Summary: Your Basis, My Basis

This chapter developed the mathematics needed to express one set of basis vectors in terms of another. This is important in QIS because measuring a quantum state in terms of a basis that is different from the one in which it was prepared can yield additional information about the state.

• The following equations express a quantum state originally in basis $A$ in terms of basis $B$. The two basis vector sets differ, in state space, by an angle $\theta_{AB}$,
  $\vert A_0 \rangle = \cos\theta_{AB}\vert{B_0}\rangle+\sin\theta_{AB}\vert{B_1}\rangle$
  $\vert A_1 \rangle = -\sin\theta_{AB}\vert{B_0}\rangle+\cos\theta_{AB}\vert{B_1}\rangle$.

• Changing the basis from $A$ to $B$ expresses the quantum basis states $\vert A_0 \rangle$ and $\vert A_1 \rangle$ as a superposition state in the other basis $B$. If a qubit is prepared in one of the $A$ basis states, the coefficients of the $B$ basis states give, according to the Born rule, the probabilities of obtaining measurement outcomes of that state with a measurement device designed to observe $B$ basis states. These changes in the probabilities can be leveraged for applications.

• The power of measuring in a basis different from the preparation basis was illustrated using three linear polarizers. The light passes through a horizontal polarizer first, putting it into a completely horizontal polarization state. If that light were to encounter a vertical polarizer, none of the light would get through because there is no projection of the completely horizontal basis state vector onto the vertical state vector. If, however, a $45\degree$ polarizer is inserted between the horizontal and vertical filters, light does indeed pass through the vertical filter. Seen from the $45\degree$ state space, a $\text{vlp}$ state is an equal superposition of $45\degree$ basis states. This tells us that light will pass through with some probability given by the coefficients of that superposition. Next, according to the third polarizing sheet’s state space, the $45\degree$ basis states are superpositions of the horizontal/vertical basis states. Therefore, there is a coefficient in the vertical component of that state vector, and some light will pass through the third filter. This probability explanation is necessary for single-photon measurements because the classical wave explanation involving an electric field vector is not applicable to a single photon.

• Quantum cryptography also utilizes the superposition that results from changing basis states. Alice and Bob both randomly choose from two basis sets to prepare (Alice) and read (Bob) the photons they are using to communicate. If they both choose the same set, they will both agree 100% on the quantum state of the photon and they can use their measurements to prepare a key with which to encrypt their message. They can openly communicate which basis they used because knowing the measurement basis alone does not tell us the actual measurement result.

• The danger in sending information is that it might be intercepted by an eavesdropper named Eve in quantum information parlance. To eavesdrop undetected, Eve needs to intercept Alice’s photon, read its state, and then send a photon in the same state to Bob. If Eve reads and prepares the photon using a basis different from Alice’s, she will not have a 100% probability of sending Bob a photon in the exact state that she received it in because she cannot know the exact state. Since Eve is receiving a superposition state in terms of her basis states, her measurement cannot tell her the amplitudes of the components of the superposition. Hence, what she sends to Bob will lead, in general, to a change in outcome probabilities for Bob’s measurement. To check for eavesdropping, Alice and Bob compare not only their basis sets but also the preparation and measurement results. If eavesdropping has occurred, they will not get the same results 100% of the time, even when they choose the same basis states.