"""Extended nonlocal games ========================== In this tutorial, we will define the concept of an *extended nonlocal game*. Extended nonlocal games are a more general abstraction of nonlocal games wherein the referee, who previously only provided questions and answers to the players, now share a state with the players and is able to perform a measurement on that shared state. """ # %% # Every extended nonlocal game has a *value* associated to it. Analogously to # nonlocal games, this value is a quantity that dictates how well the players can # perform a task in the extended nonlocal game model when given access to certain # resources. We will be using :code:`|toqito⟩` to calculate these quantities. # %% # We will also look at existing results in the literature on these values and be # able to replicate them using :code:`|toqito⟩`. Much of the written content in # this tutorial will be directly taken from :footcite:`Russo_2017_Extended`. # %% # Extended nonlocal games have a natural physical interpretation in the setting # of tripartite steering :footcite:`Cavalcanti_2015_Detection` and in device-independent quantum scenarios # :footcite:`Tomamichel_2013_AMonogamy`. For more information on extended nonlocal games, please refer to # :footcite:`Johnston_2016_Extended` and :footcite:`Russo_2017_Extended`. # %% # The extended nonlocal game model # -------------------------------- # An *extended nonlocal game* is similar to a nonlocal game in the sense that it # is a cooperative game played between two players Alice and Bob against a # referee. The game begins much like a nonlocal game, with the referee selecting # and sending a pair of questions :math:`(x,y)` according to a fixed probability # distribution. Once Alice and Bob receive :math:`x` and :math:`y`, they respond # with respective answers :math:`a` and :math:`b`. Unlike a nonlocal game, the # outcome of an extended nonlocal game is determined by measurements performed by # the referee on its share of the state initially provided to it by Alice and # Bob. # # .. figure:: ../../figures/extended_nonlocal_game.svg # :alt: extended nonlocal game # :align: center # # An extended nonlocal game. # # Specifically, Alice and Bob's winning probability is determined by # collections of measurements, :math:`V(a,b|x,y) \in \text{Pos}(\mathcal{R})`, # where :math:`\mathcal{R} = \mathbb{C}^m` is a complex Euclidean space with # :math:`m` denoting the dimension of the referee's quantum system--so if Alice # and Bob's response :math:`(a,b)` to the question pair :math:`(x,y)` leaves the # referee's system in the quantum state # # .. math:: # \sigma_{a,b}^{x,y} \in \text{D}(\mathcal{R}), # # then their winning and losing probabilities are given by # # .. math:: # \left\langle V(a,b|x,y), \sigma_{a,b}^{x,y} \right\rangle # \quad \text{and} \quad # \left\langle \mathbb{I} - V(a,b|x,y), \sigma_{a,b}^{x,y} \right\rangle. # # # Strategies for extended nonlocal games # --------------------------------------- # # An extended nonlocal game :math:`G` is defined by a pair :math:`(\pi, V)`, # where :math:`\pi` is a probability distribution of the form # # .. math:: # \pi : \Sigma_A \times \Sigma_B \rightarrow [0, 1] # # on the Cartesian product of two alphabets :math:`\Sigma_A` and # :math:`\Sigma_B`, and :math:`V` is a function of the form # # .. math:: # V : \Gamma_A \times \Gamma_B \times \Sigma_A \times \Sigma_B \rightarrow \text{Pos}(\mathcal{R}) # # for :math:`\Sigma_A` and :math:`\Sigma_B` as above, :math:`\Gamma_A` and # :math:`\Gamma_B` being alphabets, and :math:`\mathcal{R}` refers to the # referee's space. Just as in the case for nonlocal games, we shall use the # convention that # # .. math:: # \Sigma = \Sigma_A \times \Sigma_B \quad \text{and} \quad \Gamma = \Gamma_A \times \Gamma_B # # to denote the respective sets of questions asked to Alice and Bob and the sets # of answers sent from Alice and Bob to the referee. # # When analyzing a strategy for Alice and Bob, it may be convenient to define a # function # # .. math:: # K : \Gamma_A \times \Gamma_B \times \Sigma_A \times \Sigma_B \rightarrow \text{Pos}(\mathcal{R}). # # We can represent Alice and Bob's winning probability for an extended nonlocal # game as # # .. math:: # \sum_{(x,y) \in \Sigma} \pi(x,y) \sum_{(a,b) \in \Gamma} \left\langle V(a,b|x,y), K(a,b|x,y) \right\rangle. # # Standard quantum strategies for extended nonlocal games # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # A *standard quantum strategy* for an extended nonlocal game consists of # finite-dimensional complex Euclidean spaces :math:`\mathcal{U}` for Alice and # :math:`\mathcal{V}` for Bob, a quantum state :math:`\sigma \in # \text{D}(\mathcal{U} \otimes \mathcal{R} \otimes \mathcal{V})`, and two # collections of measurements # # .. math:: # \{ A_a^x : a \in \Gamma_A \} \subset \text{Pos}(\mathcal{U}) # \quad \text{and} \quad # \{ B_b^y : b \in \Gamma_B \} \subset \text{Pos}(\mathcal{V}), # # for each :math:`x \in \Sigma_A` and :math:`y \in \Sigma_B` respectively. As # usual, the measurement operators satisfy the constraint that # # .. math:: # \sum_{a \in \Gamma_A} A_a^x = \mathbb{I}_{\mathcal{U}} # \quad \text{and} \quad # \sum_{b \in \Gamma_B} B_b^y = \mathbb{I}_{\mathcal{V}}, # # for each :math:`x \in \Sigma_A` and :math:`y \in \Sigma_B`. # # When the game is played, Alice and Bob present the referee with a quantum # system so that the three parties share the state :math:`\sigma \in # \text{D}(\mathcal{U} \otimes \mathcal{R} \otimes \mathcal{V})`. The referee # selects questions :math:`(x,y) \in \Sigma` according to the distribution # :math:`\pi` that is known to all participants in the game. # # The referee then sends :math:`x` to Alice and :math:`y` to Bob. At this point, # Alice and Bob make measurements on their respective portions of the state # :math:`\sigma` using their measurement operators to yield an outcome to send # back to the referee. Specifically, Alice measures her portion of the state # :math:`\sigma` with respect to her set of measurement operators :math:`\{A_a^x # : a \in \Gamma_A\}`, and sends the result :math:`a \in \Gamma_A` of this # measurement to the referee. Likewise, Bob measures his portion of the state # :math:`\sigma` with respect to his measurement operators # :math:`\{B_b^y : b \in \Gamma_B\}` to yield the outcome :math:`b \in \Gamma_B`, # that is then sent back to the referee. # # At the end of the protocol, the referee measures its quantum system with # respect to the measurement :math:`\{V(a,b|x,y), \mathbb{I}-V(a,b|x,y)\}`. # # The winning probability for such a strategy in this game :math:`G = (\pi,V)` is # given by # # .. math:: # \sum_{(x,y) \in \Sigma} \pi(x,y) \sum_{(a,b) \in \Gamma} # \left \langle A_a^x \otimes V(a,b|x,y) \otimes B_b^y, # \sigma # \right \rangle. # # For a given extended nonlocal game :math:`G = (\pi,V)`, we write # :math:`\omega^*(G)` to denote the *standard quantum value* of :math:`G`, which # is the supremum value of Alice and Bob's winning probability over all standard # quantum strategies for :math:`G`. # # Unentangled strategies for extended nonlocal games # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # An *unentangled strategy* for an extended nonlocal game is simply a standard # quantum strategy for which the state :math:`\sigma \in \text{D}(\mathcal{U} # \otimes \mathcal{R} \otimes \mathcal{V})` initially prepared by Alice and Bob # is fully separable. # # Any unentangled strategy is equivalent to a strategy where Alice and Bob store # only classical information after the referee's quantum system has been provided # to it. # # For a given extended nonlocal game :math:`G = (\pi, V)` we write # :math:`\omega(G)` to denote the *unentangled value* of :math:`G`, which is the # supremum value for Alice and Bob's winning probability in :math:`G` over all # unentangled strategies. The unentangled value of any extended nonlocal game, # :math:`G`, may be written as # # .. math:: # \omega(G) = \max_{f, g} # \lVert # \sum_{(x,y) \in \Sigma} \pi(x,y) # V(f(x), g(y)|x, y) # \rVert # # where the maximum is over all functions :math:`f : \Sigma_A \rightarrow # \Gamma_A` and :math:`g : \Sigma_B \rightarrow \Gamma_B`. # # Non-signaling strategies for extended nonlocal games # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # A *non-signaling strategy* for an extended nonlocal game consists of a function # # .. math:: # K : \Gamma_A \times \Gamma_B \times \Sigma_A \times \Sigma_B \rightarrow \text{Pos}(\mathcal{R}) # # such that # # .. math:: # \sum_{a \in \Gamma_A} K(a,b|x,y) = \rho_b^y \quad \text{and} \quad \sum_{b \in \Gamma_B} K(a,b|x,y) = \sigma_a^x, # # for all :math:`x \in \Sigma_A` and :math:`y \in \Sigma_B` where # :math:`\{\rho_b^y : y \in \Sigma_B, b \in \Gamma_B\}` and :math:`\{\sigma_a^x: # x \in \Sigma_A, a \in \Gamma_A\}` are collections of operators satisfying # # .. math:: # \sum_{a \in \Gamma_A} \sigma_a^x = \tau = \sum_{b \in \Gamma_B} \rho_b^y, # # for every choice of :math:`x \in \Sigma_A` and :math:`y \in \Sigma_B` and where # :math:`\tau \in \text{D}(\mathcal{R})` is a density operator. # # For any extended nonlocal game, :math:`G = (\pi, V)`, the winning probability # for a non-signaling strategy is given by # # .. math:: # \sum_{(x,y) \in \Sigma} \pi(x,y) \sum_{(a,b) \in \Gamma} \left\langle V(a,b|x,y) K(a,b|x,y) \right\rangle. # # We denote the *non-signaling value* of :math:`G` as :math:`\omega_{ns}(G)` # which is the supremum value of the winning probability of :math:`G` taken over # all non-signaling strategies for Alice and Bob. # # Relationships between different strategies and values # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # For an extended nonlocal game, :math:`G`, the values have the following relationship: # # # .. note:: # .. math:: # 0 \leq \omega(G) \leq \omega^*(G) \leq \omega_{ns}(G) \leq 1. # # # .. _ref-label-bb84_extended_nl_example: # # Example: The BB84 extended nonlocal game # ----------------------------------------- # # The *BB84 extended nonlocal game* is defined as follows. Let :math:`\Sigma_A = # \Sigma_B = \Gamma_A = \Gamma_B = \{0,1\}`, define # # .. math:: # \begin{equation} # \begin{aligned} # V(0,0|0,0) = \begin{pmatrix} # 1 & 0 \\ # 0 & 0 # \end{pmatrix}, &\quad # V(1,1|0,0) = \begin{pmatrix} # 0 & 0 \\ # 0 & 1 # \end{pmatrix}, \\ # V(0,0|1,1) = \frac{1}{2}\begin{pmatrix} # 1 & 1 \\ # 1 & 1 # \end{pmatrix}, &\quad # V(1,1|1,1) = \frac{1}{2}\begin{pmatrix} # 1 & -1 \\ # -1 & 1 # \end{pmatrix}, # \end{aligned} # \end{equation} # # define # # .. math:: # V(a,b|x,y) = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} # # for all :math:`a \not= b` or :math:`x \not= y`, define :math:`\pi(0,0) = # \pi(1,1) = 1/2`, and define :math:`\pi(x,y) = 0` if :math:`x \not=y`. # # We can encode the BB84 game, :math:`G_{BB84} = (\pi, V)`, in :code:`numpy` # arrays where :code:`prob_mat` corresponds to the probability distribution # :math:`\pi` and where :code:`pred_mat` corresponds to the operator :math:`V`. # sphinx_gallery_start_ignore # sphinx_gallery_thumbnail_path = 'figures/extended_nonlocal_game.svg' # sphinx_gallery_end_ignore # Define the BB84 extended nonlocal game. import numpy as np from toqito.states import basis # The basis: {|0>, |1>}: e_0, e_1 = basis(2, 0), basis(2, 1) # The basis: {|+>, |->}: e_p = (e_0 + e_1) / np.sqrt(2) e_m = (e_0 - e_1) / np.sqrt(2) # The dimension of referee's measurement operators: dim = 2 # The number of outputs for Alice and Bob: a_out, b_out = 2, 2 # The number of inputs for Alice and Bob: a_in, b_in = 2, 2 # Define the predicate matrix V(a,b|x,y) \in Pos(R) bb84_pred_mat = np.zeros([dim, dim, a_out, b_out, a_in, b_in]) # V(0,0|0,0) = |0><0| bb84_pred_mat[:, :, 0, 0, 0, 0] = e_0 @ e_0.conj().T # V(1,1|0,0) = |1><1| bb84_pred_mat[:, :, 1, 1, 0, 0] = e_1 @ e_1.conj().T # V(0,0|1,1) = |+><+| bb84_pred_mat[:, :, 0, 0, 1, 1] = e_p @ e_p.conj().T # V(1,1|1,1) = |-><-| bb84_pred_mat[:, :, 1, 1, 1, 1] = e_m @ e_m.conj().T # The probability matrix encode \pi(0,0) = \pi(1,1) = 1/2 bb84_prob_mat = 1 / 2 * np.identity(2) # %% # The unentangled value of the BB84 extended nonlocal game # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # It was shown in :footcite:`Tomamichel_2013_AMonogamy` and :footcite:`Johnston_2016_Extended` that # # .. math:: # \omega(G_{BB84}) = \cos^2(\pi/8). # # This can be verified in :code:`|toqito⟩` as follows. # Calculate the unentangled value of the BB84 extended nonlocal game. from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame import numpy as np # Define an ExtendedNonlocalGame object based on the BB84 game. bb84 = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat) # The unentangled value is cos(pi/8)**2 \approx 0.85356 print("The unentangled value is ", np.around(bb84.unentangled_value(), decimals=2)) # %% # The BB84 game also exhibits strong parallel repetition. We can specify how many # parallel repetitions for :code:`|toqito⟩` to run. The example below provides an # example of two parallel repetitions for the BB84 game. # The unentangled value of BB84 under parallel repetition. from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame import numpy as np # Define the bb84 game for two parallel repetitions. bb84_2_reps = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat, 2) # The unentangled value for two parallel repetitions is cos(pi/8)**4 \approx 0.72855 print("The unentangled value for two parallel repetitions is ", np.around(bb84_2_reps.unentangled_value(), decimals=2)) # %% # It was shown in :footcite:`Johnston_2016_Extended` that the BB84 game possesses the property of strong # parallel repetition. That is, # # .. math:: # \omega(G_{BB84}^r) = \omega(G_{BB84})^r # # for any integer :math:`r`. # # The standard quantum value of the BB84 extended nonlocal game # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # We can calculate lower bounds on the standard quantum value of the BB84 game # using :code:`|toqito⟩` as well. # Calculate lower bounds on the standard quantum value of the BB84 extended nonlocal game. from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame import numpy as np # Define an ExtendedNonlocalGame object based on the BB84 game. bb84_lb = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat) # The standard quantum value is cos(pi/8)**2 \approx 0.85356 print("The standard quantum value is ", np.around(bb84_lb.quantum_value_lower_bound(), decimals=2)) # %% # From :footcite:`Johnston_2016_Extended`, it is known that :math:`\omega(G_{BB84}) = # \omega^*(G_{BB84})`, however, if we did not know this beforehand, we could # attempt to calculate upper bounds on the standard quantum value. # # There are a few methods to do this, but one easy way is to simply calculate the # non-signaling value of the game as this provides a natural upper bound on the # standard quantum value. Typically, this bound is not tight and usually not all # that useful in providing tight upper bounds on the standard quantum value, # however, in this case, it will prove to be useful. # # The non-signaling value of the BB84 extended nonlocal game # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # Using :code:`|toqito⟩`, we can see that :math:`\omega_{ns}(G) = \cos^2(\pi/8)`. # Calculate the non-signaling value of the BB84 extended nonlocal game. from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame import numpy as np # Define an ExtendedNonlocalGame object based on the BB84 game. bb84 = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat) # The non-signaling value is cos(pi/8)**2 \approx 0.85356 print("The non-signaling value is ", np.around(bb84.nonsignaling_value(), decimals=2)) # %% # So we have the relationship that # # .. math:: # \omega(G_{BB84}) = \omega^*(G_{BB84}) = \omega_{ns}(G_{BB84}) = \cos^2(\pi/8). # # It turns out that strong parallel repetition does *not* hold in the # non-signaling scenario for the BB84 game. This was shown in :footcite:`Russo_2017_Extended`, and we # can observe this by the following snippet. # The non-signaling value of BB84 under parallel repetition. from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame import numpy as np # Define the bb84 game for two parallel repetitions. bb84_2_reps = ExtendedNonlocalGame(bb84_prob_mat, bb84_pred_mat, 2) # The non-signaling value for two parallel repetitions is cos(pi/8)**4 \approx 0.73825 print( "The non-signaling value for two parallel repetitions is ", np.around(bb84_2_reps.nonsignaling_value(), decimals=2) ) # %% # Note that :math:`0.73825 \geq \cos(\pi/8)^4 \approx 0.72855` and therefore we # have that # # .. math:: # \omega_{ns}(G^r_{BB84}) \not= \omega_{ns}(G_{BB84})^r # # for :math:`r = 2`. # # Example: The CHSH extended nonlocal game # ----------------------------------------- # # Let us now define another extended nonlocal game, :math:`G_{CHSH}`. # # Let :math:`\Sigma_A = \Sigma_B = \Gamma_A = \Gamma_B = \{0,1\}`, define a # collection of measurements :math:`\{V(a,b|x,y) : a \in \Gamma_A, b \in # \Gamma_B, x \in \Sigma_A, y \in \Sigma_B\} \subset \text{Pos}(\mathcal{R})` # such that # # .. math:: # \begin{equation} # \begin{aligned} # V(0,0|0,0) = V(0,0|0,1) = V(0,0|1,0) = \begin{pmatrix} # 1 & 0 \\ # 0 & 0 # \end{pmatrix}, \\ # V(1,1|0,0) = V(1,1|0,1) = V(1,1|1,0) = \begin{pmatrix} # 0 & 0 \\ # 0 & 1 # \end{pmatrix}, \\ # V(0,1|1,1) = \frac{1}{2}\begin{pmatrix} # 1 & 1 \\ # 1 & 1 # \end{pmatrix}, \\ # V(1,0|1,1) = \frac{1}{2} \begin{pmatrix} # 1 & -1 \\ # -1 & 1 # \end{pmatrix}, # \end{aligned} # \end{equation} # # define # # .. math:: # V(a,b|x,y) = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} # # for all :math:`a \oplus b \not= x \land y`, and define :math:`\pi(0,0) = # \pi(0,1) = \pi(1,0) = \pi(1,1) = 1/4`. # # In the event that :math:`a \oplus b \not= x \land y`, the referee's measurement # corresponds to the zero matrix. If instead it happens that :math:`a \oplus b = # x \land y`, the referee then proceeds to measure with respect to one of the # measurement operators. This winning condition is reminiscent of the standard # CHSH nonlocal game. # # We can encode :math:`G_{CHSH}` in a similar way using :code:`numpy` arrays as # we did for :math:`G_{BB84}`. # Define the CHSH extended nonlocal game. import numpy as np # The dimension of referee's measurement operators: dim = 2 # The number of outputs for Alice and Bob: a_out, b_out = 2, 2 # The number of inputs for Alice and Bob: a_in, b_in = 2, 2 # Define the predicate matrix V(a,b|x,y) \in Pos(R) chsh_pred_mat = np.zeros([dim, dim, a_out, b_out, a_in, b_in]) # V(0,0|0,0) = V(0,0|0,1) = V(0,0|1,0). chsh_pred_mat[:, :, 0, 0, 0, 0] = np.array([[1, 0], [0, 0]]) chsh_pred_mat[:, :, 0, 0, 0, 1] = np.array([[1, 0], [0, 0]]) chsh_pred_mat[:, :, 0, 0, 1, 0] = np.array([[1, 0], [0, 0]]) # V(1,1|0,0) = V(1,1|0,1) = V(1,1|1,0). chsh_pred_mat[:, :, 1, 1, 0, 0] = np.array([[0, 0], [0, 1]]) chsh_pred_mat[:, :, 1, 1, 0, 1] = np.array([[0, 0], [0, 1]]) chsh_pred_mat[:, :, 1, 1, 1, 0] = np.array([[0, 0], [0, 1]]) # V(0,1|1,1) chsh_pred_mat[:, :, 0, 1, 1, 1] = 1 / 2 * np.array([[1, 1], [1, 1]]) # V(1,0|1,1) chsh_pred_mat[:, :, 1, 0, 1, 1] = 1 / 2 * np.array([[1, -1], [-1, 1]]) # The probability matrix encode \pi(0,0) = \pi(0,1) = \pi(1,0) = \pi(1,1) = 1/4. chsh_prob_mat = np.array([[1 / 4, 1 / 4], [1 / 4, 1 / 4]]) # %% # Example: The unentangled value of the CHSH extended nonlocal game # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # Similar to what we did for the BB84 extended nonlocal game, we can also compute # the unentangled value of :math:`G_{CHSH}`. # Calculate the unentangled value of the CHSH extended nonlocal game from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame import numpy as np # Define an ExtendedNonlocalGame object based on the CHSH game. chsh = ExtendedNonlocalGame(chsh_prob_mat, chsh_pred_mat) # The unentangled value is 3/4 = 0.75 print("The unentangled value is ", np.around(chsh.unentangled_value(), decimals=2)) # %% # We can also run multiple repetitions of :math:`G_{CHSH}`. # The unentangled value of CHSH under parallel repetition. from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame import numpy as np # Define the CHSH game for two parallel repetitions. chsh_2_reps = ExtendedNonlocalGame(chsh_prob_mat, chsh_pred_mat, 2) # The unentangled value for two parallel repetitions is (3/4)**2 \approx 0.5625 print("The unentangled value for two parallel repetitions is ", np.around(chsh_2_reps.unentangled_value(), decimals=2)) # %% # Note that strong parallel repetition holds as # # .. math:: # \omega(G_{CHSH})^2 = \omega(G_{CHSH}^2) = \left(\frac{3}{4}\right)^2. # # Example: The non-signaling value of the CHSH extended nonlocal game # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # To obtain an upper bound for :math:`G_{CHSH}`, we can calculate the # non-signaling value. # Calculate the non-signaling value of the CHSH extended nonlocal game. from toqito.nonlocal_games.extended_nonlocal_game import ExtendedNonlocalGame import numpy as np # Define an ExtendedNonlocalGame object based on the CHSH game. chsh = ExtendedNonlocalGame(chsh_prob_mat, chsh_pred_mat) # The non-signaling value is 3/4 = 0.75 print("The non-signaling value is ", np.around(chsh.nonsignaling_value(), decimals=2)) # %% # As we know that :math:`\omega(G_{CHSH}) = \omega_{ns}(G_{CHSH}) = 3/4` and that # # .. math:: # \omega(G) \leq \omega^*(G) \leq \omega_{ns}(G) # # for any extended nonlocal game, :math:`G`, we may also conclude that # :math:`\omega^*(G) = 3/4`. # # # Example: An extended nonlocal game with quantum advantage # ---------------------------------------------------------- # # So far, we have only seen examples of extended nonlocal games where the # standard quantum and unentangled values are equal. Here we'll see an example of # an extended nonlocal game where the standard quantum value is *strictly higher* # than the unentangled value. # # # Example: A monogamy-of-entanglement game with mutually unbiased bases # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # Let :math:`\zeta = \exp(\frac{2 \pi i}{3})` and consider the following four # mutually unbiased bases: # # .. math:: # \begin{equation}\label{eq:MUB43} # \begin{aligned} # \mathcal{B}_0 &= \left\{ e_0,\: e_1,\: e_2 \right\}, \\ # \mathcal{B}_1 &= \left\{ \frac{e_0 + e_1 + e_2}{\sqrt{3}},\: # \frac{e_0 + \zeta^2 e_1 + \zeta e_2}{\sqrt{3}},\: # \frac{e_0 + \zeta e_1 + \zeta^2 e_2}{\sqrt{3}} \right\}, \\ # \mathcal{B}_2 &= \left\{ \frac{e_0 + e_1 + \zeta e_2}{\sqrt{3}},\: # \frac{e_0 + \zeta^2 e_1 + \zeta^2 e_2}{\sqrt{3}},\: # \frac{e_0 + \zeta e_1 + e_2}{\sqrt{3}} \right\}, \\ # \mathcal{B}_3 &= \left\{ \frac{e_0 + e_1 + \zeta^2 e_2}{\sqrt{3}},\: # \frac{e_0 + \zeta^2 e_1 + e_2}{\sqrt{3}},\: # \frac{e_0 + \zeta e_1 + \zeta e_2}{\sqrt{3}} \right\}. # \end{aligned} # \end{equation} # # Define an extended nonlocal game :math:`G_{MUB} = (\pi,R)` so that # # .. math:: # # \pi(0) = \pi(1) = \pi(2) = \pi(3) = \frac{1}{4} # # and :math:`R` is such that # # .. math:: # { R(0|x), R(1|x), R(2|x) } # # represents a measurement with respect to the basis :math:`\mathcal{B}_x`, for # each :math:`x \in \{0,1,2,3\}`. # # Taking the description of :math:`G_{MUB}`, we can encode this as follows. # Define the monogamy-of-entanglement game defined by MUBs. prob_mat = 1 / 4 * np.identity(4) dim = 3 e_0, e_1, e_2 = basis(dim, 0), basis(dim, 1), basis(dim, 2) eta = np.exp((2 * np.pi * 1j) / dim) mub_0 = [e_0, e_1, e_2] mub_1 = [ (e_0 + e_1 + e_2) / np.sqrt(3), (e_0 + eta**2 * e_1 + eta * e_2) / np.sqrt(3), (e_0 + eta * e_1 + eta**2 * e_2) / np.sqrt(3), ] mub_2 = [ (e_0 + e_1 + eta * e_2) / np.sqrt(3), (e_0 + eta**2 * e_1 + eta**2 * e_2) / np.sqrt(3), (e_0 + eta * e_1 + e_2) / np.sqrt(3), ] mub_3 = [ (e_0 + e_1 + eta**2 * e_2) / np.sqrt(3), (e_0 + eta**2 * e_1 + e_2) / np.sqrt(3), (e_0 + eta * e_1 + eta * e_2) / np.sqrt(3), ] # List of measurements defined from mutually unbiased basis. mubs = [mub_0, mub_1, mub_2, mub_3] num_in = 4 num_out = 3 pred_mat = np.zeros([dim, dim, num_out, num_out, num_in, num_in], dtype=complex) pred_mat[:, :, 0, 0, 0, 0] = mubs[0][0] @ mubs[0][0].conj().T pred_mat[:, :, 1, 1, 0, 0] = mubs[0][1] @ mubs[0][1].conj().T pred_mat[:, :, 2, 2, 0, 0] = mubs[0][2] @ mubs[0][2].conj().T pred_mat[:, :, 0, 0, 1, 1] = mubs[1][0] @ mubs[1][0].conj().T pred_mat[:, :, 1, 1, 1, 1] = mubs[1][1] @ mubs[1][1].conj().T pred_mat[:, :, 2, 2, 1, 1] = mubs[1][2] @ mubs[1][2].conj().T pred_mat[:, :, 0, 0, 2, 2] = mubs[2][0] @ mubs[2][0].conj().T pred_mat[:, :, 1, 1, 2, 2] = mubs[2][1] @ mubs[2][1].conj().T pred_mat[:, :, 2, 2, 2, 2] = mubs[2][2] @ mubs[2][2].conj().T pred_mat[:, :, 0, 0, 3, 3] = mubs[3][0] @ mubs[3][0].conj().T pred_mat[:, :, 1, 1, 3, 3] = mubs[3][1] @ mubs[3][1].conj().T pred_mat[:, :, 2, 2, 3, 3] = mubs[3][2] @ mubs[3][2].conj().T # %% # Now that we have encoded :math:`G_{MUB}`, we can calculate the unentangled value. import numpy as np g_mub = ExtendedNonlocalGame(prob_mat, pred_mat) unent_val = g_mub.unentangled_value() print("The unentangled value is ", np.around(unent_val, decimals=2)) # %% # That is, we have that # # .. math:: # # \omega(G_{MUB}) = \frac{3 + \sqrt{5}}{8} \approx 0.65409. # # However, if we attempt to run a lower bound on the standard quantum value, we # obtain. import numpy as np g_mub = ExtendedNonlocalGame(prob_mat, pred_mat) q_val = g_mub.quantum_value_lower_bound() print("The standard quantum value lower bound is ", np.around(q_val, decimals=2)) # %% # Note that as we are calculating a lower bound, it is possible that a value this # high will not be obtained, or in other words, the algorithm can get stuck in a # local maximum that prevents it from finding the global maximum. # # It is uncertain what the optimal standard quantum strategy is for this game, # but the value of such a strategy is bounded as follows # # .. math:: # # 2/3 \geq \omega^*(G) \geq 0.6609. # # For further information on the :math:`G_{MUB}` game, consult :footcite:`Russo_2017_Extended`. # %% # # # References # ---------- # # .. footbibliography::