## EPR-Redux; (We’re so sorry, Uncle Albert!)

### July 7, 2015

#### 2015 marks the 80th. anniversary of the EPR paradox.

“One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory for the representation of reality. But nobody knows how to find the basis for such a theory.” A. Einstein; Appendix 2: The Meaning of Relativity, 1935

In 1935 Albert Einstein, along with two colleagues, Boris Podolsky and Nathan Rosen, published a work known as the EPR paper, challenging the foundations of quantum mechanics. The three scientists argued that extra pieces of information – hidden variables – were necessary to perform the unitary evolution of the wave function and make quantum mechanics a complete theory. In the following, we show that the idea of hidden variables can be understood from the measurement-based implementation of a controlled-NOT (CNOT) quantum gate.

In quantum mechanics, the EPR paradox is a thought experiment which challenges long-held ideas about the completeness of quantum theory. The core of the EPR paradox is that quantum mechanics cannot be both consistent and complete unless hidden variables exist.

The EPR paradox has been strongly debated over for eighty years and almost all of the arguments, including the work of John Bell, have refuted the hidden variables theory. Here we show in a fairly simple way, how the concept of hidden variables advocated by Einstein, Podolsky and Rosen can be understood with the aid of a most common quantum device: the controlled-NOT gate.

The controlled-NOT gate (CNOT) is used to entangle and disentangle EPR states. Its unitary $U_{CNOT}$ operator  can be written on two-bits operationally, $|a\rangle$ and $|b\rangle \in GF_{2}$ , where the former is the control bit, the latter is the target bit, and $GF_{2}$ is the Galois field of two elements $F_{2}= \{0,1\}$:

$U_{CNOT} (|a\rangle \otimes |b\rangle) = |a\rangle \otimes |a \oplus b\rangle$ where, $a \oplus b = (a + b)mod2$

The matrix $U_{CNOT}$ for this operation is:

 $|0,0\rangle$ $|0,1\rangle$ $|1,0\rangle$ $|1,1\rangle$ $|0,0\rangle$ 1 0 0 0 $|0,1\rangle$ 0 1 0 0 $|1,0\rangle$ 0 0 0 1 $|1,1\rangle$ 0 0 1 0

Now, note that the state transformation $|1,0\rangle = |1,1\rangle$ can be replaced with $|1,b\rangle = |1,NOT(b)\oplus b\rangle$, where $b=0$.
In $GF_{2}$, the inverter gate corresponds to the polynomial representation $NOT(b) = b \oplus 1$, and every element $b$ satisfies the property $b = b^2$.
Hence, the logical negation can also be represented by the polynomial $NOT(b)=b^2 \oplus 1$.

The exclusive disjunction $NOT(b)\oplus b$ corresponds to the field’s addition (mod2) operation in Galois field of two elements, therefore, $NOT(b) \oplus b = b^2 \oplus b \oplus 1$.
It follows that the mapping $|1,b\rangle = |1,NOT(b)\oplus b$ becomes $|1,b\rangle = |1,b^2\oplus b \oplus 1\rangle$ for $b = 0$.

As quantum logic gates are reversible, the action of a CNOT gate must be undone when/if a second CNOT is applied. Hence, the transformation $|1,b\rangle = |1,b^2\oplus b \oplus 1\rangle$ must be a bijective map that is its own inverse, so that there exits the involution $CNOT|1,0\rangle \rightleftharpoons |1,1\rangle$ for $b = \left\{0,1\right\}$.

But, the polynomial in one variable $P=b^2\oplus b\oplus 1$ is irreducible over $GF_{2}$, namely, it always outputs 1 for $b$ equal to 0 or 1. Therefore, $P$ requires some hidden variable for the state transformation $|1,b\rangle = |1,b^2\oplus b \oplus 1\rangle$ to be two-way.

This is exactly the core of the EPR paradox advocated by Einstein, Podolsky and Rosen, because, if a unitary operation needs one extra piece of information outside the reach of the binary system, then the quantum theory cannot be complete.

A. DeCastro