>Now consider the description of an EPR-entangled pair of photons:
(|↑↓> + |↓↑>)/√2
At first glance this looks very much like the single-photon case, except that where before we had ΨU and ΨL we now have |↑↓> and |↓↑>, representing respectively photon 1 being in the upper slit and photon 2 being in the lower slit and viceversa. But this distinction is crucial because it turns out that there is some notational sleight-of-hand going on here. First, |↑↓> is shorthand for |↑>|↓>. Second, the arrow symbols have no semantic significance; they are just compact mnemonic identifiers. We could just as well have written |UL> and |LU> (which of course is shorthand for |U>|L> and |L>|U>) as |↑↓> and |↓↑>. Finally, ΨU is just another way of writing |U>.
So if we employ alternative notation we get the following description of two entangled photons:
(ΨU |U> + ΨL |L>)/√2
As I have probably demonstrated other places, IANAQM. But I don't follow that last transformation. If |↑↓> == |↑>|↓> == |U>|L> == |UL> and ΨU == |U>, how does (|↑↓> + |↓↑>)/√2 == (ΨU |U> + ΨL |L>)/√2 and not (ΨU |L> + ΨL |U>)/√2 ? Shouldn't the |U> and |L> be reversed?
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From the article:
>Now consider the description of an EPR-entangled pair of photons:
(|↑↓> + |↓↑>)/√2
At first glance this looks very much like the single-photon case, except that where before we had ΨU and ΨL we now have |↑↓> and |↓↑>, representing respectively photon 1 being in the upper slit and photon 2 being in the lower slit and viceversa. But this distinction is crucial because it turns out that there is some notational sleight-of-hand going on here. First, |↑↓> is shorthand for |↑>|↓>. Second, the arrow symbols have no semantic significance; they are just compact mnemonic identifiers. We could just as well have written |UL> and |LU> (which of course is shorthand for |U>|L> and |L>|U>) as |↑↓> and |↓↑>. Finally, ΨU is just another way of writing |U>.
So if we employ alternative notation we get the following description of two entangled photons:
(ΨU |U> + ΨL |L>)/√2
As I have probably demonstrated other places, IANAQM. But I don't follow that last transformation. If |↑↓> == |↑>|↓> == |U>|L> == |UL> and ΨU == |U>, how does (|↑↓> + |↓↑>)/√2 == (ΨU |U> + ΨL |L>)/√2 and not (ΨU |L> + ΨL |U>)/√2 ? Shouldn't the |U> and |L> be reversed?