Separate Alice and Bob quantum gate circuits for Brassard's "Teleportation as a quantum computation" [GB] were run on a Panvia Quantum Computer [PQC] and functionally mapped for different Alice input qubit states. The output of the Alice teleportation circuits is an entangled state of of the three qubits which was saved as a Quantum Possibility Space, QPS, in a JSON text file.
The QPS intermediate was processed by running Bob's teleportation circuit for each Alice circuit output. The results confirm that Alice's first qubit is teleported to Bob's third qubit, however this is only in the case of Alice's third qubit being input in |0> state. The teleportation formula obtained from the run-time results shows that Alice's third qubit in |1> state causes anti-sense teleportation in which the |0> and |1> states are swapped. When Alice's third qubit is in a 50% 50% superposition of |0> and |1> states, Bob's third qubit is a nonsense version of Alice's input qubit where each state has been equally mixed |0> and |1> so no information is teleported.
Distributed quantum computing uses quantum teleportation to link multiple quantum processing units, QPU's, together in a network and so provides an alternative and potentially more scalable paradigm for large quantum computing systems. This application case study provides a worked example of how to share quantum state between two separate systems by building on existing teleportation circuitry [Method].
The Panvia Quantum Computer [PQC] uses dual-domain qubits based on Quantum Field Theory wavefunctions. Brassard's 'Teleportation as a quantum computation' [GB] provides a reference benchmark for quantum gate circuits functionality and a practical challenge to represent the intermediate entangled state as it is teleported from Alice to Bob.
Brassard's method works for a SU(2) qubit represented in the Pauli basis which includes complex coefficients for the qubit states |0> and |1>. In this implementation an S-Gate is applied to Alice's top qubit 0 which transforms the ground state |0> from real to imaginary.
Following the S-Gate on qubit 0, Brassard's Alice circuit is four quantum gates comprising a single qubit L-Gate on the 2nd qubit, two CNOT two qubit gates and a single qubit R-Gate on the 1st qubit. The function of Alice's circuit is to (a) entangle the 1st qubit with the 2nd and 3rd qubits, and (b) create ambiguity so that each qubit will be a superposition with 50% in each of |0> and |1> states.
The innovation is the ability to share the entangled qubit state as an internet friendly text file. The Panvia Quantum Computer is able to generate a precise description of the entangled state of each of Alice's circuits in an unobserved way i.e. the output is not an observation and so does not 'collapse' to a |0> or |1> state but remains in entangled superpositions of the multi-qubit state.
The Quantum Possibility Space is a new way to compactly and accurately represent quantum tomography i.e. the composite state vectors and complex magnitudes that define wavefunctions of the multi-qubit entangled state.
The QPS intermediates between Alice and Bob are sent in an opaque form where each qubit in the entangled state is ambiguously |0> or |1>. As a result the QPS intermediate has a first level of security in that the information of Alice's first qubit is now delocalized and so not directly measurable.
A second level of security is provided by Alice's third qubit which determines the state in which Bob's third qubit will materialize at the output of Bob's circuit.
The QPS output of each of Alice's circuits is processed by Bob's circuit of quantum gates. This arrangement disambiguates the entangled state input on the left of the circuit to an output in qubit 2 which is the teleported SU(2) qubit state that was in Alice's qubit 0 in each circuit.
The role of Alice's third qubit as a 'channel sense' input makes Bob's teleported qubit 100% ambiguous in this end to end scheme. Sharing the state of the channel sense between Alice and Bob via a secure parallel channel therefore becomes the basis of a secure teleportation protocol. The JavaScript Object Notation, JSON, form of the QPS is 100% compatible with existing web based signing, authetication and encryption internet standards for JSON Web Signature and JSON Web Token.
Web based Quantum Teleportation is shown in the Figure below connecting Alices's Quantum Computing System with Bob's Quantum Computing System via a QPS as the intermediate media sent securely via the HTTP protocol. The Quantum Encoding Circuit in the Figure is Alice's circuit, the first half of Brassard's teleportation method that performs ambiguation.
The Quantum Possibility Space in JSON, also shown below, is output by the Quantum Encoding Circuit.
The Quantum Possibility Space in JSON becomes the JWS Payload component of a JSON Web Signature (JWS) according to [ RFC7515]. This defines the JWS as a data structure representing a digitally signed or MACed message, where the Message Authentication Codes are computable hash codes uniquely defined by the byte sequence of the JOSE header and JWS Payload.
The JSON Web Signature (JWS) is combined with a Claim Set that is structured data also encapsulated in a JWS, JSON Web Signature, of the claims fields according to [ RFC7519] to create a JSON Web Token. The JSON Web Token comprising the QPS and the claims fields is published via the HyperText Transfer Protocol, HTTP, and the content delivered in response to a GET request.
Successful validation of the JWT claims will allow the content to pass to the Read step, yeilding the JSON Quantum Possibility Space structured teleportation data.
The Quantum Possibility Space recovered from the JWT is input to the Quantum Decoding Circuit comprising an arrangements of quantum gates. Bob's Circuit shown above is the Quantum Decoding Circuit employed in the Brassard teleportation method. The output of the Quantum Decoding Circuit is a teleported qubit that is input to the receiver Bob's Quantum Computing System.
The Key is any binary sequence or seed for a binary sequence generator and is shared between the sender Alice and receiver Bob. The binary sequence is used to determine the state of Alice's qubit 2 in the Quantum Encoding Circuit and therefore the sense or anti-sense qubit teleportation according to the Figure 8 Teleportation Formula.
JSON code listing
[{"teleport": {"manifest": [{"index": 0, "qubit": "zero"},
{"index": 1, "qubit": "yeong"},
{"index": 2, "qubit": "nihil"}],
"quantumPossibilitySpace": [{"magnitude": {"imaginary": 0.5},
"vector": 0},
{"magnitude": {"imaginary": -0.5},
"vector": 1},
{"magnitude": {"imaginary": 0.5},
"vector": 6},
{"magnitude": {"imaginary": -0.5},
"vector": 7}]}},
{"teleport": {"manifest": [{"index": 0, "qubit": "one"},
{"index": 1, "qubit": "il"},
{"index": 2, "qubit": "unus"}],
"quantumPossibilitySpace": [{"magnitude": {"imaginary": 0.46194},
"vector": 0},
{"magnitude": {"imaginary": -0.46194},
"vector": 1},
{"magnitude": {"real": 0.191342},
"vector": 2},
{"magnitude": {"real": 0.191342},
"vector": 3},
{"magnitude": {"real": 0.191342},
"vector": 4},
{"magnitude": {"real": 0.191342},
"vector": 5},
{"magnitude": {"imaginary": 0.46194},
"vector": 6},
{"magnitude": {"imaginary": -0.46194},
"vector": 7}]}},
{"teleport": {"manifest": [{"index": 0, "qubit": "two"},
{"index": 1, "qubit": "i"},
{"index": 2, "qubit": "duo"}],
"quantumPossibilitySpace": [{"magnitude": {"imaginary": 0.353553},
"vector": 0},
{"magnitude": {"imaginary": -0.353553},
"vector": 1},
{"magnitude": {"real": 0.353553},
"vector": 2},
{"magnitude": {"real": 0.353553},
"vector": 3},
{"magnitude": {"real": 0.353553},
"vector": 4},
{"magnitude": {"real": 0.353553},
"vector": 5},
{"magnitude": {"imaginary": 0.353553},
"vector": 6},
{"magnitude": {"imaginary": -0.353553},
"vector": 7}]}}] ]]
The functional integrity of the QPS can be computationally tested using the property described above: the probability density function, pdf, of |0> versus |1> for each qubit equals the total sum of the squares of the real and imaginary components of the complex magnitudes of each state vector in the QPS with the designated qubit in |0> or |1> state.
The two probabilities calculated in the pdf for each qubit MUST add together to total 1.0 to pass this verification test. Verification failure is de facto evidence that the QPS has been disturbed and has lost its integrity.
The QPS changes irreversably with the measurement of any qubit in the connected ensemble. This change is known as a 'collapse' in that the designated qubit transitions from being a superposition of |0> and |1> states to being 100% of just one. In the Quantum Field Theory model, Dirac's 2nd Quantization is a dual domain where quanta of energy exist as a wave or a particle [ PD]. For the principle of Least Action to be able to operate in such a dual domain regime it is logical to conclude that energy transitions between the two domains as time moves forward. In the Quantum Field Theory view the collapse is therefore seen as a normal transition from delocalized wave with a pdf (probability density function) to a particle with discrete state |0> or |1>.
The state of the QPS following measurement of a single qubit reflects the measurement outcome. A measurement outcome can be calculated using a uniform probability distribution partitioned according to the QPS pdf for the particular qubit. A random sampling from the uniform distribution is a |0> state measurement if it is below the |0>/|1> pdf partition, or a |1> state measurement if it is above the |0>/|1> pdf partition.
Measurement randomness is equivalent to the quantum field oscillation sampling at the observer's chosen time instant. Since the quantum field oscillations are asynchronous to the observer's sampling time there is an inherent randomness in transitioning from the frequency domain to the time domain.
The probability density function, pdf, of |0> versus |1> for each qubit equals the total sum of the squares of the real and imaginary components of the complex magnitudes of each state vector in the QPS with the designated qubit in |0> or |1> state.
The QPS state vectors with the non-observed state are combined with their counterparts with the observed state according to root-sum-of-squares addition. The QPS observed qubit state vectors new magnitude will equal the square root of the sum of the square of its previous magnitude and the square of the magnitude of the non-observed state vector counterpart. The QPS does not record zero magnitude state vectors so the non-observed counterparts are eliminated by the collapse.
The von Neumann density matrix is expressed as a real product of adjoint operators for diagonal eigenvector states in a square matrix [ JvN]. The diagonal sum of the von Neumann density matrix directly equates to the dot product of conjugate QPS pairs described below in Authentication of entangled QPS pairs below. The entropy of the quantum possibility space can be evaluated using von Neumann's entropy formula in the following way.
For each state vector in the QPS:
The entropy S of a QPS is a measure of its information content and the complexity of the entangled quantum state.
The Aspect Experiment [ AA] measured entangled green and blue photon pairs emitted by a calcium atom with net zero angular momentum. To conserve net zero angular momentum the green and blue photons must be oppositely polarized i.e. their momenta are opposite and they are as a result termed 'entangled'. By experimentally measuring four polarization components and adding them to a combined correlation metric of S = 2.7, Aspect proved violation of Bell's equation for the 'Hidden Variables' model that predicted S to be 2 or less [ AE].
Dirac created a photon directly from complex Fourier space by his 2nd Quantization of Maxwell's Field Equation for the electromagnetic field [ PD]. The Aspect experiment's blue and green photons have net zero conserved quantized angular momentum by being conjugate phase pair 'opposites'. Their entanglement can be expressed mathematically in complex numbers by saying they have opposite phase quantized angular momenta.
The Quantum Possibility Space has the mathematical property of having a complex conjugate QPS* where each magnitude in the QPS has the sign of its imaginary component reversed. The dot product of a QPS with its complex conjugate equals real unity and imaginary zero:
QPS . QPS* = Σ θ.θ† = e^0 = 1
In Euler form real unity is a zero value of the imaginary exponent of e, the base of the natural logarithm, so 1 = e^0 = e^xi . e^-xi = e^(x-x)i for any x. As a result unity may be decomposed to a pairs of conjugate phases, opposite signed imaginary exponents that add to zero when the conjugates are multiplied together.
The QPS is able to represent a multi-qubit entangled state because Dirac's 2nd quantization applies not only to single bosons (photons) and fermions (charged particles e.g. electron) but also to multiple quantum particles. By quantizing the occupancy of energy levels in Fock space the statistical mechanics is able to accurately model properties of the combined system. The classical physics ground-up approach based on tracking physical particle trajectories is radically simplified as a result, thereby solving the classically insolvable 'many body' problem.
The dot product between two Quantum Possibility Spaces is computed as the sum of the complex multiplication products of the complex magnitudes of matching state vectors. If the two QPS are an entangled conjugate pair they will have the exactly matching set of state vectors and the complex multiplication of their magnitudes will sum to unity.
The ability to create multi-qubit entangled states as Quantum Possibility Space conjugate pairs is the foundation of a quantum key distribution protocol. A key distributor creates entangled QPS pairs comprising a Quantum Possibility Space X and its complex conjugate counterpart X* which are sent to separate parties. Authentication is tested by computing the dot product of any two candidates as described directly above
In the Authentication process that evaluates two QPS inputs, one is the 'question' and the other is the complementary 'answer' that is by design specific to only that question. The computed observation tests if the answer exactly matches the question i.e. the complex dot product of the QPS pair is unity real and zero imaginary. If that test condition is 'true' the result is a quantized binary variable i.e. a digital bit in memory, such as a 1 bit for a hit versus 0 bit for a miss. The 0 bit is the output in the absence of an adjoint entity, resulting in the case of a complex dot product of the QPS pair not equal to unity real and zero imaginary. If it is known that squares of two QPS are both unity as calculated by the Verification test and the result of their Authentication is also unity real it is a mathematical certainty that the two QPS are a conjugate pair and so represent entangled quantum states shared by two parties.
Quantum state information exchange in a network is a foundational technology for distributed quantum computation. Multi-qubit entangled state quantum information is represented as structured data in the application/json media type [ RFC4627]. The collective state of the connected qubits is represented in a multiverse superposition described mathematically as a quantum possibility space, QPS, of different qubit state combinations. The QPS is designed as an intermediate between quantum gate circuits that preserves qubit superposition and entanglement states and allows their subsequent quantum algorithmic processing without any state collapsing observations.
roger@panviaquantum.com
[GB] Brassard, G. Teleportation as a quantum computation, PhysComp96 Extended Abstract, https://arxiv.org/pdf/quant-ph/9605035
[ AA] Aspect, A. Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities, Physical Review Letters. 49 (2): 91–94, https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.91
[ PD] Dirac, P.A.M. The Quantum Theory of the Emission and Absorption of Radiation, Proceedings of the Royal Society Volume 114 Issue 767, https://royalsocietypublishing.org/doi/10.1098/rspa.1927.0039
[ JvN] vonNeumann, J. Thermodynamik quantummechanischer, Gesamheiten Gott. Nach. 1(1927), pages 273-291, https://arxiv.org/pdf/math-ph/0102013
[ AE] The Aspect Experiment, http://www.roxanne.org/epr/experiment.html
[PQC] Panvia Quantum Computer, https://www.panviaquantum.com
[Method] A practical demonstration of Quantum Teleportation, https://panvia.us/pubsite/content.html
[ RFC4627] Crockford, D. The application/json Media Type for JavaScript Object Notation (JSON) https://www.rfc-editor.org/info/rfc4627
[ RFC7515] Jones, Bradley, Sakimura JSON Web Signature (JWS) https://www.rfc-editor.org/info/rfc7515
[ RFC7519] Jones, Bradley, Sakimura JSON Web Token (JWT) https://www.rfc-editor.org/info/rfc7519