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

albert-einsteinIn 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

One Response to “EPR-Redux; (We’re so sorry, Uncle Albert!)”

  1. […] page in September. I suppose it represents a rigorous development of the work mentioned here & here previously, whilst also extending the scope to demonstrate quantum information theory […]


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: