (representing $5$,$6$ and $7$) of the 3-bit number would be unused (which is fine). Three are unused instead of two because we can use $0$ to represent a color. This means, $111_2$ ($7$), $110_2$ ($6$) and $101_2$ ($5$) would never show up.

Next, we would need separate binary numbers to represent the state, or color, of each ball.

If red was $0$ then it would be represented by $000_2$ and if yellow was $5$ then it would be represented by $101_2$. Then for two balls with these results we would get:

$000_2$ (red)

$101_2$ (yellow)

...

and so on if we had more balls and assuming order didn't matter (meaning the balls are identical apart from color). Thus, we'd get a string of binary numbers representing the outcome of your experiment measuring the color of a system of balls.