Editing Quantum key distribution (section)

=== BB84 protocol: Charles H. Bennett and Gilles Brassard (1984) ===
{{Main|BB84}}
This protocol, known as [[BB84]] after its inventors and year of publication, was originally described using [[photon polarization]] states to transmit the information.<ref>{{cite book|first1=C. H. |last1=Bennett |first2=G. |last2=Brassard |chapter=Quantum cryptography: Public key distribution and coin tossing |title=Proceedings of the International Conference on Computers, Systems & Signal Processing, Bangalore, India |volume=1 |pages=175–179 |publisher=IEEE |year=1984 |location=New York }} Reprinted as {{cite journal|first1=C. H. |last1=Bennett |first2=G. |last2=Brassard |title=Quantum cryptography: Public key distribution and coin tossing |journal=Theoretical Computer Science |series=Theoretical Aspects of Quantum Cryptography – celebrating 30 years of BB84 |volume=560 |number=1 |date=4 December 2014 |pages=7–11 |doi=10.1016/j.tcs.2014.05.025 |doi-access=free|arxiv=2003.06557 }}</ref> However, any two pairs of [[Conjugate variables|conjugate]] states can be used for the protocol, and many [[optical fibre|optical-fibre]]-based implementations described as BB84 use phase encoded states. The sender (traditionally referred to as [[Alice and Bob|Alice]]) and the receiver (Bob) are connected by a [[quantum communication channel]] which allows [[quantum states]] to be transmitted. In the case of photons this channel is generally either an optical fibre or simply [[free space]]. In addition they communicate via a public classical channel, for example using broadcast radio or the internet. The protocol is designed with the assumption that an [[eavesdropper]] (referred to as Eve) can interfere in any way with the quantum channel, while the classical channel needs to be [[authenticated]].<ref>{{cite journal |title=A largely self-contained and complete security proof for quantum key distribution |first1=Marco |last1=Tomamichel |first2=Anthony |last2=Leverrier |date=2017 |arxiv=1506.08458 |doi=10.22331/q-2017-07-14-14 |journal=Quantum |volume=1 |pages=14|bibcode=2017Quant...1...14T |s2cid=56465385 }}</ref><ref>{{cite arXiv |title=Cryptographic security of quantum key distribution |first1=Christopher |last1=Portmann |first2=Renato |last2=Renner |eprint=1409.3525|year=2014 |class=quant-ph }}</ref>

The security of the protocol comes from encoding the information in [[Orthogonality|non-orthogonal states]]. [[Quantum indeterminacy]] means that these states cannot in general be measured without disturbing the original state (see ''[[No-cloning theorem]]''). BB84 uses two pairs of states, with each pair [[Conjugate variables|conjugate]] to the other pair, and the two states within a pair orthogonal to each other. Pairs of orthogonal states are referred to as a [[Basis (linear algebra)|basis]]. The usual polarization state pairs used are either the [[linear polarization|rectilinear basis]] of vertical (0°) and horizontal (90°), the [[linear polarization|diagonal basis]] of 45° and 135° or the [[circular polarization|circular basis]] of left- and right-handedness. Any two of these bases are conjugate to each other, and so any two can be used in the protocol. Below the rectilinear and diagonal bases are used.

{| class="wikitable" style="float:left; text-align:center;"
|-
 ! Basis
 ! 0
 ! 1
|-
 | [[File:PlusCM128.svg|15x15px]]
 | [[File:Arrow north.svg|20x20px]]
 | [[File:Arrow east.svg|20x20px]]
|-
 | [[File:Multiplication Sign.svg|15x15px]]
 | [[File:Arrow northeast.svg|15x15px]]
 | [[File:Arrow southeast.svg|15x15px]]
|}

The first step in BB84 is quantum transmission. Alice creates a random [[bit]] (0 or 1) and then randomly selects one of her two bases (rectilinear or diagonal in this case) to transmit it in. She then prepares a photon polarization state depending both on the bit value and basis, as shown in the adjacent table. So for example a 0 is encoded in the rectilinear basis (+) as a vertical polarization state, and a 1 is encoded in the diagonal basis (x) as a 135° state. Alice then transmits a single photon in the state specified to Bob, using the quantum channel. This process is then repeated from the random bit stage, with Alice recording the state, basis and time of each photon sent.

According to quantum mechanics (particularly quantum indeterminacy), no possible measurement distinguishes between the 4 different polarization states, as they are not all orthogonal. The only possible measurement is between any two orthogonal states (an orthonormal basis). So, for example, measuring in the rectilinear basis gives a result of horizontal or vertical. If the photon was created as horizontal or vertical (as a rectilinear [[eigenstate]]) then this measures the correct state, but if it was created as 45° or 135° (diagonal eigenstates) then the rectilinear measurement instead returns either horizontal or vertical at random. Furthermore, after this measurement the photon is polarized in the state it was measured in (horizontal or vertical), with all information about its initial polarization lost.

As Bob does not know the basis the photons were encoded in, all he can do is to select a basis at random to measure in, either rectilinear or diagonal. He does this for each photon he receives, recording the time, measurement basis used and measurement result. After Bob has measured all the photons, he communicates with Alice over the public classical channel. Alice broadcasts the basis each photon was sent in, and Bob the basis each was measured in. They both discard photon measurements (bits) where Bob used a different basis, which is half on average, leaving half the bits as a shared key.

{| class="wikitable" style="width:75%; text-align: center; margin: 1em auto 1em auto"
|-
 ! Alice's random bit
 | style="width:40pt;"| 0 || style="width:40pt;"| 1 || style="width:40pt;"| 1 || style="width:40pt;"| 0 || style="width:40pt;"| 1 || style="width:40pt;"| 0 || style="width:40pt;"| 0 || style="width:40pt;"| 1
|-
 ! Alice's random sending basis
 | style="width:40pt;"| [[File:PlusCM128.svg|15x15px]] || style="width:40pt;" | [[File:PlusCM128.svg|15x15px]] || style="width:40pt;" | [[File:Multiplication Sign.svg|15x15px]] || style="width:40pt;" | [[File:PlusCM128.svg|15x15px]] || style="width:40pt;" | [[File:Multiplication Sign.svg|15x15px]] || style="width:40pt;" | [[File:Multiplication Sign.svg|15x15px]] || style="width:40pt;" | [[File:Multiplication Sign.svg|15x15px]] || style="width:40pt;" | [[File:PlusCM128.svg|15x15px]]
|-
 ! Photon polarization Alice sends
 | style="width:40pt;"| [[File:Arrow north.svg|20x20px]] || style="width:40pt;" | [[File:Arrow east.svg|20x20px]] || style="width:40pt;" | [[File:Arrow southeast.svg|15x15px]] || style="width:40pt;" | [[File:Arrow north.svg|20x20px]] || style="width:40pt;" | [[File:Arrow southeast.svg|15x15px]] || style="width:40pt;" | [[File:Arrow northeast.svg|15x15px]] || style="width:40pt;" | [[File:Arrow northeast.svg|15x15px]] || style="width:40pt;" | [[File:Arrow east.svg|20x20px]]
|-
 ! Bob's random measuring basis
 | style="width:40pt;"| [[File:PlusCM128.svg|15x15px]] || style="width:40pt;" | [[File:Multiplication Sign.svg|15x15px]] || style="width:40pt;" | [[File:Multiplication Sign.svg|15x15px]] || style="width:40pt;" | [[File:Multiplication Sign.svg|15x15px]] || style="width:40pt;" | [[File:PlusCM128.svg|15x15px]] || style="width:40pt;" | [[File:Multiplication Sign.svg|15x15px]] || style="width:40pt;" | [[File:PlusCM128.svg|15x15px]] || style="width:40pt;" | [[File:PlusCM128.svg|15x15px]]
|-
 ! Photon polarization Bob measures
 | style="width:40pt;"| [[File:Arrow north.svg|20x20px]] || style="width:40pt;" | [[File:Arrow northeast.svg|15x15px]] || style="width:40pt;" | [[File:Arrow southeast.svg|15x15px]] || style="width:40pt;" | [[File:Arrow northeast.svg|15x15px]] || style="width:40pt;" | [[File:Arrow east.svg|20x20px]] || style="width:40pt;" | [[File:Arrow northeast.svg|15x15px]] || style="width:40pt;" | [[File:Arrow east.svg|20x20px]] || style="width:40pt;" | [[File:Arrow east.svg|20x20px]]
|-
 ! PUBLIC DISCUSSION OF BASIS
 | colspan=8 |

|-
 ! Shared secret key
 | style="width:40pt;"| 0 ||  || style="width:40pt;"| 1 ||  ||  || style="width:40pt;"| 0 ||  || style="width:40pt;"| 1
|-
|}

To check for the presence of an eavesdropper, Alice and Bob now compare a predetermined subset of their remaining bit strings. If a third party (usually referred to as Eve, for "eavesdropper") has gained any information about the photons' polarization, this introduces errors in Bob's measurements. Other environmental conditions can cause errors in a similar fashion. If more than <math>p</math> bits differ they abort the key and try again, possibly with a different quantum channel, as the security of the key cannot be guaranteed. <math>p</math> is chosen so that if the number of bits known to Eve is less than this, [[privacy amplification]] can be used to reduce Eve's knowledge of the key to an arbitrarily small amount at the cost of reducing the length of the key.